Optimal. Leaf size=322 \[ \frac{2 b \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{a^{2/3}+b^{2/3}}}+\frac{2 (-1)^{2/3} b \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 (-1)^{2/3} b \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d} \]
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Rubi [A] time = 0.457326, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3220, 3768, 3770, 3213, 2660, 618, 204} \[ \frac{2 b \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{a^{2/3}+b^{2/3}}}+\frac{2 (-1)^{2/3} b \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac{2 (-1)^{2/3} b \tan ^{-1}\left (\frac{\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} d \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3768
Rule 3770
Rule 3213
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\text{csch}^3(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\left (i \int \left (\frac{i \text{csch}^3(c+d x)}{a}-\frac{i b}{a \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\right )\\ &=\frac{\int \text{csch}^3(c+d x) \, dx}{a}-\frac{b \int \frac{1}{a+b \sinh ^3(c+d x)} \, dx}{a}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{\int \text{csch}(c+d x) \, dx}{2 a}-\frac{b \int \left (\frac{\sqrt [6]{-1}}{3 a^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}+\frac{\sqrt [6]{-1}}{3 a^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)\right )}+\frac{\sqrt [6]{-1}}{3 a^{2/3} \left (\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)\right )}\right ) \, dx}{a}\\ &=\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{\left (\sqrt [6]{-1} b\right ) \int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{5/3}}-\frac{\left (\sqrt [6]{-1} b\right ) \int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{5/3}}-\frac{\left (\sqrt [6]{-1} b\right ) \int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{5/3}}\\ &=\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{\left (2 (-1)^{2/3} b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}-2 \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{5/3} d}+\frac{\left (2 (-1)^{2/3} b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{5/3} d}+\frac{\left (2 (-1)^{2/3} b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{-1} \sqrt [3]{a}-2 (-1)^{2/3} \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{5/3} d}\\ &=\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{\left (4 (-1)^{2/3} b\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{5/3} d}-\frac{\left (4 (-1)^{2/3} b\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \sqrt [3]{-1} \left (a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 (-1)^{2/3} \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{5/3} d}-\frac{\left (4 (-1)^{2/3} b\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{-1} \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{3 a^{5/3} d}\\ &=-\frac{2 (-1)^{2/3} b \tan ^{-1}\left (\frac{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} \sqrt{\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}+\frac{2 (-1)^{2/3} b \tan ^{-1}\left (\frac{\sqrt [3]{-1} \sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} \sqrt{\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac{\tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac{2 b \tanh ^{-1}\left (\frac{\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^{2/3}+b^{2/3}}}\right )}{3 a^{5/3} \sqrt{a^{2/3}+b^{2/3}} d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d}\\ \end{align*}
Mathematica [C] time = 0.497428, size = 178, normalized size = 0.55 \[ -\frac{16 b \text{RootSum}\left [8 \text{$\#$1}^3 a+\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b-b\& ,\frac{2 \text{$\#$1} \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+\text{$\#$1} c+\text{$\#$1} d x}{\text{$\#$1}^4 b-2 \text{$\#$1}^2 b+4 \text{$\#$1} a+b}\& \right ]+3 \left (\text{csch}^2\left (\frac{1}{2} (c+d x)\right )+\text{sech}^2\left (\frac{1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )}{24 a d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.089, size = 146, normalized size = 0.5 \begin{align*}{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{b}{3\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}-3\,a{{\it \_Z}}^{4}-8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}-a \right ) }{\frac{{{\it \_R}}^{4}-2\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{5}a-2\,{{\it \_R}}^{3}a-4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -8 \, b \int \frac{e^{\left (3 \, d x + 3 \, c\right )}}{a b e^{\left (6 \, d x + 6 \, c\right )} - 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a b e^{\left (2 \, d x + 2 \, c\right )} - a b}\,{d x} - \frac{e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}}{a d e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d e^{\left (2 \, d x + 2 \, c\right )} + a d} + \frac{\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{2 \, a d} - \frac{\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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