Optimal. Leaf size=150 \[ -\frac {3^{-1/n} e^{3 a} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{8 n}-\frac {3^{-1/n} e^{-3 a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 b x^n\right )}{8 n} \]
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Rubi [A]
time = 0.06, 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 = {5417, 5415,
2239} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 5415
Rule 5417
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*}
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Mathematica [A]
time = 1.34, size = 138, normalized size = 0.92 \begin {gather*} -\frac {3^{-1/n} e^{-3 a} x \left (-b^2 x^{2 n}\right )^{-1/n} \left (e^{6 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-3 b x^n\right )+3^{1+\frac {1}{n}} e^{4 a} \left (b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-b x^n\right )+\left (-b x^n\right )^{\frac {1}{n}} \left (3^{1+\frac {1}{n}} e^{2 a} \Gamma \left (\frac {1}{n},b x^n\right )+\Gamma \left (\frac {1}{n},3 b x^n\right )\right )\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.01, size = 0, normalized size = 0.00 \[\int \cosh ^{3}\left (a +b \,x^{n}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.09, size = 125, normalized size = 0.83 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \cosh ^{3}{\left (a + b x^{n} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {cosh}\left (a+b\,x^n\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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