Optimal. Leaf size=98 \[ -\frac{4 \sinh ^{\frac{3}{2}}(a+b x) \cosh (a+b x)}{25 b^2}-\frac{12 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt{i \sinh (a+b x)}}+\frac{2 x \sinh ^{\frac{5}{2}}(a+b x)}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0524617, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5372, 2635, 2640, 2639} \[ -\frac{4 \sinh ^{\frac{3}{2}}(a+b x) \cosh (a+b x)}{25 b^2}-\frac{12 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt{i \sinh (a+b x)}}+\frac{2 x \sinh ^{\frac{5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5372
Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int x \cosh (a+b x) \sinh ^{\frac{3}{2}}(a+b x) \, dx &=\frac{2 x \sinh ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{2 \int \sinh ^{\frac{5}{2}}(a+b x) \, dx}{5 b}\\ &=-\frac{4 \cosh (a+b x) \sinh ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \sinh ^{\frac{5}{2}}(a+b x)}{5 b}+\frac{6 \int \sqrt{\sinh (a+b x)} \, dx}{25 b}\\ &=-\frac{4 \cosh (a+b x) \sinh ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \sinh ^{\frac{5}{2}}(a+b x)}{5 b}+\frac{\left (6 \sqrt{\sinh (a+b x)}\right ) \int \sqrt{i \sinh (a+b x)} \, dx}{25 b \sqrt{i \sinh (a+b x)}}\\ &=-\frac{12 i E\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{\sinh (a+b x)}}{25 b^2 \sqrt{i \sinh (a+b x)}}-\frac{4 \cosh (a+b x) \sinh ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \sinh ^{\frac{5}{2}}(a+b x)}{5 b}\\ \end{align*}
Mathematica [C] time = 2.1644, size = 143, normalized size = 1.46 \[ \frac{e^{-3 (a+b x)} \left (48 e^{2 (a+b x)} \sqrt{1-e^{2 (a+b x)}} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 (a+b x)}\right )+\left (e^{2 (a+b x)}-1\right ) \left ((24-10 b x) e^{2 (a+b x)}+(5 b x-2) e^{4 (a+b x)}+5 b x+2\right )\right )}{50 \sqrt{2} b^2 \sqrt{e^{a+b x}-e^{-a-b x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int x\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]