Optimal. Leaf size=34 \[ \frac {5 \cosh ^3(x)}{6}+\frac {5 \cosh (x)}{2}-\frac {1}{2} \cosh ^3(x) \coth ^2(x)-\frac {5}{2} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4397, 2592, 288, 302, 206} \[ \frac {5 \cosh ^3(x)}{6}+\frac {5 \cosh (x)}{2}-\frac {1}{2} \cosh ^3(x) \coth ^2(x)-\frac {5}{2} \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 302
Rule 2592
Rule 4397
Rubi steps
\begin {align*} \int (\text {csch}(x)+\sinh (x))^3 \, dx &=\int \cosh ^3(x) \coth ^3(x) \, dx\\ &=\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{2} \cosh ^3(x) \coth ^2(x)-\frac {5}{2} \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{2} \cosh ^3(x) \coth ^2(x)-\frac {5}{2} \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cosh (x)\right )\\ &=\frac {5 \cosh (x)}{2}+\frac {5 \cosh ^3(x)}{6}-\frac {1}{2} \cosh ^3(x) \coth ^2(x)-\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {5}{2} \tanh ^{-1}(\cosh (x))+\frac {5 \cosh (x)}{2}+\frac {5 \cosh ^3(x)}{6}-\frac {1}{2} \cosh ^3(x) \coth ^2(x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 45, normalized size = 1.32 \[ \frac {1}{48} \text {csch}^2(x) \left (-50 \cosh (x)+25 \cosh (3 x)+\cosh (5 x)-60 \log \left (\tanh \left (\frac {x}{2}\right )\right )+60 \cosh (2 x) \log \left (\tanh \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 616, normalized size = 18.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 62, normalized size = 1.82 \[ \frac {1}{24} \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - \frac {e^{\left (-x\right )} + e^{x}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} + e^{\left (-x\right )} + e^{x} - \frac {5}{4} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac {5}{4} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 28, normalized size = 0.82 \[ -\frac {\coth \relax (x ) \mathrm {csch}\relax (x )}{2}-5 \arctanh \left ({\mathrm e}^{x}\right )+3 \cosh \relax (x )+\left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\relax (x )\right )}{3}\right ) \cosh \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 67, normalized size = 1.97 \[ \frac {e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {9}{8} \, e^{\left (-x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {9}{8} \, e^{x} - \frac {5}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {5}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 71, normalized size = 2.09 \[ \frac {5\,\ln \left (5-5\,{\mathrm {e}}^x\right )}{2}-\frac {5\,\ln \left (-5\,{\mathrm {e}}^x-5\right )}{2}+\frac {9\,{\mathrm {e}}^{-x}}{8}+\frac {{\mathrm {e}}^{-3\,x}}{24}+\frac {{\mathrm {e}}^{3\,x}}{24}+\frac {9\,{\mathrm {e}}^x}{8}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1} \]
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 \[ \int \left (\sinh {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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