Optimal. Leaf size=285 \[ \frac {\left (2 a^2+b^2\right ) (e x)^{1+m}}{2 e (1+m)}+\frac {i a b e^{i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-i d x^3\right )}{3 e}-\frac {i a b e^{-i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},i d x^3\right )}{3 e}+\frac {2^{-\frac {7}{3}-\frac {m}{3}} b^2 e^{2 i c} (e x)^{1+m} \left (-i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},-2 i d x^3\right )}{3 e}+\frac {2^{-\frac {7}{3}-\frac {m}{3}} b^2 e^{-2 i c} (e x)^{1+m} \left (i d x^3\right )^{\frac {1}{3} (-1-m)} \Gamma \left (\frac {1+m}{3},2 i d x^3\right )}{3 e} \]
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Rubi [A]
time = 0.16, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3484, 6, 3471,
2250, 3470} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 2250
Rule 3470
Rule 3471
Rule 3484
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*}
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Mathematica [A]
time = 5.22, size = 556, normalized size = 1.95 \begin {gather*} \frac {2^{\frac {1}{3} (-7-m)} x (e x)^m \left (d^2 x^6\right )^{\frac {1}{3} (-1-m)} \left (3\ 2^{\frac {7+m}{3}} a^2 \left (d^2 x^6\right )^{\frac {1+m}{3}}+3\ 2^{\frac {4+m}{3}} b^2 \left (d^2 x^6\right )^{\frac {1+m}{3}}+b^2 \left (i d x^3\right )^{\frac {1+m}{3}} \cos (2 c) \Gamma \left (\frac {1+m}{3},-2 i d x^3\right )+b^2 m \left (i d x^3\right )^{\frac {1+m}{3}} \cos (2 c) \Gamma \left (\frac {1+m}{3},-2 i d x^3\right )+b^2 \left (-i d x^3\right )^{\frac {1+m}{3}} \cos (2 c) \Gamma \left (\frac {1+m}{3},2 i d x^3\right )+b^2 m \left (-i d x^3\right )^{\frac {1+m}{3}} \cos (2 c) \Gamma \left (\frac {1+m}{3},2 i d x^3\right )-i 2^{\frac {7+m}{3}} a b (1+m) \left (-i d x^3\right )^{\frac {1+m}{3}} \Gamma \left (\frac {1+m}{3},i d x^3\right ) (\cos (c)-i \sin (c))+i 2^{\frac {7+m}{3}} a b (1+m) \left (i d x^3\right )^{\frac {1+m}{3}} \Gamma \left (\frac {1+m}{3},-i d x^3\right ) (\cos (c)+i \sin (c))+i b^2 \left (i d x^3\right )^{\frac {1+m}{3}} \Gamma \left (\frac {1+m}{3},-2 i d x^3\right ) \sin (2 c)+i b^2 m \left (i d x^3\right )^{\frac {1+m}{3}} \Gamma \left (\frac {1+m}{3},-2 i d x^3\right ) \sin (2 c)-i b^2 \left (-i d x^3\right )^{\frac {1+m}{3}} \Gamma \left (\frac {1+m}{3},2 i d x^3\right ) \sin (2 c)-i b^2 m \left (-i d x^3\right )^{\frac {1+m}{3}} \Gamma \left (\frac {1+m}{3},2 i d x^3\right ) \sin (2 c)\right )}{3 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.13, size = 187, normalized size = 0.66 \begin {gather*} \frac {12 \, {\left (2 \, a^{2} + b^{2}\right )} \left (x e\right )^{m} d x - i \, {\left (b^{2} m + b^{2}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (2 i \, d e^{\left (-3\right )}\right ) - 2 i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, 2 i \, d x^{3}\right ) - 8 \, {\left (a b m + a b\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (i \, d e^{\left (-3\right )}\right ) - i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, i \, d x^{3}\right ) - 8 \, {\left (a b m + a b\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-i \, d e^{\left (-3\right )}\right ) + i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -i \, d x^{3}\right ) + i \, {\left (b^{2} m + b^{2}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-2 i \, d e^{\left (-3\right )}\right ) + 2 i \, c + 2\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -2 i \, d x^{3}\right )}{24 \, {\left (d m + d\right )}} \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*} \int \left (e x\right )^{m} \left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (e\,x\right )}^m\,{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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