Optimal. Leaf size=88 \[ \frac{x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}-\frac{2 b n x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}+\frac{2 b^2 n^2 x}{1-4 b^2 n^2} \]
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Rubi [A] time = 0.0206036, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5519, 8} \[ \frac{x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}-\frac{2 b n x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}+\frac{2 b^2 n^2 x}{1-4 b^2 n^2} \]
Antiderivative was successfully verified.
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Rule 5519
Rule 8
Rubi steps
\begin{align*} \int \sinh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{2 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}+\frac{x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}+\frac{\left (2 b^2 n^2\right ) \int 1 \, dx}{1-4 b^2 n^2}\\ &=\frac{2 b^2 n^2 x}{1-4 b^2 n^2}-\frac{2 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}+\frac{x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}\\ \end{align*}
Mathematica [A] time = 0.100934, size = 55, normalized size = 0.62 \[ -\frac{x \left (-2 b n \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b^2 n^2-1\right )}{8 b^2 n^2-2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.095, size = 0, normalized size = 0. \begin{align*} \int \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16068, size = 90, normalized size = 1.02 \begin{align*} \frac{c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}}{4 \,{\left (2 \, b n + 1\right )}} - \frac{1}{2} \, x - \frac{x e^{\left (-2 \, a\right )}}{4 \,{\left (2 \, b c^{2 \, b} n - c^{2 \, b}\right )}{\left (x^{n}\right )}^{2 \, b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02648, size = 258, normalized size = 2.93 \begin{align*} \frac{4 \, b n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} -{\left (4 \, b^{2} n^{2} - 1\right )} x}{2 \,{\left (4 \, b^{2} n^{2} - 1\right )}} \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 = 1.20731, size = 228, normalized size = 2.59 \begin{align*} \frac{b c^{2 \, b} n x x^{2 \, b n} e^{\left (2 \, a\right )}}{2 \,{\left (4 \, b^{2} n^{2} - 1\right )}} - \frac{2 \, b^{2} n^{2} x}{4 \, b^{2} n^{2} - 1} - \frac{c^{2 \, b} x x^{2 \, b n} e^{\left (2 \, a\right )}}{4 \,{\left (4 \, b^{2} n^{2} - 1\right )}} - \frac{b n x e^{\left (-2 \, a\right )}}{2 \,{\left (4 \, b^{2} n^{2} - 1\right )} c^{2 \, b} x^{2 \, b n}} + \frac{x}{2 \,{\left (4 \, b^{2} n^{2} - 1\right )}} - \frac{x e^{\left (-2 \, a\right )}}{4 \,{\left (4 \, b^{2} n^{2} - 1\right )} c^{2 \, b} x^{2 \, b n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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