Optimal. Leaf size=71 \[ \frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
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Rubi [A] time = 0.0904974, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3666, 3768, 3770} \[ \frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3666
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \text{csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx &=-\left (i \int \left (i a \text{csch}^3(c+d x)+i b \text{sech}^3(c+d x)\right ) \, dx\right )\\ &=a \int \text{csch}^3(c+d x) \, dx+b \int \text{sech}^3(c+d x) \, dx\\ &=-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \text{sech}(c+d x) \tanh (c+d x)}{2 d}-\frac{1}{2} a \int \text{csch}(c+d x) \, dx+\frac{1}{2} b \int \text{sech}(c+d x) \, dx\\ &=\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \text{sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0259812, size = 95, normalized size = 1.34 \[ -\frac{a \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 62, normalized size = 0.9 \begin{align*} -{\frac{{\rm coth} \left (dx+c\right )a{\rm csch} \left (dx+c\right )}{2\,d}}+{\frac{a{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.53103, size = 211, normalized size = 2.97 \begin{align*} -b{\left (\frac{\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac{e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac{1}{2} \, a{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.79467, size = 3216, normalized size = 45.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \operatorname{csch}^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22179, size = 193, normalized size = 2.72 \begin{align*} \frac{2 \, b \arctan \left (e^{\left (d x + c\right )}\right ) + a \log \left (e^{\left (d x + c\right )} + 1\right ) - a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac{2 \,{\left (a e^{\left (7 \, d x + 7 \, c\right )} - b e^{\left (7 \, d x + 7 \, c\right )} + 3 \, a e^{\left (5 \, d x + 5 \, c\right )} + 3 \, b e^{\left (5 \, d x + 5 \, c\right )} + 3 \, a e^{\left (3 \, d x + 3 \, c\right )} - 3 \, b e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )} + b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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