Optimal. Leaf size=128 \[ -\frac {\log \left (1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (\frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+1\right )}{4 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+1}{\sqrt {3}}\right )}{2 b} \]
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Rubi [A] time = 0.09, antiderivative size = 128, normalized size of antiderivative = 1.00, 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 = {2574, 275, 292, 31, 634, 618, 204, 628} \[ -\frac {\log \left (1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (\frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+1\right )}{4 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+1}{\sqrt {3}}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 618
Rule 628
Rule 634
Rule 2574
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}} \, dx &=-\frac {3 \operatorname {Subst}\left (\int \frac {x^3}{-1+x^6} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}\\ &=-\frac {3 \operatorname {Subst}\left (\int \frac {x}{-1+x^3} \, dx,x,\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {-1+x}{1+x+x^2} \, dx,x,\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}\\ &=-\frac {\log \left (1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\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 {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{4 b}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{4 b}\\ &=-\frac {\log \left (1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (1+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)}\right )}{4 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}}{\sqrt {3}}\right )}{2 b}-\frac {\log \left (1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (1+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)}\right )}{4 b}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 59, normalized size = 0.46 \[ \frac {3 \sinh ^{\frac {4}{3}}(a+b x) \cosh ^2(a+b x)^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};-\sinh ^2(a+b x)\right )}{4 b \cosh ^{\frac {4}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 572, normalized size = 4.47 \[ -\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {3} \sinh \left (b x + a\right )^{2} + 4 \, {\left (\sqrt {3} \cosh \left (b x + a\right ) + \sqrt {3} \sinh \left (b x + a\right )\right )} \cosh \left (b x + a\right )^{\frac {1}{3}} \sinh \left (b x + a\right )^{\frac {2}{3}} + \sqrt {3}}{3 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )}}\right ) - \log \left (\frac {\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \cosh \left (b x + a\right )^{\frac {2}{3}} \sinh \left (b x + a\right )^{\frac {1}{3}} + 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )} \cosh \left (b x + a\right )^{\frac {1}{3}} \sinh \left (b x + a\right )^{\frac {2}{3}} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}\right ) + 2 \, \log \left (-\frac {\cosh \left (b x + a\right )^{2} - 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \cosh \left (b x + a\right )^{\frac {1}{3}} \sinh \left (b x + a\right )^{\frac {2}{3}} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {1}{3}}}{\cosh \left (b x + a\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{\frac {1}{3}}\left (b x +a \right )}{\cosh \left (b x +a \right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {1}{3}}}{\cosh \left (b x + a\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{1/3}}{{\mathrm {cosh}\left (a+b\,x\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{\sinh {\left (a + b x \right )}}}{\sqrt [3]{\cosh {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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