Optimal. Leaf size=104 \[ \frac{b^3 \log (a+b \sinh (c+d x))}{a^2 d \left (a^2+b^2\right )}-\frac{a \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac{b \log (\cosh (c+d x))}{d \left (a^2+b^2\right )}-\frac{b \log (\sinh (c+d x))}{a^2 d}-\frac{\text{csch}(c+d x)}{a d} \]
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Rubi [A] time = 0.169645, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2837, 12, 894, 635, 203, 260} \[ \frac{b^3 \log (a+b \sinh (c+d x))}{a^2 d \left (a^2+b^2\right )}-\frac{a \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac{b \log (\cosh (c+d x))}{d \left (a^2+b^2\right )}-\frac{b \log (\sinh (c+d x))}{a^2 d}-\frac{\text{csch}(c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(c+d x) \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac{b \operatorname{Subst}\left (\int \frac{b^2}{x^2 (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{x^2 (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac{b^3 \operatorname{Subst}\left (\int \left (-\frac{1}{a b^2 x^2}+\frac{1}{a^2 b^2 x}-\frac{1}{a^2 \left (a^2+b^2\right ) (a+x)}+\frac{a-x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac{\text{csch}(c+d x)}{a d}-\frac{b \log (\sinh (c+d x))}{a^2 d}+\frac{b^3 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}-\frac{b \operatorname{Subst}\left (\int \frac{a-x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{\text{csch}(c+d x)}{a d}-\frac{b \log (\sinh (c+d x))}{a^2 d}+\frac{b^3 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac{(a b) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{a \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d}-\frac{\text{csch}(c+d x)}{a d}+\frac{b \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}-\frac{b \log (\sinh (c+d x))}{a^2 d}+\frac{b^3 \log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 0.593664, size = 160, normalized size = 1.54 \[ -\frac{b^3 \left (\frac{\log (\sinh (c+d x))}{a^2 b^2}-\frac{\left (a \sqrt{-b^2}+b^2\right ) \log \left (\sqrt{-b^2}-b \sinh (c+d x)\right )}{2 b^4 \left (a^2+b^2\right )}-\frac{\log (a+b \sinh (c+d x))}{a^2 \left (a^2+b^2\right )}-\frac{\left (\frac{a}{\sqrt{-b^2}}+1\right ) \log \left (\sqrt{-b^2}+b \sinh (c+d x)\right )}{2 b^2 \left (a^2+b^2\right )}+\frac{\text{csch}(c+d x)}{a b^3}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 159, normalized size = 1.5 \begin{align*}{\frac{1}{2\,da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{2\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{b}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{{b}^{3}}{d{a}^{2} \left ({a}^{2}+{b}^{2} \right ) }\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a-2\,\tanh \left ( 1/2\,dx+c/2 \right ) b-a \right ) }+{\frac{b}{d \left ({a}^{2}+{b}^{2} \right ) }\ln \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) }-2\,{\frac{a\arctan \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74286, size = 234, normalized size = 2.25 \begin{align*} \frac{b^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + a^{2} b^{2}\right )} d} + \frac{2 \, a \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{b \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} - \frac{b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac{b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.8194, size = 1098, normalized size = 10.56 \begin{align*} -\frac{2 \,{\left (a^{3} \cosh \left (d x + c\right )^{2} + 2 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{3} \sinh \left (d x + c\right )^{2} - a^{3}\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 2 \,{\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right ) -{\left (b^{3} \cosh \left (d x + c\right )^{2} + 2 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} \sinh \left (d x + c\right )^{2} - b^{3}\right )} \log \left (\frac{2 \,{\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) -{\left (a^{2} b \cosh \left (d x + c\right )^{2} + 2 \, a^{2} b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} b \sinh \left (d x + c\right )^{2} - a^{2} b\right )} \log \left (\frac{2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) -{\left (a^{2} b + b^{3} -{\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2} - 2 \,{\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) -{\left (a^{2} b + b^{3}\right )} \sinh \left (d x + c\right )^{2}\right )} \log \left (\frac{2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \,{\left (a^{3} + a b^{2}\right )} \sinh \left (d x + c\right )}{{\left (a^{4} + a^{2} b^{2}\right )} d \cosh \left (d x + c\right )^{2} + 2 \,{\left (a^{4} + a^{2} b^{2}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a^{4} + a^{2} b^{2}\right )} d \sinh \left (d x + c\right )^{2} -{\left (a^{4} + a^{2} b^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] time = 1.31908, size = 284, normalized size = 2.73 \begin{align*} \frac{b^{4} \log \left ({\left | b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{4} b d + a^{2} b^{3} d} - \frac{{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} a}{2 \,{\left (a^{2} d + b^{2} d\right )}} + \frac{b \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{2 \,{\left (a^{2} d + b^{2} d\right )}} - \frac{b \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{2} d} + \frac{b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, a}{a^{2} d{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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