Optimal. Leaf size=155 \[ -\frac {3 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}-\frac {\log \left (1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (\frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)}+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1\right )}{4 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1}{\sqrt {3}}\right )}{2 b} \]
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Rubi [A] time = 0.18, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2566, 2575, 275, 292, 31, 634, 618, 204, 628} \[ -\frac {3 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}-\frac {\log \left (1-\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (\frac {\cosh ^{\frac {4}{3}}(a+b x)}{\sinh ^{\frac {4}{3}}(a+b x)}+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1\right )}{4 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+1}{\sqrt {3}}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 618
Rule 628
Rule 634
Rule 2566
Rule 2575
Rubi steps
\begin {align*} \int \frac {\sinh ^{\frac {5}{3}}(a+b x)}{\cosh ^{\frac {5}{3}}(a+b x)} \, dx &=-\frac {3 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}+\int \frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}} \, dx\\ &=-\frac {3 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}+\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 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}+\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 {3 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}+\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 {3 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}+\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 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}+\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}-\frac {3 \sinh ^{\frac {2}{3}}(a+b x)}{2 b \cosh ^{\frac {2}{3}}(a+b x)}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 59, normalized size = 0.38 \[ \frac {3 \sinh ^{\frac {8}{3}}(a+b x) \sqrt [3]{\cosh ^2(a+b x)} \, _2F_1\left (\frac {4}{3},\frac {4}{3};\frac {7}{3};-\sinh ^2(a+b x)\right )}{8 b \cosh ^{\frac {2}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 751, normalized size = 4.85 \[ \text {result too large to display} \]
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 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {5}{3}}}{\cosh \left (b x + a\right )^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {\sinh ^{\frac {5}{3}}\left (b x +a \right )}{\cosh \left (b x +a \right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {\sinh \left (b x + a\right )^{\frac {5}{3}}}{\cosh \left (b x + a\right )^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^{5/3}}{{\mathrm {cosh}\left (a+b\,x\right )}^{5/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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