Integrand size = 11, antiderivative size = 54 \[ \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {b n x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}+\frac {x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]
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Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5628} \[ \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}-\frac {b n x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]
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Rule 5628
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.76 \[ \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (b n \cosh \left (a+b \log \left (c x^n\right )\right )-\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{-1+b^2 n^2} \]
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Time = 0.92 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(\frac {x \left (b n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b^{2} n^{2}-1}\) | \(42\) |
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Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} - 1} \]
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\[ \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int \sinh {\left (a - \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {1}{n} \\\int \sinh {\left (a + \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {1}{n} \\\frac {b n x \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} - 1} - \frac {x \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} - 1} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96 \[ \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {c^{b} x e^{\left (b \log \left (x^{n}\right ) + a\right )}}{2 \, {\left (b n + 1\right )}} + \frac {x e^{\left (-b \log \left (x^{n}\right ) - a\right )}}{2 \, {\left (b c^{b} n - c^{b}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {c^{b} x x^{b n} e^{a}}{2 \, {\left (b n + 1\right )}} + \frac {x e^{\left (-a\right )}}{2 \, {\left (b n - 1\right )} c^{b} x^{b n}} \]
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Time = 1.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (2\,b\,n-2\right )}+\frac {x\,{\mathrm {e}}^a\,{\left (c\,x^n\right )}^b}{2\,b\,n+2} \]
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