Optimal. Leaf size=73 \[ -\frac {b n x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-b^2 n^2}+\frac {(1+m) x^{1+m} \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-b^2 n^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.01, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {5638}
\begin {gather*} \frac {(m+1) x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right )}{(-b n+m+1) (b n+m+1)}-\frac {b n x^{m+1} \cosh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-b^2 n^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 5638
Rubi steps
\begin {align*} \int x^m \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {b n x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-b^2 n^2}+\frac {(1+m) x^{1+m} \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-b^2 n^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 54, normalized size = 0.74 \begin {gather*} \frac {x^{1+m} \left (-b n \cosh \left (a+b \log \left (c x^n\right )\right )+(1+m) \sinh \left (a+b \log \left (c x^n\right )\right )\right )}{(1+m-b n) (1+m+b n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int x^{m} \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 64, normalized size = 0.88 \begin {gather*} \frac {c^{b} x e^{\left (b \log \left (x^{n}\right ) + m \log \left (x\right ) + a\right )}}{2 \, {\left (b n + m + 1\right )}} + \frac {x e^{\left (-b \log \left (x^{n}\right ) + m \log \left (x\right ) - a\right )}}{2 \, {\left (b c^{b} n - c^{b} {\left (m + 1\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 98, normalized size = 1.34 \begin {gather*} \frac {b n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \cosh \left (m \log \left (x\right )\right ) + b n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (m \log \left (x\right )\right ) - {\left ({\left (m + 1\right )} x \cosh \left (m \log \left (x\right )\right ) + {\left (m + 1\right )} x \sinh \left (m \log \left (x\right )\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} - m^{2} - 2 \, m - 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} \log {\left (x \right )} \sinh {\left (a \right )} & \text {for}\: b = 0 \wedge m = -1 \\- \int x^{m} \sinh {\left (- a + \frac {m \log {\left (c x^{n} \right )}}{n} + \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {- m - 1}{n} \\\int x^{m} \sinh {\left (a + \frac {m \log {\left (c x^{n} \right )}}{n} + \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {m + 1}{n} \\\frac {b n x x^{m} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} - m^{2} - 2 m - 1} - \frac {m x x^{m} \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} - m^{2} - 2 m - 1} - \frac {x x^{m} \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} - m^{2} - 2 m - 1} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 235 vs.
\(2 (75) = 150\).
time = 0.43, size = 235, normalized size = 3.22 \begin {gather*} \frac {b c^{b} n x x^{b n} x^{m} e^{a}}{2 \, {\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )}} - \frac {c^{b} m x x^{b n} x^{m} e^{a}}{2 \, {\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )}} - \frac {c^{b} x x^{b n} x^{m} e^{a}}{2 \, {\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )}} + \frac {b n x x^{m} e^{\left (-a\right )}}{2 \, {\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )} c^{b} x^{b n}} + \frac {m x x^{m} e^{\left (-a\right )}}{2 \, {\left (b^{2} n^{2} - m^{2} - 2 \, m - 1\right )} c^{b} x^{b n}} + \frac {x x^{m} e^{\left (-a\right )}}{2 \, {\left (b^{2} n^{2} - m^{2} - 2 \, m - 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 = 0.70, size = 56, normalized size = 0.77 \begin {gather*} \frac {x\,x^m\,{\mathrm {e}}^a\,{\left (c\,x^n\right )}^b}{2\,m+2\,b\,n+2}-\frac {x\,x^m\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (2\,m-2\,b\,n+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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