Optimal. Leaf size=152 \[ \frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right )}{6 a^{5/3}}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sinh (x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a} \]
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Rubi [A] time = 0.251657, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733, Rules used = {3230, 1834, 1871, 12, 200, 31, 634, 617, 204, 628, 260} \[ \frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right )}{6 a^{5/3}}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sinh (x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3230
Rule 1834
Rule 1871
Rule 12
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{\coth ^3(x)}{a+b \sinh ^3(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^2}{x^3 \left (a+b x^3\right )} \, dx,x,\sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{a x^3}+\frac{1}{a x}+\frac{-b-b x^2}{a \left (a+b x^3\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}+\frac{\operatorname{Subst}\left (\int \frac{-b-b x^2}{a+b x^3} \, dx,x,\sinh (x)\right )}{a}\\ &=-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}-\frac{\operatorname{Subst}\left (\int \frac{b}{a+b x^3} \, dx,x,\sinh (x)\right )}{a}-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,\sinh (x)\right )}{a}\\ &=-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}-\frac{\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,\sinh (x)\right )}{a}\\ &=-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}-\frac{\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sinh (x)\right )}{3 a^{5/3}}-\frac{b \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sinh (x)\right )}{3 a^{5/3}}\\ &=-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}-\frac{\log \left (a+b \sinh ^3(x)\right )}{3 a}+\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sinh (x)\right )}{6 a^{5/3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sinh (x)\right )}{2 a^{4/3}}\\ &=-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right )}{6 a^{5/3}}-\frac{\log \left (a+b \sinh ^3(x)\right )}{3 a}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sinh (x)}{\sqrt [3]{a}}\right )}{a^{5/3}}\\ &=\frac{b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sinh (x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{5/3}}-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a}-\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sinh (x)+b^{2/3} \sinh ^2(x)\right )}{6 a^{5/3}}-\frac{\log \left (a+b \sinh ^3(x)\right )}{3 a}\\ \end{align*}
Mathematica [A] time = 0.338182, size = 136, normalized size = 0.89 \[ -\frac{\left (a^{2/3}+(-1)^{2/3} b^{2/3}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{a}-\sqrt [3]{b} \sinh (x)\right )+\left (a^{2/3}+b^{2/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sinh (x)\right )+\left (a^{2/3}-\sqrt [3]{-1} b^{2/3}\right ) \log \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sinh (x)\right )}{3 a^{5/3}}-\frac{\text{csch}^2(x)}{2 a}+\frac{\log (\sinh (x))}{a} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.06, size = 132, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+{\frac{1}{3\,a}\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}}^{5}a-{{\it \_R}}^{4}b+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b-{\it \_R}\,a+b}{{{\it \_R}}^{5}a-2\,{{\it \_R}}^{3}a-4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [C] time = 11.7566, size = 2762, normalized size = 18.17 \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 [A] time = 1.23755, size = 282, normalized size = 1.86 \begin{align*} \frac{b \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | -2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} - e^{\left (-x\right )} + e^{x} \right |}\right )}{3 \, a^{2}} - \frac{\log \left ({\left | -b{\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 8 \, a \right |}\right )}{3 \, a} + \frac{\log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{a} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (\left (-\frac{a}{b}\right )^{\frac{1}{3}} - e^{\left (-x\right )} + e^{x}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2}} - \frac{3 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4}{2 \, a{\left (e^{\left (-x\right )} - e^{x}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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