Optimal. Leaf size=54 \[ -\frac {b n x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2}+\frac {x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-b^2 n^2} \]
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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 = {5628}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 5628
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*}
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Mathematica [A]
time = 0.04, size = 41, normalized size = 0.76 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 52, normalized size = 0.96 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 44, normalized size = 0.81 \begin {gather*} \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 {gather*}
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 {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 47, normalized size = 0.87 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.65, size = 43, normalized size = 0.80 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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