Optimal. Leaf size=218 \[ -\frac {\log \left (\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )}{4 b}+\frac {\log \left (\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )}{4 b}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\sqrt {3}}\right )}{2 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\sqrt {3}}\right )}{2 b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b} \]
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Rubi [A] time = 0.22, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2574, 296, 634, 618, 204, 628, 206} \[ -\frac {\log \left (\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )}{4 b}+\frac {\log \left (\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1\right )}{4 b}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\sqrt {3}}\right )}{2 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+1}{\sqrt {3}}\right )}{2 b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 296
Rule 618
Rule 628
Rule 634
Rule 2574
Rubi steps
\begin {align*} \int \frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)} \, dx &=-\frac {3 \operatorname {Subst}\left (\int \frac {x^4}{-1+x^6} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2}-\frac {x}{2}}{1-x+x^2} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2}+\frac {x}{2}}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}-\frac {\operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{4 b}+\frac {\operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{4 b}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{4 b}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{4 b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}-\frac {\log \left (1-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{4 b}+\frac {\log \left (1+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b 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}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{2 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{2 b}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\sqrt {3}}\right )}{2 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}}{\sqrt {3}}\right )}{2 b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}\right )}{b}-\frac {\log \left (1-\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{4 b}+\frac {\log \left (1+\frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{4 b}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 59, normalized size = 0.27 \[ \frac {3 \sinh ^{\frac {5}{3}}(a+b x) \cosh ^2(a+b x)^{5/6} \, _2F_1\left (\frac {5}{6},\frac {5}{6};\frac {11}{6};-\sinh ^2(a+b x)\right )}{5 b \cosh ^{\frac {5}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 727, normalized size = 3.33 \[ \text {result too large to display} \]
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 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {2}{3}}}{\cosh \left (b x + a\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{\frac {2}{3}}\left (b x +a \right )}{\cosh \left (b x +a \right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {2}{3}}}{\cosh \left (b x + a\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{2/3}}{{\mathrm {cosh}\left (a+b\,x\right )}^{2/3}} \,d x \]
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 \[ \int \frac {\sinh ^{\frac {2}{3}}{\left (a + b x \right )}}{\cosh ^{\frac {2}{3}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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