Optimal. Leaf size=150 \[ -\frac{e^{3 a} 3^{-1/n} x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-3 b x^n\right )}{8 n}-\frac{3 e^a x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},b x^n\right )}{8 n}-\frac{e^{-3 a} 3^{-1/n} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},3 b x^n\right )}{8 n} \]
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Rubi [A] time = 0.0795542, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5309, 5307, 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 5309
Rule 5307
Rule 2208
Rubi steps
\begin{align*} \int \cosh ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac{3}{4} \cosh \left (a+b x^n\right )+\frac{1}{4} \cosh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac{1}{4} \int \cosh \left (3 a+3 b x^n\right ) \, dx+\frac{3}{4} \int \cosh \left (a+b x^n\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 a-3 b x^n} \, dx+\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*}
Mathematica [A] time = 0.600603, size = 138, normalized size = 0.92 \[ -\frac{e^{-3 a} 3^{-1/n} x \left (-b^2 x^{2 n}\right )^{-1/n} \left (\left (-b x^n\right )^{\frac{1}{n}} \left (e^{2 a} 3^{\frac{1}{n}+1} \text{Gamma}\left (\frac{1}{n},b x^n\right )+\text{Gamma}\left (\frac{1}{n},3 b x^n\right )\right )+e^{6 a} \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-3 b x^n\right )+e^{4 a} 3^{\frac{1}{n}+1} \left (b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.101, size = 0, normalized size = 0. \begin{align*} \int \left ( \cosh \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 = 1.20309, size = 169, normalized size = 1.13 \begin{align*} -\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} \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 (\cosh \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 \cosh ^{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 \cosh \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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