Numerical Integration with MATLAB for two different functions

Question

I have two numerical integration functions in MATLAB as follows:

fun1 = @(x) log2(1+x).*(4*exp(2*lambda*(d*sqrt(- d^2 + (a./x).^(2/alp))-(a./x).^(2/alp).*...
    acos(((a./x).^(1/alp)./d).^(-1)))).*(a./x).^(2/alp).*lambda.*acos(((a./x).^(1/alp)./d).^(-1)))./(x.*alp);

num1=integral(@(x)fun1(x),0,a/(d^alp));

fun2 = @(x) exp(1./x).*expint(1./x).*(4*exp(2*lambda*(d*sqrt(- d^2 + (a./x).^(2/alp))-(a./x).^(2/alp).*...
    acos(((a./x).^(1/alp)./d).^(-1)))).*(a./x).^(2/alp).*lambda.*acos(((a./x).^(1/alp)./d).^(-1)))./(x.*alp);

num2=integral(@(x)fun2(x),0,a/(d^alp));

In fun1, I have log2(1+x) (rest of the terms are same for fun1 and fun2) and it gives numerical answer.

In fun2, I have exp(1./x).*expint(1./x) and it does not give numerical value.

for d=1.2; lambda=4.5; alp=2.7;f=1;a=0.5;

num1 =

0.3078

Warning: Infinite or Not-a-Number value encountered. 

num2 =

   NaN

I noticed that this can be calculated with MATHEMATICA. But I need it in MATLAB as my simulation run in it.

Can anyone help?

Answer 1

I guess the integral function causes overflow or underflow since it uses double-precision arithmetic. You can try vpaintegral, which uses variable-precision arithmetic.

The modified code:

d=1.2; lambda=4.5; alp=2.7;f=1;a=0.5;

fun1 = @(x) log2(1+x).*(4*exp(2*lambda*(d*sqrt(- d^2 + (a./x).^(2/alp))-(a./x).^(2/alp).*...
    acos(((a./x).^(1/alp)./d).^(-1)))).*(a./x).^(2/alp).*lambda.*acos(((a./x).^(1/alp)./d).^(-1)))./(x.*alp);

num1=integral(@(x)fun1(x),0,a/(d^alp))

syms x

fun2 =  exp(1./x).*expint(1./x).*(4*exp(2*lambda*(d*sqrt(- d^2 + (a./x).^(2/alp))-(a./x).^(2/alp).*...
    acos(((a./x).^(1/alp)./d).^(-1)))).*(a./x).^(2/alp).*lambda.*acos(((a./x).^(1/alp)./d).^(-1)))./(x.*alp);

num2=vpaintegral(fun2,0,a/(d^alp))

Output:

num1 =
    0.3078
num2 =
0.197608