Optimal. Leaf size=54 \[ \frac{x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}-\frac{b n x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]
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Rubi [A] time = 0.0112868, antiderivative size = 54, normalized size of antiderivative = 1., 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 = {5517} \[ \frac{x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}-\frac{b n x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]
Antiderivative was successfully verified.
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Rule 5517
Rubi steps
\begin{align*} \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{b n x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}+\frac{x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}\\ \end{align*}
Mathematica [A] time = 0.0586636, size = 41, normalized size = 0.76 \[ \frac{x \left (b n \cosh \left (a+b \log \left (c x^n\right )\right )-\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b^2 n^2-1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11891, size = 70, normalized size = 1.3 \begin{align*} \frac{c^{b} x e^{\left (b \log \left (x^{n}\right ) + a\right )}}{2 \,{\left (b n + 1\right )}} + \frac{x e^{\left (-b \log \left (x^{n}\right ) - a\right )}}{2 \,{\left (b c^{b} n - c^{b}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04283, size = 123, normalized size = 2.28 \begin{align*} \frac{b n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13202, size = 63, normalized size = 1.17 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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