Optimal. Leaf size=200 \[ -\frac {3^{-\frac {1+m}{n}} e^{3 a} x^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{8 n}-\frac {3^{-\frac {1+m}{n}} e^{-3 a} x^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 b x^n\right )}{8 n} \]
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Rubi [A]
time = 0.14, antiderivative size = 200, normalized size of antiderivative = 1.00, 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 = {5471, 5469,
2250} \begin {gather*} -\frac {e^{3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},b x^n\right )}{8 n}-\frac {e^{-3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},3 b x^n\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 5469
Rule 5471
Rubi steps
\begin {align*} \int x^m \cosh ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac {3}{4} x^m \cosh \left (a+b x^n\right )+\frac {1}{4} x^m \cosh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} \int x^m \cosh \left (3 a+3 b x^n\right ) \, dx+\frac {3}{4} \int x^m \cosh \left (a+b x^n\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 a-3 b x^n} x^m \, dx+\frac {1}{8} \int e^{3 a+3 b x^n} x^m \, dx+\frac {3}{8} \int e^{-a-b x^n} x^m \, dx+\frac {3}{8} \int e^{a+b x^n} x^m \, dx\\ &=-\frac {3^{-\frac {1+m}{n}} e^{3 a} x^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{8 n}-\frac {3^{-\frac {1+m}{n}} e^{-3 a} x^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 b x^n\right )}{8 n}\\ \end {align*}
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Mathematica [A]
time = 2.66, size = 182, normalized size = 0.91 \begin {gather*} -\frac {3^{-\frac {1+m}{n}} e^{-3 a} x^{1+m} \left (-b^2 x^{2 n}\right )^{-\frac {1+m}{n}} \left (e^{6 a} \left (b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )+3^{\frac {1+m+n}{n}} e^{4 a} \left (b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )+\left (-b x^n\right )^{\frac {1+m}{n}} \left (3^{\frac {1+m+n}{n}} e^{2 a} \Gamma \left (\frac {1+m}{n},b x^n\right )+\Gamma \left (\frac {1+m}{n},3 b x^n\right )\right )\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.37, size = 0, normalized size = 0.00 \[\int x^{m} \left (\cosh ^{3}\left (a +b \,x^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.16, size = 173, normalized size = 0.86 \begin {gather*} -\frac {x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (\frac {m + 1}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {3 \, x^{m + 1} e^{\left (-a\right )} \Gamma \left (\frac {m + 1}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {3 \, x^{m + 1} e^{a} \Gamma \left (\frac {m + 1}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (\frac {m + 1}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\frac {m + 1}{n}} 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 x^{m} \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 = -0.00 \begin {gather*} \int x^m\,{\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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