Optimal. Leaf size=84 \[ -\frac{4 \sinh (a+b x)}{25 b^2 \text{sech}^{\frac{3}{2}}(a+b x)}+\frac{12 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{25 b^2}+\frac{2 x}{5 b \text{sech}^{\frac{5}{2}}(a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0534484, antiderivative size = 84, 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 = {5444, 3769, 3771, 2639} \[ -\frac{4 \sinh (a+b x)}{25 b^2 \text{sech}^{\frac{3}{2}}(a+b x)}+\frac{12 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{25 b^2}+\frac{2 x}{5 b \text{sech}^{\frac{5}{2}}(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5444
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{x \sinh (a+b x)}{\text{sech}^{\frac{3}{2}}(a+b x)} \, dx &=\frac{2 x}{5 b \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{2 \int \frac{1}{\text{sech}^{\frac{5}{2}}(a+b x)} \, dx}{5 b}\\ &=\frac{2 x}{5 b \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{4 \sinh (a+b x)}{25 b^2 \text{sech}^{\frac{3}{2}}(a+b x)}-\frac{6 \int \frac{1}{\sqrt{\text{sech}(a+b x)}} \, dx}{25 b}\\ &=\frac{2 x}{5 b \text{sech}^{\frac{5}{2}}(a+b x)}-\frac{4 \sinh (a+b x)}{25 b^2 \text{sech}^{\frac{3}{2}}(a+b x)}-\frac{\left (6 \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)}\right ) \int \sqrt{\cosh (a+b x)} \, dx}{25 b}\\ &=\frac{2 x}{5 b \text{sech}^{\frac{5}{2}}(a+b x)}+\frac{12 i \sqrt{\cosh (a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right ) \sqrt{\text{sech}(a+b x)}}{25 b^2}-\frac{4 \sinh (a+b x)}{25 b^2 \text{sech}^{\frac{3}{2}}(a+b x)}\\ \end{align*}
Mathematica [C] time = 2.13397, size = 125, normalized size = 1.49 \[ \frac{e^{-3 (a+b x)} \left (48 e^{2 (a+b x)} \sqrt{e^{2 (a+b x)}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 (a+b x)}\right )+\left (e^{2 (a+b x)}+1\right ) \left (2 (5 b x-12) e^{2 (a+b x)}+(5 b x-2) e^{4 (a+b x)}+5 b x+2\right )\right ) \sqrt{\text{sech}(a+b x)}}{100 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{x\sinh \left ( bx+a \right ) \left ({\rm sech} \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 \frac{x \sinh \left (b x + a\right )}{\operatorname{sech}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sinh{\left (a + b x \right )}}{\operatorname{sech}^{\frac{3}{2}}{\left (a + b x \right )}}\, dx \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 \frac{x \sinh \left (b x + a\right )}{\operatorname{sech}\left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]