Optimal. Leaf size=54 \[ \frac {x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}-\frac {b n x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]
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Rubi [A] time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5518} \[ \frac {x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}-\frac {b n x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]
Antiderivative was successfully verified.
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Rule 5518
Rubi steps
\begin {align*} \int \cosh \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}-\frac {b n x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 41, normalized size = 0.76 \[ \frac {x \left (b n \sinh \left (a+b \log \left (c x^n\right )\right )-\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b^2 n^2-1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 44, normalized size = 0.81 \[ \frac {b n x \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - x \cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{b^{2} n^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 47, normalized size = 0.87 \[ \frac {c^{b} x x^{b n} e^{a}}{2 \, {\left (b n + 1\right )}} - \frac {x e^{\left (-a\right )}}{2 \, {\left (b n - 1\right )} c^{b} x^{b n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 51, normalized size = 0.94 \[ \frac {c^{b} x e^{\left (b \log \left (x^{n}\right ) + a\right )}}{2 \, {\left (b n + 1\right )}} - \frac {x e^{\left (-a\right )}}{2 \, {\left (b c^{b} n - c^{b}\right )} {\left (x^{n}\right )}^{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 44, normalized size = 0.81 \[ \frac {x\,{\mathrm {e}}^a\,{\left (c\,x^n\right )}^b}{2\,b\,n+2}-\frac {x\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (2\,b\,n-2\right )} \]
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 \[ \begin {cases} \int \cosh {\left (a - \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {1}{n} \\\int \cosh {\left (a + \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {1}{n} \\\frac {b n x \sinh {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b^{2} n^{2} - 1} - \frac {x \cosh {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b^{2} n^{2} - 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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