Optimal. Leaf size=237 \[ \frac {1}{8} x^4 \left (2 a^2+b^2\right )-\frac {2 a b x \cos \left (c+d x^3\right )}{3 d}-\frac {a b e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{9 d \sqrt [3]{-i d x^3}}-\frac {a b e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{9 d \sqrt [3]{i d x^3}}-\frac {b^2 x \sin \left (2 c+2 d x^3\right )}{12 d}+\frac {i b^2 e^{2 i c} x \Gamma \left (\frac {1}{3},-2 i d x^3\right )}{72 \sqrt [3]{2} d \sqrt [3]{-i d x^3}}-\frac {i b^2 e^{-2 i c} x \Gamma \left (\frac {1}{3},2 i d x^3\right )}{72 \sqrt [3]{2} d \sqrt [3]{i d x^3}} \]
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Rubi [A] time = 0.15, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3403, 6, 3386, 3355, 2208, 3385, 3356} \[ -\frac {a b e^{i c} x \text {Gamma}\left (\frac {1}{3},-i d x^3\right )}{9 d \sqrt [3]{-i d x^3}}-\frac {a b e^{-i c} x \text {Gamma}\left (\frac {1}{3},i d x^3\right )}{9 d \sqrt [3]{i d x^3}}+\frac {i b^2 e^{2 i c} x \text {Gamma}\left (\frac {1}{3},-2 i d x^3\right )}{72 \sqrt [3]{2} d \sqrt [3]{-i d x^3}}-\frac {i b^2 e^{-2 i c} x \text {Gamma}\left (\frac {1}{3},2 i d x^3\right )}{72 \sqrt [3]{2} d \sqrt [3]{i d x^3}}+\frac {1}{8} x^4 \left (2 a^2+b^2\right )-\frac {2 a b x \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x \sin \left (2 c+2 d x^3\right )}{12 d} \]
Antiderivative was successfully verified.
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Rule 6
Rule 2208
Rule 3355
Rule 3356
Rule 3385
Rule 3386
Rule 3403
Rubi steps
\begin {align*} \int x^3 \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\int \left (a^2 x^3+\frac {b^2 x^3}{2}-\frac {1}{2} b^2 x^3 \cos \left (2 c+2 d x^3\right )+2 a b x^3 \sin \left (c+d x^3\right )\right ) \, dx\\ &=\int \left (\left (a^2+\frac {b^2}{2}\right ) x^3-\frac {1}{2} b^2 x^3 \cos \left (2 c+2 d x^3\right )+2 a b x^3 \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac {1}{8} \left (2 a^2+b^2\right ) x^4+(2 a b) \int x^3 \sin \left (c+d x^3\right ) \, dx-\frac {1}{2} b^2 \int x^3 \cos \left (2 c+2 d x^3\right ) \, dx\\ &=\frac {1}{8} \left (2 a^2+b^2\right ) x^4-\frac {2 a b x \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x \sin \left (2 c+2 d x^3\right )}{12 d}+\frac {(2 a b) \int \cos \left (c+d x^3\right ) \, dx}{3 d}+\frac {b^2 \int \sin \left (2 c+2 d x^3\right ) \, dx}{12 d}\\ &=\frac {1}{8} \left (2 a^2+b^2\right ) x^4-\frac {2 a b x \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x \sin \left (2 c+2 d x^3\right )}{12 d}+\frac {(a b) \int e^{-i c-i d x^3} \, dx}{3 d}+\frac {(a b) \int e^{i c+i d x^3} \, dx}{3 d}+\frac {\left (i b^2\right ) \int e^{-2 i c-2 i d x^3} \, dx}{24 d}-\frac {\left (i b^2\right ) \int e^{2 i c+2 i d x^3} \, dx}{24 d}\\ &=\frac {1}{8} \left (2 a^2+b^2\right ) x^4-\frac {2 a b x \cos \left (c+d x^3\right )}{3 d}-\frac {a b e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{9 d \sqrt [3]{-i d x^3}}-\frac {a b e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{9 d \sqrt [3]{i d x^3}}+\frac {i b^2 e^{2 i c} x \Gamma \left (\frac {1}{3},-2 i d x^3\right )}{72 \sqrt [3]{2} d \sqrt [3]{-i d x^3}}-\frac {i b^2 e^{-2 i c} x \Gamma \left (\frac {1}{3},2 i d x^3\right )}{72 \sqrt [3]{2} d \sqrt [3]{i d x^3}}-\frac {b^2 x \sin \left (2 c+2 d x^3\right )}{12 d}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 339, normalized size = 1.43 \[ \frac {d x^7 \left (36 a^2 d x^3 \sqrt [3]{d^2 x^6}-96 a b \sqrt [3]{d^2 x^6} \cos \left (c+d x^3\right )-16 a b \sqrt [3]{-i d x^3} (\cos (c)-i \sin (c)) \Gamma \left (\frac {1}{3},i d x^3\right )-16 a b \sqrt [3]{i d x^3} (\cos (c)+i \sin (c)) \Gamma \left (\frac {1}{3},-i d x^3\right )-12 b^2 \sqrt [3]{d^2 x^6} \sin \left (2 \left (c+d x^3\right )\right )+i 2^{2/3} b^2 \cos (2 c) \sqrt [3]{i d x^3} \Gamma \left (\frac {1}{3},-2 i d x^3\right )-i 2^{2/3} b^2 \cos (2 c) \sqrt [3]{-i d x^3} \Gamma \left (\frac {1}{3},2 i d x^3\right )-2^{2/3} b^2 \sin (2 c) \sqrt [3]{i d x^3} \Gamma \left (\frac {1}{3},-2 i d x^3\right )-2^{2/3} b^2 \sin (2 c) \sqrt [3]{-i d x^3} \Gamma \left (\frac {1}{3},2 i d x^3\right )+18 b^2 d x^3 \sqrt [3]{d^2 x^6}\right )}{144 \left (d^2 x^6\right )^{4/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 146, normalized size = 0.62 \[ \frac {18 \, {\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{4} - 24 \, b^{2} d x \cos \left (d x^{3} + c\right ) \sin \left (d x^{3} + c\right ) - 96 \, a b d x \cos \left (d x^{3} + c\right ) - b^{2} \left (2 i \, d\right )^{\frac {2}{3}} e^{\left (-2 i \, c\right )} \Gamma \left (\frac {1}{3}, 2 i \, d x^{3}\right ) + 16 i \, a b \left (i \, d\right )^{\frac {2}{3}} e^{\left (-i \, c\right )} \Gamma \left (\frac {1}{3}, i \, d x^{3}\right ) - 16 i \, a b \left (-i \, d\right )^{\frac {2}{3}} e^{\left (i \, c\right )} \Gamma \left (\frac {1}{3}, -i \, d x^{3}\right ) - b^{2} \left (-2 i \, d\right )^{\frac {2}{3}} e^{\left (2 i \, c\right )} \Gamma \left (\frac {1}{3}, -2 i \, d x^{3}\right )}{144 \, d^{2}} \]
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 \[ \int {\left (b \sin \left (d x^{3} + c\right ) + a\right )}^{2} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 240, normalized size = 1.01 \[ \frac {1}{4} \, a^{2} x^{4} - \frac {2^{\frac {2}{3}} {\left ({\left ({\left ({\left (i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {1}{3}, 2 i \, d x^{3}\right ) + {\left (-i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {1}{3}, -2 i \, d x^{3}\right )\right )} \cos \left (2 \, c\right ) + {\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, 2 i \, d x^{3}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -2 i \, d x^{3}\right )\right )} \sin \left (2 \, c\right )\right )} x - 6 \cdot 2^{\frac {1}{3}} {\left (3 \, d x^{4} - 2 \, x \sin \left (2 \, d x^{3} + 2 \, c\right )\right )} \left (d x^{3}\right )^{\frac {1}{3}}\right )} b^{2}}{288 \, \left (d x^{3}\right )^{\frac {1}{3}} d} - \frac {{\left (12 \, \left (d x^{3}\right )^{\frac {1}{3}} x \cos \left (d x^{3} + c\right ) + {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -i \, d x^{3}\right )\right )} \cos \relax (c) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, i \, d x^{3}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -i \, d x^{3}\right )\right )} \sin \relax (c)\right )} x\right )} a b}{18 \, \left (d x^{3}\right )^{\frac {1}{3}} d} \]
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 \[ \int x^3\,{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2 \,d x \]
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 \[ \int x^{3} \left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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