Optimal. Leaf size=166 \[ -\frac{e^{3 a} 3^{-3/n} x^3 \left (-b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-3 b x^n\right )}{8 n}+\frac{3 e^a x^3 \left (-b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^3 \left (b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},b x^n\right )}{8 n}+\frac{e^{-3 a} 3^{-3/n} x^3 \left (b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},3 b x^n\right )}{8 n} \]
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Rubi [A] time = 0.195389, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5362, 5360, 2218} \[ -\frac{e^{3 a} 3^{-3/n} x^3 \left (-b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-3 b x^n\right )}{8 n}+\frac{3 e^a x^3 \left (-b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^3 \left (b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},b x^n\right )}{8 n}+\frac{e^{-3 a} 3^{-3/n} x^3 \left (b x^n\right )^{-3/n} \text{Gamma}\left (\frac{3}{n},3 b x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Rule 5362
Rule 5360
Rule 2218
Rubi steps
\begin{align*} \int x^2 \sinh ^3\left (a+b x^n\right ) \, dx &=\int \left (-\frac{3}{4} x^2 \sinh \left (a+b x^n\right )+\frac{1}{4} x^2 \sinh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac{1}{4} \int x^2 \sinh \left (3 a+3 b x^n\right ) \, dx-\frac{3}{4} \int x^2 \sinh \left (a+b x^n\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 a-3 b x^n} x^2 \, dx\right )+\frac{1}{8} \int e^{3 a+3 b x^n} x^2 \, dx+\frac{3}{8} \int e^{-a-b x^n} x^2 \, dx-\frac{3}{8} \int e^{a+b x^n} x^2 \, dx\\ &=-\frac{3^{-3/n} e^{3 a} x^3 \left (-b x^n\right )^{-3/n} \Gamma \left (\frac{3}{n},-3 b x^n\right )}{8 n}+\frac{3 e^a x^3 \left (-b x^n\right )^{-3/n} \Gamma \left (\frac{3}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^3 \left (b x^n\right )^{-3/n} \Gamma \left (\frac{3}{n},b x^n\right )}{8 n}+\frac{3^{-3/n} e^{-3 a} x^3 \left (b x^n\right )^{-3/n} \Gamma \left (\frac{3}{n},3 b x^n\right )}{8 n}\\ \end{align*}
Mathematica [A] time = 1.5339, size = 161, normalized size = 0.97 \[ -\frac{e^{-3 a} 27^{-1/n} x^3 \left (-b^2 x^{2 n}\right )^{-3/n} \left (\left (-b x^n\right )^{3/n} \left (e^{2 a} 3^{\frac{n+3}{n}} \text{Gamma}\left (\frac{3}{n},b x^n\right )-\text{Gamma}\left (\frac{3}{n},3 b x^n\right )\right )+e^{6 a} \left (b x^n\right )^{3/n} \text{Gamma}\left (\frac{3}{n},-3 b x^n\right )-e^{4 a} 3^{\frac{n+3}{n}} \left (b x^n\right )^{3/n} \text{Gamma}\left (\frac{3}{n},-b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \sinh \left ( a+b{x}^{n} \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.34138, size = 201, normalized size = 1.21 \begin{align*} \frac{x^{3} e^{\left (-3 \, a\right )} \Gamma \left (\frac{3}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\frac{3}{n}} n} - \frac{3 \, x^{3} e^{\left (-a\right )} \Gamma \left (\frac{3}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\frac{3}{n}} n} + \frac{3 \, x^{3} e^{a} \Gamma \left (\frac{3}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\frac{3}{n}} n} - \frac{x^{3} e^{\left (3 \, a\right )} \Gamma \left (\frac{3}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\frac{3}{n}} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \sinh \left (b x^{n} + a\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sinh ^{3}{\left (a + b x^{n} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sinh \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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