Optimal. Leaf size=46 \[ -\frac{\sinh (a+b x)}{b^2}-\frac{\tan ^{-1}(\sinh (a+b x))}{b^2}+\frac{x \cosh (a+b x)}{b}+\frac{x \text{sech}(a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0508757, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5449, 3296, 2637, 5418, 3770} \[ -\frac{\sinh (a+b x)}{b^2}-\frac{\tan ^{-1}(\sinh (a+b x))}{b^2}+\frac{x \cosh (a+b x)}{b}+\frac{x \text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5449
Rule 3296
Rule 2637
Rule 5418
Rule 3770
Rubi steps
\begin{align*} \int x \sinh (a+b x) \tanh ^2(a+b x) \, dx &=\int x \sinh (a+b x) \, dx-\int x \text{sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac{x \cosh (a+b x)}{b}+\frac{x \text{sech}(a+b x)}{b}-\frac{\int \cosh (a+b x) \, dx}{b}-\frac{\int \text{sech}(a+b x) \, dx}{b}\\ &=-\frac{\tan ^{-1}(\sinh (a+b x))}{b^2}+\frac{x \cosh (a+b x)}{b}+\frac{x \text{sech}(a+b x)}{b}-\frac{\sinh (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.11489, size = 50, normalized size = 1.09 \[ -\frac{\sinh (a+b x)}{b^2}-\frac{2 \tan ^{-1}\left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{b^2}+\frac{x \cosh (a+b x)}{b}+\frac{x \text{sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.082, size = 94, normalized size = 2. \begin{align*}{\frac{ \left ( bx-1 \right ){{\rm e}^{bx+a}}}{2\,{b}^{2}}}+{\frac{ \left ( bx+1 \right ){{\rm e}^{-bx-a}}}{2\,{b}^{2}}}+2\,{\frac{{{\rm e}^{bx+a}}x}{b \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }}+{\frac{i\ln \left ({{\rm e}^{bx+a}}-i \right ) }{{b}^{2}}}-{\frac{i\ln \left ({{\rm e}^{bx+a}}+i \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.85291, size = 109, normalized size = 2.37 \begin{align*} \frac{6 \, b x e^{\left (b x + 2 \, a\right )} +{\left (b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} +{\left (b x + 1\right )} e^{\left (-b x\right )}}{2 \,{\left (b^{2} e^{\left (2 \, b x + 3 \, a\right )} + b^{2} e^{a}\right )}} - \frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.13181, size = 779, normalized size = 16.93 \begin{align*} \frac{{\left (b x - 1\right )} \cosh \left (b x + a\right )^{4} + 4 \,{\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} +{\left (b x - 1\right )} \sinh \left (b x + a\right )^{4} + 6 \, b x \cosh \left (b x + a\right )^{2} + 6 \,{\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + b x\right )} \sinh \left (b x + a\right )^{2} + b x - 4 \,{\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} +{\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 4 \,{\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{3} + 3 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{2 \,{\left (b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3} + b^{2} \cosh \left (b x + a\right ) +{\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} + b^{2}\right )} \sinh \left (b x + a\right )\right )}} \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 [B] time = 1.2364, size = 138, normalized size = 3. \begin{align*} \frac{b x e^{\left (4 \, b x + 4 \, a\right )} + 6 \, b x e^{\left (2 \, b x + 2 \, a\right )} + b x - 4 \, \arctan \left (e^{\left (b x + a\right )}\right ) e^{\left (3 \, b x + 3 \, a\right )} - 4 \, \arctan \left (e^{\left (b x + a\right )}\right ) e^{\left (b x + a\right )} - e^{\left (4 \, b x + 4 \, a\right )} + 1}{2 \,{\left (b^{2} e^{\left (3 \, b x + 3 \, a\right )} + b^{2} e^{\left (b x + a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]