Optimal. Leaf size=67 \[ -\frac {e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{2 n}-\frac {e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{2 n} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5415, 2239}
\begin {gather*} -\frac {e^a x \left (-b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-b x^n\right )}{2 n}-\frac {e^{-a} x \left (b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},b x^n\right )}{2 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 5415
Rubi steps
\begin {align*} \int \cosh \left (a+b x^n\right ) \, dx &=\frac {1}{2} \int e^{-a-b x^n} \, dx+\frac {1}{2} \int e^{a+b x^n} \, dx\\ &=-\frac {e^a x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b x^n\right )}{2 n}-\frac {e^{-a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 77, normalized size = 1.15 \begin {gather*} -\frac {x \left (-b^2 x^{2 n}\right )^{-1/n} \left (\left (-b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},b x^n\right ) (\cosh (a)-\sinh (a))+\left (b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-b x^n\right ) (\cosh (a)+\sinh (a))\right )}{2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.45, size = 74, normalized size = 1.10
method | result | size |
meijerg | \(x \hypergeom \left (\left [\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \cosh \left (a \right )+\frac {x^{1+n} b \hypergeom \left (\left [\frac {1}{2}+\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {1}{2 n}\right ], \frac {x^{2 n} b^{2}}{4}\right ) \sinh \left (a \right )}{1+n}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.07, size = 61, normalized size = 0.91 \begin {gather*} -\frac {x e^{\left (-a\right )} \Gamma \left (\frac {1}{n}, b x^{n}\right )}{2 \, \left (b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {x e^{a} \Gamma \left (\frac {1}{n}, -b x^{n}\right )}{2 \, \left (-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 {\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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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