Optimal. Leaf size=52 \[ -\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b (2 a-b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0657536, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3186, 390, 206} \[ -\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b (2 a-b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 390
Rule 206
Rubi steps
\begin{align*} \int \text{csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-(2 a-b) b-b^2 x^2+\frac{a^2}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{(2 a-b) b \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{(2 a-b) b \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0339139, size = 104, normalized size = 2. \[ \frac{a^2 \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a^2 \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{2 a b \sinh (c) \sinh (d x)}{d}+\frac{2 a b \cosh (c) \cosh (d x)}{d}-\frac{3 b^2 \cosh (c+d x)}{4 d}+\frac{b^2 \cosh (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 50, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -2\,{a}^{2}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +2\,ab\cosh \left ( dx+c \right ) +{b}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04856, size = 138, normalized size = 2.65 \begin{align*} \frac{1}{24} \, b^{2}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{a^{2} \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85076, size = 1280, normalized size = 24.62 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + 3 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} - 24 \,{\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 24 \,{\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 6 \,{\left (b^{2} \cosh \left (d x + c\right )^{5} + 2 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} +{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \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 = 1.28008, size = 174, normalized size = 3.35 \begin{align*} -\frac{a^{2} \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} + \frac{{\left (24 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} + \frac{b^{2} d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b d^{2} e^{\left (d x + c\right )} - 9 \, b^{2} d^{2} e^{\left (d x + c\right )}}{24 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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