Optimal. Leaf size=160 \[ -\frac{b^4 \log (a+b \sinh (c+d x))}{a d \left (a^2+b^2\right )^2}-\frac{b^3 \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}-\frac{a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac{\log (\sinh (c+d x))}{a d} \]
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Rubi [A] time = 0.270547, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2837, 12, 894, 639, 203, 635, 260} \[ -\frac{b^4 \log (a+b \sinh (c+d x))}{a d \left (a^2+b^2\right )^2}-\frac{b^3 \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )}-\frac{a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac{\log (\sinh (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rule 639
Rule 203
Rule 635
Rule 260
Rubi steps
\begin{align*} \int \frac{\text{csch}(c+d x) \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{b^3 \operatorname{Subst}\left (\int \frac{b}{x (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac{b^4 \operatorname{Subst}\left (\int \frac{1}{x (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac{b^4 \operatorname{Subst}\left (\int \left (\frac{1}{a b^4 x}-\frac{1}{a \left (a^2+b^2\right )^2 (a+x)}+\frac{-b^2-a x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}+\frac{-b^4-a \left (a^2+2 b^2\right ) x}{b^4 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac{\log (\sinh (c+d x))}{a d}-\frac{b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac{\operatorname{Subst}\left (\int \frac{-b^4-a \left (a^2+2 b^2\right ) x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{-b^2-a x}{\left (b^2+x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{\log (\sinh (c+d x))}{a d}-\frac{b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{b^4 \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}-\frac{\left (a \left (a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac{b^3 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{b \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac{a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{\log (\sinh (c+d x))}{a d}-\frac{b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac{\text{sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 0.793089, size = 196, normalized size = 1.22 \[ -\frac{-a^2 \left (a^2+b^2\right ) \text{sech}^2(c+d x)-2 \left (a^2+b^2\right )^2 \log (\sinh (c+d x))+a \left (a^3+2 a b^2+\left (-b^2\right )^{3/2}\right ) \log \left (\sqrt{-b^2}-b \sinh (c+d x)\right )+a \left (a^3+2 a b^2-\left (-b^2\right )^{3/2}\right ) \log \left (\sqrt{-b^2}+b \sinh (c+d x)\right )+a b \left (a^2+b^2\right ) \tan ^{-1}(\sinh (c+d x))+a b \left (a^2+b^2\right ) \tanh (c+d x) \text{sech}(c+d x)+2 b^4 \log (a+b \sinh (c+d x))}{2 a d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 530, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.81423, size = 358, normalized size = 2.24 \begin{align*} -\frac{b^{4} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d} + \frac{{\left (a^{2} b + 3 \, b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac{{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac{b e^{\left (-d x - c\right )} - 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \,{\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.2508, size = 3066, normalized size = 19.16 \begin{align*} \text{result too large to display} \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 = 1.76585, size = 485, normalized size = 3.03 \begin{align*} -\frac{b^{5} \log \left ({\left | b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{5} b d + 2 \, a^{3} b^{3} d + a b^{5} 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 )}{\left (a^{2} b + 3 \, b^{3}\right )}}{4 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac{{\left (a^{3} + 2 \, a b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{2 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} + \frac{a^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 2 \, a b^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{2} b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, b^{3}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 8 \, a^{3} + 12 \, a b^{2}}{2 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}} + \frac{\log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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