Optimal. Leaf size=285 \[ \frac {\left (2 a^2+b^2\right ) (e x)^{m+1}}{2 e (m+1)}+\frac {i a b e^{i c} \left (-i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{3},-i d x^3\right )}{3 e}-\frac {i a b e^{-i c} \left (i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{3},i d x^3\right )}{3 e}+\frac {b^2 e^{2 i c} 2^{-\frac {m}{3}-\frac {7}{3}} \left (-i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{3},-2 i d x^3\right )}{3 e}+\frac {b^2 e^{-2 i c} 2^{-\frac {m}{3}-\frac {7}{3}} \left (i d x^3\right )^{\frac {1}{3} (-m-1)} (e x)^{m+1} \Gamma \left (\frac {m+1}{3},2 i d x^3\right )}{3 e} \]
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Rubi [A] time = 0.23, 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 = {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 6
Rule 2218
Rule 3389
Rule 3390
Rule 3403
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 = 6.67, size = 556, normalized size = 1.95 \[ \frac {2^{\frac {1}{3} (-m-7)} x \left (d^2 x^6\right )^{\frac {1}{3} (-m-1)} (e x)^m \left (3 a^2 2^{\frac {m+7}{3}} \left (d^2 x^6\right )^{\frac {m+1}{3}}-i a b 2^{\frac {m+7}{3}} (m+1) (\cos (c)-i \sin (c)) \left (-i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},i d x^3\right )+i a b 2^{\frac {m+7}{3}} (m+1) (\cos (c)+i \sin (c)) \left (i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},-i d x^3\right )+b^2 \cos (2 c) \left (-i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},2 i d x^3\right )+b^2 m \cos (2 c) \left (-i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},2 i d x^3\right )+b^2 \cos (2 c) \left (i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},-2 i d x^3\right )+b^2 m \cos (2 c) \left (i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},-2 i d x^3\right )-i b^2 \sin (2 c) \left (-i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},2 i d x^3\right )-i b^2 m \sin (2 c) \left (-i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},2 i d x^3\right )+i b^2 \sin (2 c) \left (i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},-2 i d x^3\right )+i b^2 m \sin (2 c) \left (i d x^3\right )^{\frac {m+1}{3}} \Gamma \left (\frac {m+1}{3},-2 i d x^3\right )+3 b^2 2^{\frac {m+4}{3}} \left (d^2 x^6\right )^{\frac {m+1}{3}}\right )}{3 (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 214, normalized size = 0.75 \[ \frac {12 \, {\left (2 \, a^{2} + b^{2}\right )} \left (e x\right )^{m} d x + {\left (-i \, b^{2} e^{2} m - i \, b^{2} e^{2}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (\frac {2 i \, d}{e^{3}}\right ) - 2 i \, c\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, 2 i \, d x^{3}\right ) - 8 \, {\left (a b e^{2} m + a b e^{2}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (\frac {i \, d}{e^{3}}\right ) - i \, c\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, i \, d x^{3}\right ) - 8 \, {\left (a b e^{2} m + a b e^{2}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-\frac {i \, d}{e^{3}}\right ) + i \, c\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -i \, d x^{3}\right ) + {\left (i \, b^{2} e^{2} m + i \, b^{2} e^{2}\right )} e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-\frac {2 i \, d}{e^{3}}\right ) + 2 i \, c\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -2 i \, d x^{3}\right )}{24 \, {\left (d m + d\right )}} \]
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 \[ \int {\left (b \sin \left (d x^{3} + c\right ) + a\right )}^{2} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.70, 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 \[ \frac {\left (e x\right )^{m + 1} a^{2}}{e {\left (m + 1\right )}} + \frac {b^{2} e^{m} x x^{m} - {\left (b^{2} e^{m} m + b^{2} e^{m}\right )} \int x^{m} \cos \left (2 \, d x^{3} + 2 \, c\right )\,{d x} + 4 \, {\left (a b e^{m} m + a b e^{m}\right )} \int x^{m} \sin \left (d x^{3} + c\right )\,{d x}}{2 \, {\left (m + 1\right )}} \]
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 \[ \int {\left (e\,x\right )}^m\,{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2 \,d x \]
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 \[ \int \left (e x\right )^{m} \left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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