Optimal. Leaf size=113 \[ -\frac {3 b \cosh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}-\frac {3 b \sinh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n}+\frac {3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n} \]
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Rubi [A] time = 0.22, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5362, 5320, 3297, 3303, 3298, 3301} \[ -\frac {3 b \cosh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}-\frac {3 b \sinh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n}+\frac {3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5320
Rule 5362
Rubi steps
\begin {align*} \int x^{-1-n} \sinh ^3\left (a+b x^n\right ) \, dx &=\int \left (-\frac {3}{4} x^{-1-n} \sinh \left (a+b x^n\right )+\frac {1}{4} x^{-1-n} \sinh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} \int x^{-1-n} \sinh \left (3 a+3 b x^n\right ) \, dx-\frac {3}{4} \int x^{-1-n} \sinh \left (a+b x^n\right ) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{4 n}-\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (a+b x)}{x^2} \, dx,x,x^n\right )}{4 n}\\ &=\frac {3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\cosh (a+b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\cosh (3 a+3 b x)}{x} \, dx,x,x^n\right )}{4 n}\\ &=\frac {3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {(3 b \cosh (a)) \operatorname {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac {(3 b \cosh (3 a)) \operatorname {Subst}\left (\int \frac {\cosh (3 b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \sinh (a)) \operatorname {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^n\right )}{4 n}+\frac {(3 b \sinh (3 a)) \operatorname {Subst}\left (\int \frac {\sinh (3 b x)}{x} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {3 b \cosh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {3 b \cosh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}+\frac {3 x^{-n} \sinh \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \sinh \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {3 b \sinh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {3 b \sinh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 95, normalized size = 0.84 \[ -\frac {x^{-n} \left (3 b \cosh (a) x^n \text {Chi}\left (b x^n\right )-3 b \cosh (3 a) x^n \text {Chi}\left (3 b x^n\right )+3 b \sinh (a) x^n \text {Shi}\left (b x^n\right )-3 b \sinh (3 a) x^n \text {Shi}\left (3 b x^n\right )-3 \sinh \left (a+b x^n\right )+\sinh \left (3 \left (a+b x^n\right )\right )\right )}{4 n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 303, normalized size = 2.68 \[ -\frac {2 \, \sinh \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )^{3} - 3 \, {\left ({\left (b \cosh \left (3 \, a\right ) + b \sinh \left (3 \, a\right )\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \left (3 \, a\right ) + b \sinh \left (3 \, a\right )\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (3 \, b \cosh \left (n \log \relax (x)\right ) + 3 \, b \sinh \left (n \log \relax (x)\right )\right ) + 3 \, {\left ({\left (b \cosh \relax (a) + b \sinh \relax (a)\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \relax (a) + b \sinh \relax (a)\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right ) + 3 \, {\left ({\left (b \cosh \relax (a) - b \sinh \relax (a)\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \relax (a) - b \sinh \relax (a)\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (-b \cosh \left (n \log \relax (x)\right ) - b \sinh \left (n \log \relax (x)\right )\right ) - 3 \, {\left ({\left (b \cosh \left (3 \, a\right ) - b \sinh \left (3 \, a\right )\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \left (3 \, a\right ) - b \sinh \left (3 \, a\right )\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (-3 \, b \cosh \left (n \log \relax (x)\right ) - 3 \, b \sinh \left (n \log \relax (x)\right )\right ) + 6 \, {\left (\cosh \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )^{2} - 1\right )} \sinh \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )}{8 \, {\left (n \cosh \left (n \log \relax (x)\right ) + n \sinh \left (n \log \relax (x)\right )\right )}} \]
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 \[ \int x^{-n - 1} \sinh \left (b x^{n} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 152, normalized size = 1.35 \[ \frac {{\mathrm e}^{-3 a -3 b \,x^{n}} x^{-n}}{8 n}-\frac {3 b \,{\mathrm e}^{-3 a} \Ei \left (1, 3 b \,x^{n}\right )}{8 n}-\frac {3 \,{\mathrm e}^{-a -b \,x^{n}} x^{-n}}{8 n}+\frac {3 b \,{\mathrm e}^{-a} \Ei \left (1, b \,x^{n}\right )}{8 n}-\frac {x^{-n} {\mathrm e}^{3 a +3 b \,x^{n}}}{8 n}-\frac {3 b \,{\mathrm e}^{3 a} \Ei \left (1, -3 b \,x^{n}\right )}{8 n}+\frac {3 \,{\mathrm e}^{a +b \,x^{n}} x^{-n}}{8 n}+\frac {3 b \,{\mathrm e}^{a} \Ei \left (1, -b \,x^{n}\right )}{8 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 70, normalized size = 0.62 \[ \frac {3 \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{n}\right )}{8 \, n} - \frac {3 \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{n}\right )}{8 \, n} - \frac {3 \, b e^{a} \Gamma \left (-1, -b x^{n}\right )}{8 \, n} + \frac {3 \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{n}\right )}{8 \, n} \]
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 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x^n\right )}^3}{x^{n+1}} \,d x \]
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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