Optimal. Leaf size=112 \[ \frac{(m+1) e^{\frac{a \sqrt{-(m+1)^2}}{m+1}} x^{m+1} \log (x) \left (c x^2\right )^{\frac{1}{2} (-m-1)}}{2 \sqrt{-(m+1)^2}}-\frac{e^{\frac{a (m+1)}{\sqrt{-(m+1)^2}}} x^{m+1} \left (c x^2\right )^{\frac{m+1}{2}}}{4 \sqrt{-(m+1)^2}} \]
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Rubi [A] time = 0.194204, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4493, 4489} \[ \frac{(m+1) e^{\frac{a \sqrt{-(m+1)^2}}{m+1}} x^{m+1} \log (x) \left (c x^2\right )^{\frac{1}{2} (-m-1)}}{2 \sqrt{-(m+1)^2}}-\frac{e^{\frac{a (m+1)}{\sqrt{-(m+1)^2}}} x^{m+1} \left (c x^2\right )^{\frac{m+1}{2}}}{4 \sqrt{-(m+1)^2}} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int x^m \sin \left (a+\frac{1}{2} \sqrt{-(1+m)^2} \log \left (c x^2\right )\right ) \, dx &=\frac{1}{2} \left (x^{1+m} \left (c x^2\right )^{\frac{1}{2} (-1-m)}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{2}} \sin \left (a+\frac{1}{2} \sqrt{-(1+m)^2} \log (x)\right ) \, dx,x,c x^2\right )\\ &=\frac{\left ((1+m) x^{1+m} \left (c x^2\right )^{\frac{1}{2} (-1-m)}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{\frac{a \sqrt{-(1+m)^2}}{1+m}}}{x}-e^{\frac{a (1+m)}{\sqrt{-(1+m)^2}}} x^m\right ) \, dx,x,c x^2\right )}{4 \sqrt{-(1+m)^2}}\\ &=-\frac{e^{\frac{a (1+m)}{\sqrt{-(1+m)^2}}} x^{1+m} \left (c x^2\right )^{\frac{1+m}{2}}}{4 \sqrt{-(1+m)^2}}+\frac{e^{\frac{a \sqrt{-(1+m)^2}}{1+m}} (1+m) x^{1+m} \left (c x^2\right )^{\frac{1}{2} (-1-m)} \log (x)}{2 \sqrt{-(1+m)^2}}\\ \end{align*}
Mathematica [F] time = 0.230448, size = 0, normalized size = 0. \[ \int x^m \sin \left (a+\frac{1}{2} \sqrt{-(1+m)^2} \log \left (c x^2\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sin \left ( a+{\frac{\ln \left ( c{x}^{2} \right ) }{2}\sqrt{- \left ( 1+m \right ) ^{2}}} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06175, size = 65, normalized size = 0.58 \begin{align*} \frac{c^{m + 1} x^{2} x^{2 \, m} \sin \left (a\right ) + 2 \,{\left (m \sin \left (a\right ) + \sin \left (a\right )\right )} \log \left (x\right )}{4 \,{\left (c^{\frac{1}{2} \, m} m + c^{\frac{1}{2} \, m}\right )} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.486322, size = 149, normalized size = 1.33 \begin{align*} \frac{{\left (i \, x^{2} x^{2 \, m} +{\left (-2 i \, m - 2 i\right )} e^{\left (-{\left (m + 1\right )} \log \left (c\right ) + 2 i \, a\right )} \log \left (x\right )\right )} e^{\left (\frac{1}{2} \,{\left (m + 1\right )} \log \left (c\right ) - i \, a\right )}}{4 \,{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sin{\left (a + \frac{\sqrt{- m^{2} - 2 m - 1} \log{\left (c x^{2} \right )}}{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.4687, size = 255, normalized size = 2.28 \begin{align*} -\frac{i \, m x x^{m} e^{\left (\frac{1}{2} \,{\left | m + 1 \right |} \log \left (c\right ) +{\left | m + 1 \right |} \log \left (x\right ) - i \, a\right )} - i \, x x^{m}{\left | m + 1 \right |} e^{\left (\frac{1}{2} \,{\left | m + 1 \right |} \log \left (c\right ) +{\left | m + 1 \right |} \log \left (x\right ) - i \, a\right )} - i \, m x x^{m} e^{\left (-\frac{1}{2} \,{\left | m + 1 \right |} \log \left (c\right ) -{\left | m + 1 \right |} \log \left (x\right ) + i \, a\right )} - i \, x x^{m}{\left | m + 1 \right |} e^{\left (-\frac{1}{2} \,{\left | m + 1 \right |} \log \left (c\right ) -{\left | m + 1 \right |} \log \left (x\right ) + i \, a\right )} + i \, x x^{m} e^{\left (\frac{1}{2} \,{\left | m + 1 \right |} \log \left (c\right ) +{\left | m + 1 \right |} \log \left (x\right ) - i \, a\right )} - i \, x x^{m} e^{\left (-\frac{1}{2} \,{\left | m + 1 \right |} \log \left (c\right ) -{\left | m + 1 \right |} \log \left (x\right ) + i \, a\right )}}{2 \,{\left ({\left (m + 1\right )}^{2} - m^{2} - 2 \, m - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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