Optimal. Leaf size=25 \[ \frac {1}{2} \text {Chi}\left (b x^2\right ) \sinh (a)+\frac {1}{2} \cosh (a) \text {Shi}\left (b x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5426, 5425,
5424} \begin {gather*} \frac {1}{2} \sinh (a) \text {Chi}\left (b x^2\right )+\frac {1}{2} \cosh (a) \text {Shi}\left (b x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 5424
Rule 5425
Rule 5426
Rubi steps
\begin {align*} \int \frac {\sinh \left (a+b x^2\right )}{x} \, dx &=\cosh (a) \int \frac {\sinh \left (b x^2\right )}{x} \, dx+\sinh (a) \int \frac {\cosh \left (b x^2\right )}{x} \, dx\\ &=\frac {1}{2} \text {Chi}\left (b x^2\right ) \sinh (a)+\frac {1}{2} \cosh (a) \text {Shi}\left (b x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{2} \left (\text {Chi}\left (b x^2\right ) \sinh (a)+\cosh (a) \text {Shi}\left (b x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 27, normalized size = 1.08
method | result | size |
risch | \(\frac {{\mathrm e}^{-a} \expIntegral \left (1, x^{2} b \right )}{4}-\frac {{\mathrm e}^{a} \expIntegral \left (1, -x^{2} b \right )}{4}\) | \(27\) |
meijerg | \(\frac {\sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {2 \hyperbolicCosineIntegral \left (x^{2} b \right )-2 \ln \left (x^{2} b \right )-2 \gamma }{\sqrt {\pi }}+\frac {2 \gamma +4 \ln \left (x \right )+2 \ln \left (i b \right )}{\sqrt {\pi }}\right )}{4}+\frac {\cosh \left (a \right ) \hyperbolicSineIntegral \left (x^{2} b \right )}{2}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 24, normalized size = 0.96 \begin {gather*} -\frac {1}{4} \, {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac {1}{4} \, {\rm Ei}\left (b x^{2}\right ) e^{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.47, size = 39, normalized size = 1.56 \begin {gather*} \frac {1}{4} \, {\left ({\rm Ei}\left (b x^{2}\right ) - {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) + \frac {1}{4} \, {\left ({\rm Ei}\left (b x^{2}\right ) + {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (a + b x^{2} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.39, size = 24, normalized size = 0.96 \begin {gather*} -\frac {1}{4} \, {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + \frac {1}{4} \, {\rm Ei}\left (b x^{2}\right ) e^{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \frac {\mathrm {sinh}\left (a\right )\,\mathrm {coshint}\left (b\,x^2\right )}{2}+\frac {\mathrm {cosh}\left (a\right )\,\mathrm {sinhint}\left (b\,x^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________