Optimal. Leaf size=285 \[ \frac{i a b e^{i c} \left (-i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},-i d x^3\right )}{3 e}-\frac{i a b e^{-i c} \left (i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},i d x^3\right )}{3 e}+\frac{b^2 e^{2 i c} 2^{-\frac{m}{3}-\frac{7}{3}} \left (-i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},-2 i d x^3\right )}{3 e}+\frac{b^2 e^{-2 i c} 2^{-\frac{m}{3}-\frac{7}{3}} \left (i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},2 i d x^3\right )}{3 e}+\frac{\left (2 a^2+b^2\right ) (e x)^{m+1}}{2 e (m+1)} \]
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Rubi [A] time = 0.232444, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3403, 6, 3390, 2218, 3389} \[ \frac{i a b e^{i c} \left (-i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},-i d x^3\right )}{3 e}-\frac{i a b e^{-i c} \left (i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},i d x^3\right )}{3 e}+\frac{b^2 e^{2 i c} 2^{-\frac{m}{3}-\frac{7}{3}} \left (-i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},-2 i d x^3\right )}{3 e}+\frac{b^2 e^{-2 i c} 2^{-\frac{m}{3}-\frac{7}{3}} \left (i d x^3\right )^{\frac{1}{3} (-m-1)} (e x)^{m+1} \text{Gamma}\left (\frac{m+1}{3},2 i d x^3\right )}{3 e}+\frac{\left (2 a^2+b^2\right ) (e x)^{m+1}}{2 e (m+1)} \]
Antiderivative was successfully verified.
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Rule 3403
Rule 6
Rule 3390
Rule 2218
Rule 3389
Rubi steps
\begin{align*} \int (e x)^m \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\int \left (a^2 (e x)^m+\frac{1}{2} b^2 (e x)^m-\frac{1}{2} b^2 (e x)^m \cos \left (2 c+2 d x^3\right )+2 a b (e x)^m \sin \left (c+d x^3\right )\right ) \, dx\\ &=\int \left (\left (a^2+\frac{b^2}{2}\right ) (e x)^m-\frac{1}{2} b^2 (e x)^m \cos \left (2 c+2 d x^3\right )+2 a b (e x)^m \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac{\left (2 a^2+b^2\right ) (e x)^{1+m}}{2 e (1+m)}+(2 a b) \int (e x)^m \sin \left (c+d x^3\right ) \, dx-\frac{1}{2} b^2 \int (e x)^m \cos \left (2 c+2 d x^3\right ) \, dx\\ &=\frac{\left (2 a^2+b^2\right ) (e x)^{1+m}}{2 e (1+m)}+(i a b) \int e^{-i c-i d x^3} (e x)^m \, dx-(i a b) \int e^{i c+i d x^3} (e x)^m \, dx-\frac{1}{4} b^2 \int e^{-2 i c-2 i d x^3} (e x)^m \, dx-\frac{1}{4} b^2 \int e^{2 i c+2 i d x^3} (e x)^m \, dx\\ &=\frac{\left (2 a^2+b^2\right ) (e x)^{1+m}}{2 e (1+m)}+\frac{i a b e^{i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac{1}{3} (-1-m)} \Gamma \left (\frac{1+m}{3},-i d x^3\right )}{3 e}-\frac{i a b e^{-i c} (e x)^{1+m} \left (i d x^3\right )^{\frac{1}{3} (-1-m)} \Gamma \left (\frac{1+m}{3},i d x^3\right )}{3 e}+\frac{2^{-\frac{7}{3}-\frac{m}{3}} b^2 e^{2 i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac{1}{3} (-1-m)} \Gamma \left (\frac{1+m}{3},-2 i d x^3\right )}{3 e}+\frac{2^{-\frac{7}{3}-\frac{m}{3}} b^2 e^{-2 i c} (e x)^{1+m} \left (i d x^3\right )^{\frac{1}{3} (-1-m)} \Gamma \left (\frac{1+m}{3},2 i d x^3\right )}{3 e}\\ \end{align*}
Mathematica [A] time = 6.8193, size = 556, normalized size = 1.95 \[ \frac{2^{\frac{1}{3} (-m-7)} x \left (d^2 x^6\right )^{\frac{1}{3} (-m-1)} (e x)^m \left (-i a b 2^{\frac{m+7}{3}} (m+1) (\cos (c)-i \sin (c)) \left (-i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},i d x^3\right )+i a b 2^{\frac{m+7}{3}} (m+1) (\cos (c)+i \sin (c)) \left (i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},-i d x^3\right )+b^2 \cos (2 c) \left (-i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},2 i d x^3\right )+b^2 m \cos (2 c) \left (-i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},2 i d x^3\right )+b^2 \cos (2 c) \left (i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},-2 i d x^3\right )+b^2 m \cos (2 c) \left (i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},-2 i d x^3\right )-i b^2 \sin (2 c) \left (-i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},2 i d x^3\right )-i b^2 m \sin (2 c) \left (-i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},2 i d x^3\right )+i b^2 \sin (2 c) \left (i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},-2 i d x^3\right )+i b^2 m \sin (2 c) \left (i d x^3\right )^{\frac{m+1}{3}} \text{Gamma}\left (\frac{m+1}{3},-2 i d x^3\right )+3 a^2 2^{\frac{m+7}{3}} \left (d^2 x^6\right )^{\frac{m+1}{3}}+3 b^2 2^{\frac{m+4}{3}} \left (d^2 x^6\right )^{\frac{m+1}{3}}\right )}{3 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.381, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b\sin \left ( d{x}^{3}+c \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90572, size = 578, normalized size = 2.03 \begin{align*} \frac{12 \,{\left (2 \, a^{2} + b^{2}\right )} \left (e x\right )^{m} d x +{\left (-i \, b^{2} e^{2} m - i \, b^{2} e^{2}\right )} e^{\left (-\frac{1}{3} \,{\left (m - 2\right )} \log \left (\frac{2 i \, d}{e^{3}}\right ) - 2 i \, c\right )} \Gamma \left (\frac{1}{3} \, m + \frac{1}{3}, 2 i \, d x^{3}\right ) - 8 \,{\left (a b e^{2} m + a b e^{2}\right )} e^{\left (-\frac{1}{3} \,{\left (m - 2\right )} \log \left (\frac{i \, d}{e^{3}}\right ) - i \, c\right )} \Gamma \left (\frac{1}{3} \, m + \frac{1}{3}, i \, d x^{3}\right ) - 8 \,{\left (a b e^{2} m + a b e^{2}\right )} e^{\left (-\frac{1}{3} \,{\left (m - 2\right )} \log \left (-\frac{i \, d}{e^{3}}\right ) + i \, c\right )} \Gamma \left (\frac{1}{3} \, m + \frac{1}{3}, -i \, d x^{3}\right ) +{\left (i \, b^{2} e^{2} m + i \, b^{2} e^{2}\right )} e^{\left (-\frac{1}{3} \,{\left (m - 2\right )} \log \left (-\frac{2 i \, d}{e^{3}}\right ) + 2 i \, c\right )} \Gamma \left (\frac{1}{3} \, m + \frac{1}{3}, -2 i \, d x^{3}\right )}{24 \,{\left (d m + d\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 \left (e x\right )^{m} \left (a + b \sin{\left (c + d x^{3} \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x^{3} + c\right ) + a\right )}^{2} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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