Optimal. Leaf size=117 \[ \frac{x^{m+1} e^{-\frac{2 a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left (c x^n\right )^{\frac{m+1}{n}}}{8 (m+1)}+\frac{1}{4} x^{m+1} \log (x) e^{\frac{2 a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left (c x^n\right )^{-\frac{m+1}{n}}+\frac{x^{m+1}}{2 (m+1)} \]
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Rubi [A] time = 0.117212, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {4494, 4490} \[ \frac{x^{m+1} e^{-\frac{2 a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left (c x^n\right )^{\frac{m+1}{n}}}{8 (m+1)}+\frac{1}{4} x^{m+1} \log (x) e^{\frac{2 a n \sqrt{-\frac{(m+1)^2}{n^2}}}{m+1}} \left (c x^n\right )^{-\frac{m+1}{n}}+\frac{x^{m+1}}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 4494
Rule 4490
Rubi steps
\begin{align*} \int x^m \cos ^2\left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{n}} \cos ^2\left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{\frac{2 a \sqrt{-\frac{(1+m)^2}{n^2}} n}{1+m}}}{x}+2 x^{-1+\frac{1+m}{n}}+e^{-\frac{2 a \sqrt{-\frac{(1+m)^2}{n^2}} n}{1+m}} x^{-1+\frac{2 (1+m)}{n}}\right ) \, dx,x,c x^n\right )}{4 n}\\ &=\frac{x^{1+m}}{2 (1+m)}+\frac{e^{-\frac{2 a \sqrt{-\frac{(1+m)^2}{n^2}} n}{1+m}} x^{1+m} \left (c x^n\right )^{\frac{1+m}{n}}}{8 (1+m)}+\frac{1}{4} e^{\frac{2 a \sqrt{-\frac{(1+m)^2}{n^2}} n}{1+m}} x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \log (x)\\ \end{align*}
Mathematica [F] time = 0.340159, size = 0, normalized size = 0. \[ \int x^m \cos ^2\left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cos \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{2}\sqrt{-{\frac{ \left ( 1+m \right ) ^{2}}{{n}^{2}}}}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2687, size = 232, normalized size = 1.98 \begin{align*} \frac{4 \,{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} c^{\frac{m}{n} + \frac{1}{n}} x x^{m} + c^{\frac{2 \, m}{n} + \frac{2}{n}} x \cos \left (2 \, a\right ) e^{\left (m \log \left (x\right ) + \frac{m \log \left (x^{n}\right )}{n} + \frac{\log \left (x^{n}\right )}{n}\right )} + 2 \,{\left (\cos \left (2 \, a\right )^{3} + \cos \left (2 \, a\right ) \sin \left (2 \, a\right )^{2} +{\left (\cos \left (2 \, a\right )^{3} + \cos \left (2 \, a\right ) \sin \left (2 \, a\right )^{2}\right )} m\right )} \log \left (x\right )}{8 \,{\left ({\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} c^{\frac{m}{n} + \frac{1}{n}} m +{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} c^{\frac{m}{n} + \frac{1}{n}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.50012, size = 315, normalized size = 2.69 \begin{align*} \frac{{\left (2 \,{\left (m + 1\right )} e^{\left (-\frac{2 \,{\left ({\left (m + 1\right )} n \log \left (x\right ) - 2 i \, a n +{\left (m + 1\right )} \log \left (c\right )\right )}}{n}\right )} \log \left (x\right ) + 4 \, e^{\left (-\frac{{\left (m + 1\right )} n \log \left (x\right ) - 2 i \, a n +{\left (m + 1\right )} \log \left (c\right )}{n}\right )} + 1\right )} e^{\left (\frac{2 \,{\left ({\left (m + 1\right )} n \log \left (x\right ) - 2 i \, a n +{\left (m + 1\right )} \log \left (c\right )\right )}}{n} + \frac{2 i \, a n -{\left (m + 1\right )} \log \left (c\right )}{n}\right )}}{8 \,{\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} \cos ^{2}{\left (a + \frac{\sqrt{- \frac{m^{2}}{n^{2}} - \frac{2 m}{n^{2}} - \frac{1}{n^{2}}} \log{\left (c x^{n} \right )}}{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.71286, size = 672, normalized size = 5.74 \begin{align*} \frac{m^{2} n^{2} x x^{m} e^{\left (2 i \, a - \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + m^{2} n^{2} x x^{m} e^{\left (-2 i \, a + \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + 2 \, m^{2} n^{2} x x^{m} + 2 \, m n^{2} x x^{m} e^{\left (2 i \, a - \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + m n x x^{m}{\left | m n + n \right |} e^{\left (2 i \, a - \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + 2 \, m n^{2} x x^{m} e^{\left (-2 i \, a + \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} - m n x x^{m}{\left | m n + n \right |} e^{\left (-2 i \, a + \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + 4 \, m n^{2} x x^{m} + n^{2} x x^{m} e^{\left (2 i \, a - \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + n x x^{m}{\left | m n + n \right |} e^{\left (2 i \, a - \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} + n^{2} x x^{m} e^{\left (-2 i \, a + \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} - n x x^{m}{\left | m n + n \right |} e^{\left (-2 i \, a + \frac{n{\left | m n + n \right |} \log \left (x\right ) +{\left | m n + n \right |} \log \left (c\right )}{n^{2}}\right )} - 2 \,{\left (m n + n\right )}^{2} x x^{m} + 2 \, n^{2} x x^{m}}{4 \,{\left (m^{3} n^{2} + 3 \, m^{2} n^{2} -{\left (m n + n\right )}^{2} m + 3 \, m n^{2} -{\left (m n + n\right )}^{2} + n^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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