Optimal. Leaf size=45 \[ -\frac{\sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{b}+\frac{\cosh (a-c) \text{sech}(b x+c)}{b}+\frac{\cosh (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0551535, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5620, 5623, 2638, 3770, 2606, 8} \[ -\frac{\sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{b}+\frac{\cosh (a-c) \text{sech}(b x+c)}{b}+\frac{\cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5620
Rule 5623
Rule 2638
Rule 3770
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \sinh (a+b x) \tanh ^2(c+b x) \, dx &=-(\cosh (a-c) \int \text{sech}(c+b x) \tanh (c+b x) \, dx)+\int \cosh (a+b x) \tanh (c+b x) \, dx\\ &=\frac{\cosh (a-c) \operatorname{Subst}(\int 1 \, dx,x,\text{sech}(c+b x))}{b}-\sinh (a-c) \int \text{sech}(c+b x) \, dx+\int \sinh (a+b x) \, dx\\ &=\frac{\cosh (a+b x)}{b}+\frac{\cosh (a-c) \text{sech}(c+b x)}{b}-\frac{\tan ^{-1}(\sinh (c+b x)) \sinh (a-c)}{b}\\ \end{align*}
Mathematica [B] time = 0.10009, size = 102, normalized size = 2.27 \[ \frac{\cosh (a-c) \text{sech}(b x+c)}{b}-\frac{2 \sinh (a-c) \tan ^{-1}\left (\frac{(\cosh (c)-\sinh (c)) \left (\sinh (c) \cosh \left (\frac{b x}{2}\right )+\cosh (c) \sinh \left (\frac{b x}{2}\right )\right )}{\cosh (c) \cosh \left (\frac{b x}{2}\right )-\sinh (c) \cosh \left (\frac{b x}{2}\right )}\right )}{b}+\frac{\sinh (a) \sinh (b x)}{b}+\frac{\cosh (a) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.089, size = 205, normalized size = 4.6 \begin{align*}{\frac{{{\rm e}^{bx+a}}}{2\,b}}+{\frac{{{\rm e}^{-bx-a}}}{2\,b}}+{\frac{{{\rm e}^{bx+a}} \left ({{\rm e}^{2\,a}}+{{\rm e}^{2\,c}} \right ) }{b \left ({{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) }}+{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}-i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{a}} \right ) ^{2}}{b}}-{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}-i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{c}} \right ) ^{2}}{b}}-{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}+i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{a}} \right ) ^{2}}{b}}+{\frac{{\frac{i}{2}}\ln \left ({{\rm e}^{bx+a}}+i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{c}} \right ) ^{2}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.73557, size = 142, normalized size = 3.16 \begin{align*} \frac{{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{b} + \frac{e^{\left (-b x - a\right )}}{2 \, b} + \frac{{\left (3 \, e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (2 \, c\right )}}{2 \, b{\left (e^{\left (-b x - a + 2 \, c\right )} + e^{\left (-3 \, b x - a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.24425, size = 2491, normalized size = 55.36 \begin{align*} \text{result too large to display} \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 \sinh{\left (a + b x \right )} \tanh ^{2}{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.24064, size = 119, normalized size = 2.64 \begin{align*} -\frac{2 \,{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )} - \frac{{\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, c\right )} + 1\right )} e^{\left (-a\right )}}{e^{\left (3 \, b x + 2 \, c\right )} + e^{\left (b x\right )}} - e^{\left (b x + a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]