Optimal. Leaf size=155 \[ -\frac {3 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}-\frac {\log \left (1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (\frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+1\right )}{4 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+1}{\sqrt {3}}\right )}{2 b} \]
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Rubi [A] time = 0.12, 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 = {2567, 2574, 275, 292, 31, 634, 618, 204, 628} \[ -\frac {3 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}-\frac {\log \left (1-\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}\right )}{2 b}+\frac {\log \left (\frac {\sinh ^{\frac {4}{3}}(a+b x)}{\cosh ^{\frac {4}{3}}(a+b x)}+\frac {\sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+1\right )}{4 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sinh ^{\frac {2}{3}}(a+b x)}{\cosh ^{\frac {2}{3}}(a+b x)}+1}{\sqrt {3}}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 618
Rule 628
Rule 634
Rule 2567
Rule 2574
Rubi steps
\begin {align*} \int \frac {\cosh ^{\frac {5}{3}}(a+b x)}{\sinh ^{\frac {5}{3}}(a+b x)} \, dx &=-\frac {3 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}+\int \frac {\sqrt [3]{\sinh (a+b x)}}{\sqrt [3]{\cosh (a+b x)}} \, dx\\ &=-\frac {3 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}-\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 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}-\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 {3 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}-\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 {3 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}+\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 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}+\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}-\frac {3 \cosh ^{\frac {2}{3}}(a+b x)}{2 b \sinh ^{\frac {2}{3}}(a+b x)}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 59, normalized size = 0.38 \[ -\frac {3 \cosh ^2(a+b x)^{2/3} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};-\sinh ^2(a+b x)\right )}{2 b \sinh ^{\frac {2}{3}}(a+b x) \cosh ^{\frac {4}{3}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 749, normalized size = 4.83 \[ \text {result too large to display} \]
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 \[ \int \frac {\cosh \left (b x + a\right )^{\frac {5}{3}}}{\sinh \left (b x + a\right )^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{\frac {5}{3}}\left (b x +a \right )}{\sinh \left (b x +a \right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {\cosh \left (b x + a\right )^{\frac {5}{3}}}{\sinh \left (b x + a\right )^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^{5/3}}{{\mathrm {sinh}\left (a+b\,x\right )}^{5/3}} \,d x \]
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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