Optimal. Leaf size=75 \[ -\frac{b \left (6 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 d}+a^3 x-\frac{5 a b^2 \coth (c+d x)}{2 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))}{2 d} \]
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Rubi [A] time = 0.051388, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3782, 3770, 3767, 8} \[ -\frac{b \left (6 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 d}+a^3 x-\frac{5 a b^2 \coth (c+d x)}{2 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 3782
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \text{csch}(c+d x))^3 \, dx &=-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))}{2 d}+\frac{1}{2} \int \left (2 a^3+b \left (6 a^2-b^2\right ) \text{csch}(c+d x)+5 a b^2 \text{csch}^2(c+d x)\right ) \, dx\\ &=a^3 x-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))}{2 d}+\frac{1}{2} \left (5 a b^2\right ) \int \text{csch}^2(c+d x) \, dx+\frac{1}{2} \left (b \left (6 a^2-b^2\right )\right ) \int \text{csch}(c+d x) \, dx\\ &=a^3 x-\frac{b \left (6 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))}{2 d}-\frac{\left (5 i a b^2\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{2 d}\\ &=a^3 x-\frac{b \left (6 a^2-b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{5 a b^2 \coth (c+d x)}{2 d}-\frac{b^2 \coth (c+d x) (a+b \text{csch}(c+d x))}{2 d}\\ \end{align*}
Mathematica [A] time = 0.880053, size = 118, normalized size = 1.57 \[ -\frac{-24 a^2 b \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-8 a^3 c-8 a^3 d x+12 a b^2 \tanh \left (\frac{1}{2} (c+d x)\right )+12 a b^2 \coth \left (\frac{1}{2} (c+d x)\right )+b^3 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )+b^3 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )+4 b^3 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 66, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( dx+c \right ) -6\,{a}^{2}b{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) -3\,a{b}^{2}{\rm coth} \left (dx+c\right )+{b}^{3} \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994928, size = 184, normalized size = 2.45 \begin{align*} a^{3} x + \frac{1}{2} \, b^{3}{\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 )} + \frac{3 \, a^{2} b \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} + \frac{6 \, a b^{2}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66081, size = 1844, normalized size = 24.59 \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 \operatorname{csch}{\left (c + d x \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18527, size = 174, normalized size = 2.32 \begin{align*} \frac{{\left (d x + c\right )} a^{3}}{d} - \frac{{\left (6 \, a^{2} b - b^{3}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{2 \, d} + \frac{{\left (6 \, a^{2} b - b^{3}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{2 \, d} - \frac{b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 6 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{3} e^{\left (d x + c\right )} - 6 \, a b^{2}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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