Optimal. Leaf size=67 \[ \frac {b \sinh (2 a) \text {Chi}\left (2 b x^n\right )}{n}+\frac {b \cosh (2 a) \text {Shi}\left (2 b x^n\right )}{n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-n}}{2 n} \]
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Rubi [A] time = 0.13, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5363, 5321, 3297, 3303, 3298, 3301} \[ \frac {b \sinh (2 a) \text {Chi}\left (2 b x^n\right )}{n}+\frac {b \cosh (2 a) \text {Shi}\left (2 b x^n\right )}{n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5321
Rule 5363
Rubi steps
\begin {align*} \int x^{-1-n} \cosh ^2\left (a+b x^n\right ) \, dx &=\int \left (\frac {x^{-1-n}}{2}+\frac {1}{2} x^{-1-n} \cosh \left (2 a+2 b x^n\right )\right ) \, dx\\ &=-\frac {x^{-n}}{2 n}+\frac {1}{2} \int x^{-1-n} \cosh \left (2 a+2 b x^n\right ) \, dx\\ &=-\frac {x^{-n}}{2 n}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 a+2 b x)}{x^2} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {x^{-n}}{2 n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac {b \operatorname {Subst}\left (\int \frac {\sinh (2 a+2 b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n}}{2 n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac {(b \cosh (2 a)) \operatorname {Subst}\left (\int \frac {\sinh (2 b x)}{x} \, dx,x,x^n\right )}{n}+\frac {(b \sinh (2 a)) \operatorname {Subst}\left (\int \frac {\cosh (2 b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n}}{2 n}-\frac {x^{-n} \cosh \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac {b \text {Chi}\left (2 b x^n\right ) \sinh (2 a)}{n}+\frac {b \cosh (2 a) \text {Shi}\left (2 b x^n\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 54, normalized size = 0.81 \[ \frac {x^{-n} \left (b \sinh (2 a) x^n \text {Chi}\left (2 b x^n\right )+b \cosh (2 a) x^n \text {Shi}\left (2 b x^n\right )-\cosh ^2\left (a+b x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 182, normalized size = 2.72 \[ \frac {{\left ({\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \left (2 \, a\right ) + b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (2 \, b \cosh \left (n \log \relax (x)\right ) + 2 \, b \sinh \left (n \log \relax (x)\right )\right ) - {\left ({\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \cosh \left (n \log \relax (x)\right ) + {\left (b \cosh \left (2 \, a\right ) - b \sinh \left (2 \, a\right )\right )} \sinh \left (n \log \relax (x)\right )\right )} {\rm Ei}\left (-2 \, b \cosh \left (n \log \relax (x)\right ) - 2 \, b \sinh \left (n \log \relax (x)\right )\right ) - \cosh \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )^{2} - \sinh \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a\right )^{2} - 1}{2 \, {\left (n \cosh \left (n \log \relax (x)\right ) + n \sinh \left (n \log \relax (x)\right )\right )}} \]
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 \[ \int x^{-n - 1} \cosh \left (b x^{n} + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 90, normalized size = 1.34 \[ -\frac {x^{-n}}{2 n}-\frac {{\mathrm e}^{-2 a -2 b \,x^{n}} x^{-n}}{4 n}+\frac {b \,{\mathrm e}^{-2 a} \Ei \left (1, 2 b \,x^{n}\right )}{2 n}-\frac {x^{-n} {\mathrm e}^{2 a +2 b \,x^{n}}}{4 n}-\frac {b \,{\mathrm e}^{2 a} \Ei \left (1, -2 b \,x^{n}\right )}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 47, normalized size = 0.70 \[ -\frac {b e^{\left (-2 \, a\right )} \Gamma \left (-1, 2 \, b x^{n}\right )}{2 \, n} + \frac {b e^{\left (2 \, a\right )} \Gamma \left (-1, -2 \, b x^{n}\right )}{2 \, n} - \frac {1}{2 \, n x^{n}} \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x^n\right )}^2}{x^{n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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