Optimal. Leaf size=150 \[ -\frac {e^{3 a} 3^{-1/n} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 b x^n\right )}{8 n}+\frac {3 e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{8 n}+\frac {e^{-3 a} 3^{-1/n} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 b x^n\right )}{8 n} \]
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Rubi [A] time = 0.08, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5308, 5306, 2208} \[ -\frac {e^{3 a} 3^{-1/n} x \left (-b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-3 b x^n\right )}{8 n}+\frac {3 e^a x \left (-b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x \left (b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},b x^n\right )}{8 n}+\frac {e^{-3 a} 3^{-1/n} x \left (b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},3 b x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 5306
Rule 5308
Rubi steps
\begin {align*} \int \sinh ^3\left (a+b x^n\right ) \, dx &=\int \left (-\frac {3}{4} \sinh \left (a+b x^n\right )+\frac {1}{4} \sinh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} \int \sinh \left (3 a+3 b x^n\right ) \, dx-\frac {3}{4} \int \sinh \left (a+b x^n\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 a-3 b x^n} \, dx\right )+\frac {1}{8} \int e^{3 a+3 b x^n} \, dx+\frac {3}{8} \int e^{-a-b x^n} \, dx-\frac {3}{8} \int e^{a+b x^n} \, dx\\ &=-\frac {3^{-1/n} e^{3 a} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 b x^n\right )}{8 n}+\frac {3 e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{8 n}+\frac {3^{-1/n} e^{-3 a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 b x^n\right )}{8 n}\\ \end {align*}
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Mathematica [A] time = 1.21, size = 140, normalized size = 0.93 \[ \frac {e^{-3 a} 3^{-1/n} x \left (-b^2 x^{2 n}\right )^{-1/n} \left (\left (-b x^n\right )^{\frac {1}{n}} \left (\Gamma \left (\frac {1}{n},3 b x^n\right )-e^{2 a} 3^{\frac {1}{n}+1} \Gamma \left (\frac {1}{n},b x^n\right )\right )-e^{6 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-3 b x^n\right )+e^{4 a} 3^{\frac {1}{n}+1} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sinh \left (b x^{n} + a\right )^{3}, x\right ) \]
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 \[ \int \sinh \left (b x^{n} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \sinh ^{3}\left (a +b \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 125, normalized size = 0.83 \[ \frac {x e^{\left (-3 \, a\right )} \Gamma \left (\frac {1}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {3 \, x e^{\left (-a\right )} \Gamma \left (\frac {1}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\left (\frac {1}{n}\right )} n} + \frac {3 \, x e^{a} \Gamma \left (\frac {1}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {x e^{\left (3 \, a\right )} \Gamma \left (\frac {1}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} n} \]
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 \[ \int {\mathrm {sinh}\left (a+b\,x^n\right )}^3 \,d x \]
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 \[ \int \sinh ^{3}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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