Optimal. Leaf size=166 \[ -\frac {e^{3 a} 9^{-1/n} x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-3 b x^n\right )}{8 n}+\frac {3 e^a x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )}{8 n}+\frac {e^{-3 a} 9^{-1/n} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},3 b x^n\right )}{8 n} \]
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Rubi [A] time = 0.13, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5362, 5360, 2218} \[ -\frac {e^{3 a} 9^{-1/n} x^2 \left (-b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-3 b x^n\right )}{8 n}+\frac {3 e^a x^2 \left (-b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^2 \left (b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},b x^n\right )}{8 n}+\frac {e^{-3 a} 9^{-1/n} x^2 \left (b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},3 b x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 5360
Rule 5362
Rubi steps
\begin {align*} \int x \sinh ^3\left (a+b x^n\right ) \, dx &=\int \left (-\frac {3}{4} x \sinh \left (a+b x^n\right )+\frac {1}{4} x \sinh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} \int x \sinh \left (3 a+3 b x^n\right ) \, dx-\frac {3}{4} \int x \sinh \left (a+b x^n\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 a-3 b x^n} x \, dx\right )+\frac {1}{8} \int e^{3 a+3 b x^n} x \, dx+\frac {3}{8} \int e^{-a-b x^n} x \, dx-\frac {3}{8} \int e^{a+b x^n} x \, dx\\ &=-\frac {9^{-1/n} e^{3 a} x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-3 b x^n\right )}{8 n}+\frac {3 e^a x^2 \left (-b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},b x^n\right )}{8 n}+\frac {9^{-1/n} e^{-3 a} x^2 \left (b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},3 b x^n\right )}{8 n}\\ \end {align*}
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Mathematica [A] time = 1.60, size = 161, normalized size = 0.97 \[ -\frac {e^{-3 a} 9^{-1/n} x^2 \left (-b^2 x^{2 n}\right )^{-2/n} \left (\left (-b x^n\right )^{2/n} \left (e^{2 a} 3^{\frac {n+2}{n}} \Gamma \left (\frac {2}{n},b x^n\right )-\Gamma \left (\frac {2}{n},3 b x^n\right )\right )+e^{6 a} \left (b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},-3 b x^n\right )-e^{4 a} 3^{\frac {n+2}{n}} \left (b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},-b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \sinh \left (b x^{n} + a\right )^{3}, x\right ) \]
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 \[ \int x \sinh \left (b x^{n} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x \left (\sinh ^{3}\left (a +b \,x^{n}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 149, normalized size = 0.90 \[ \frac {x^{2} e^{\left (-3 \, a\right )} \Gamma \left (\frac {2}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\frac {2}{n}} n} - \frac {3 \, x^{2} e^{\left (-a\right )} \Gamma \left (\frac {2}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\frac {2}{n}} n} + \frac {3 \, x^{2} e^{a} \Gamma \left (\frac {2}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\frac {2}{n}} n} - \frac {x^{2} e^{\left (3 \, a\right )} \Gamma \left (\frac {2}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\frac {2}{n}} n} \]
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 \[ \int x\,{\mathrm {sinh}\left (a+b\,x^n\right )}^3 \,d x \]
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 \[ \int x \sinh ^{3}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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