Optimal. Leaf size=193 \[ \frac{i a b e^{i c} x^2 \text{Gamma}\left (\frac{2}{3},-i d x^3\right )}{3 \left (-i d x^3\right )^{2/3}}-\frac{i a b e^{-i c} x^2 \text{Gamma}\left (\frac{2}{3},i d x^3\right )}{3 \left (i d x^3\right )^{2/3}}+\frac{b^2 e^{2 i c} x^2 \text{Gamma}\left (\frac{2}{3},-2 i d x^3\right )}{12\ 2^{2/3} \left (-i d x^3\right )^{2/3}}+\frac{b^2 e^{-2 i c} x^2 \text{Gamma}\left (\frac{2}{3},2 i d x^3\right )}{12\ 2^{2/3} \left (i d x^3\right )^{2/3}}+\frac{1}{4} x^2 \left (2 a^2+b^2\right ) \]
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Rubi [A] time = 0.135917, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {3403, 6, 3390, 2218, 3389} \[ \frac{i a b e^{i c} x^2 \text{Gamma}\left (\frac{2}{3},-i d x^3\right )}{3 \left (-i d x^3\right )^{2/3}}-\frac{i a b e^{-i c} x^2 \text{Gamma}\left (\frac{2}{3},i d x^3\right )}{3 \left (i d x^3\right )^{2/3}}+\frac{b^2 e^{2 i c} x^2 \text{Gamma}\left (\frac{2}{3},-2 i d x^3\right )}{12\ 2^{2/3} \left (-i d x^3\right )^{2/3}}+\frac{b^2 e^{-2 i c} x^2 \text{Gamma}\left (\frac{2}{3},2 i d x^3\right )}{12\ 2^{2/3} \left (i d x^3\right )^{2/3}}+\frac{1}{4} x^2 \left (2 a^2+b^2\right ) \]
Antiderivative was successfully verified.
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Rule 3403
Rule 6
Rule 3390
Rule 2218
Rule 3389
Rubi steps
\begin{align*} \int x \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\int \left (a^2 x+\frac{b^2 x}{2}-\frac{1}{2} b^2 x \cos \left (2 c+2 d x^3\right )+2 a b x \sin \left (c+d x^3\right )\right ) \, dx\\ &=\int \left (\left (a^2+\frac{b^2}{2}\right ) x-\frac{1}{2} b^2 x \cos \left (2 c+2 d x^3\right )+2 a b x \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac{1}{4} \left (2 a^2+b^2\right ) x^2+(2 a b) \int x \sin \left (c+d x^3\right ) \, dx-\frac{1}{2} b^2 \int x \cos \left (2 c+2 d x^3\right ) \, dx\\ &=\frac{1}{4} \left (2 a^2+b^2\right ) x^2+(i a b) \int e^{-i c-i d x^3} x \, dx-(i a b) \int e^{i c+i d x^3} x \, dx-\frac{1}{4} b^2 \int e^{-2 i c-2 i d x^3} x \, dx-\frac{1}{4} b^2 \int e^{2 i c+2 i d x^3} x \, dx\\ &=\frac{1}{4} \left (2 a^2+b^2\right ) x^2+\frac{i a b e^{i c} x^2 \Gamma \left (\frac{2}{3},-i d x^3\right )}{3 \left (-i d x^3\right )^{2/3}}-\frac{i a b e^{-i c} x^2 \Gamma \left (\frac{2}{3},i d x^3\right )}{3 \left (i d x^3\right )^{2/3}}+\frac{b^2 e^{2 i c} x^2 \Gamma \left (\frac{2}{3},-2 i d x^3\right )}{12\ 2^{2/3} \left (-i d x^3\right )^{2/3}}+\frac{b^2 e^{-2 i c} x^2 \Gamma \left (\frac{2}{3},2 i d x^3\right )}{12\ 2^{2/3} \left (i d x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.366051, size = 283, normalized size = 1.47 \[ \frac{x^2 \left (-8 i a b \left (-i d x^3\right )^{2/3} (\cos (c)-i \sin (c)) \text{Gamma}\left (\frac{2}{3},i d x^3\right )+8 i a b \left (i d x^3\right )^{2/3} (\cos (c)+i \sin (c)) \text{Gamma}\left (\frac{2}{3},-i d x^3\right )+\sqrt [3]{2} b^2 \cos (2 c) \left (i d x^3\right )^{2/3} \text{Gamma}\left (\frac{2}{3},-2 i d x^3\right )+\sqrt [3]{2} b^2 \cos (2 c) \left (-i d x^3\right )^{2/3} \text{Gamma}\left (\frac{2}{3},2 i d x^3\right )+i \sqrt [3]{2} b^2 \sin (2 c) \left (i d x^3\right )^{2/3} \text{Gamma}\left (\frac{2}{3},-2 i d x^3\right )-i \sqrt [3]{2} b^2 \sin (2 c) \left (-i d x^3\right )^{2/3} \text{Gamma}\left (\frac{2}{3},2 i d x^3\right )+12 a^2 \left (d^2 x^6\right )^{2/3}+6 b^2 \left (d^2 x^6\right )^{2/3}\right )}{24 \left (d^2 x^6\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.173, size = 0, normalized size = 0. \begin{align*} \int x \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 [B] time = 1.25813, size = 770, normalized size = 3.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69592, size = 328, normalized size = 1.7 \begin{align*} \frac{-i \, b^{2} \left (2 i \, d\right )^{\frac{1}{3}} e^{\left (-2 i \, c\right )} \Gamma \left (\frac{2}{3}, 2 i \, d x^{3}\right ) - 8 \, a b \left (i \, d\right )^{\frac{1}{3}} e^{\left (-i \, c\right )} \Gamma \left (\frac{2}{3}, i \, d x^{3}\right ) - 8 \, a b \left (-i \, d\right )^{\frac{1}{3}} e^{\left (i \, c\right )} \Gamma \left (\frac{2}{3}, -i \, d x^{3}\right ) + i \, b^{2} \left (-2 i \, d\right )^{\frac{1}{3}} e^{\left (2 i \, c\right )} \Gamma \left (\frac{2}{3}, -2 i \, d x^{3}\right ) + 6 \,{\left (2 \, a^{2} + b^{2}\right )} d x^{2}}{24 \, d} \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 \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} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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