Optimal. Leaf size=128 \[ -\frac{\log \left (1-\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\log \left (\frac{\cosh ^{\frac{4}{3}}(a+b x)}{\sinh ^{\frac{4}{3}}(a+b x)}+\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}+1\right )}{4 b}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}+1}{\sqrt{3}}\right )}{2 b} \]
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Rubi [A] time = 0.0887767, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {2575, 275, 292, 31, 634, 618, 204, 628} \[ -\frac{\log \left (1-\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\log \left (\frac{\cosh ^{\frac{4}{3}}(a+b x)}{\sinh ^{\frac{4}{3}}(a+b x)}+\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}+1\right )}{4 b}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}+1}{\sqrt{3}}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2575
Rule 275
Rule 292
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx &=\frac{3 \operatorname{Subst}\left (\int \frac{x^3}{1-x^6} \, dx,x,\frac{\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}\\ &=\frac{3 \operatorname{Subst}\left (\int \frac{x}{1-x^3} \, dx,x,\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}-\frac{\operatorname{Subst}\left (\int \frac{1-x}{1+x+x^2} \, dx,x,\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}\\ &=-\frac{\log \left (1-\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{4 b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{4 b}\\ &=-\frac{\log \left (1-\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\log \left (1+\frac{\cosh ^{\frac{4}{3}}(a+b x)}{\sinh ^{\frac{4}{3}}(a+b x)}+\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{4 b}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right )}{2 b}-\frac{\log \left (1-\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{2 b}+\frac{\log \left (1+\frac{\cosh ^{\frac{4}{3}}(a+b x)}{\sinh ^{\frac{4}{3}}(a+b x)}+\frac{\cosh ^{\frac{2}{3}}(a+b x)}{\sinh ^{\frac{2}{3}}(a+b x)}\right )}{4 b}\\ \end{align*}
Mathematica [C] time = 0.0243457, size = 59, normalized size = 0.46 \[ \frac{3 \sinh ^{\frac{2}{3}}(a+b x) \sqrt [3]{\cosh ^2(a+b x)} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\sinh ^2(a+b x)\right )}{2 b \cosh ^{\frac{2}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{\cosh \left ( bx+a \right ) }{\frac{1}{\sqrt [3]{\sinh \left ( bx+a \right ) }}}}\, dx \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*} \int \frac{\cosh \left (b x + a\right )^{\frac{1}{3}}}{\sinh \left (b x + a\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95119, size = 1694, normalized size = 13.23 \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 \frac{\sqrt [3]{\cosh{\left (a + b x \right )}}}{\sqrt [3]{\sinh{\left (a + b x \right )}}}\, dx \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{\cosh \left (b x + a\right )^{\frac{1}{3}}}{\sinh \left (b x + a\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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