Optimal. Leaf size=249 \[ \frac {1}{10} x^5 \left (2 a^2+b^2\right )-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {2 a b e^{i c} x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac {2 a b e^{-i c} x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d}+\frac {i b^2 e^{2 i c} x^2 \Gamma \left (\frac {2}{3},-2 i d x^3\right )}{36\ 2^{2/3} d \left (-i d x^3\right )^{2/3}}-\frac {i b^2 e^{-2 i c} x^2 \Gamma \left (\frac {2}{3},2 i d x^3\right )}{36\ 2^{2/3} d \left (i d x^3\right )^{2/3}} \]
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Rubi [A] time = 0.21, antiderivative size = 249, 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, 3389, 2218, 3385, 3390} \[ -\frac {2 a b e^{i c} x^2 \text {Gamma}\left (\frac {2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac {2 a b e^{-i c} x^2 \text {Gamma}\left (\frac {2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}+\frac {i b^2 e^{2 i c} x^2 \text {Gamma}\left (\frac {2}{3},-2 i d x^3\right )}{36\ 2^{2/3} d \left (-i d x^3\right )^{2/3}}-\frac {i b^2 e^{-2 i c} x^2 \text {Gamma}\left (\frac {2}{3},2 i d x^3\right )}{36\ 2^{2/3} d \left (i d x^3\right )^{2/3}}+\frac {1}{10} x^5 \left (2 a^2+b^2\right )-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d} \]
Antiderivative was successfully verified.
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Rule 6
Rule 2218
Rule 3385
Rule 3386
Rule 3389
Rule 3390
Rule 3403
Rubi steps
\begin {align*} \int x^4 \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\int \left (a^2 x^4+\frac {b^2 x^4}{2}-\frac {1}{2} b^2 x^4 \cos \left (2 c+2 d x^3\right )+2 a b x^4 \sin \left (c+d x^3\right )\right ) \, dx\\ &=\int \left (\left (a^2+\frac {b^2}{2}\right ) x^4-\frac {1}{2} b^2 x^4 \cos \left (2 c+2 d x^3\right )+2 a b x^4 \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac {1}{10} \left (2 a^2+b^2\right ) x^5+(2 a b) \int x^4 \sin \left (c+d x^3\right ) \, dx-\frac {1}{2} b^2 \int x^4 \cos \left (2 c+2 d x^3\right ) \, dx\\ &=\frac {1}{10} \left (2 a^2+b^2\right ) x^5-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d}+\frac {(4 a b) \int x \cos \left (c+d x^3\right ) \, dx}{3 d}+\frac {b^2 \int x \sin \left (2 c+2 d x^3\right ) \, dx}{6 d}\\ &=\frac {1}{10} \left (2 a^2+b^2\right ) x^5-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d}+\frac {(2 a b) \int e^{-i c-i d x^3} x \, dx}{3 d}+\frac {(2 a b) \int e^{i c+i d x^3} x \, dx}{3 d}+\frac {\left (i b^2\right ) \int e^{-2 i c-2 i d x^3} x \, dx}{12 d}-\frac {\left (i b^2\right ) \int e^{2 i c+2 i d x^3} x \, dx}{12 d}\\ &=\frac {1}{10} \left (2 a^2+b^2\right ) x^5-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {2 a b e^{i c} x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac {2 a b e^{-i c} x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}+\frac {i b^2 e^{2 i c} x^2 \Gamma \left (\frac {2}{3},-2 i d x^3\right )}{36\ 2^{2/3} d \left (-i d x^3\right )^{2/3}}-\frac {i b^2 e^{-2 i c} x^2 \Gamma \left (\frac {2}{3},2 i d x^3\right )}{36\ 2^{2/3} d \left (i d x^3\right )^{2/3}}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 339, normalized size = 1.36 \[ \frac {d x^8 \left (72 a^2 d x^3 \left (d^2 x^6\right )^{2/3}-240 a b \left (d^2 x^6\right )^{2/3} \cos \left (c+d x^3\right )-80 a b \left (-i d x^3\right )^{2/3} (\cos (c)-i \sin (c)) \Gamma \left (\frac {2}{3},i d x^3\right )-80 a b \left (i d x^3\right )^{2/3} (\cos (c)+i \sin (c)) \Gamma \left (\frac {2}{3},-i d x^3\right )-30 b^2 \left (d^2 x^6\right )^{2/3} \sin \left (2 \left (c+d x^3\right )\right )+5 i \sqrt [3]{2} b^2 \cos (2 c) \left (i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},-2 i d x^3\right )-5 i \sqrt [3]{2} b^2 \cos (2 c) \left (-i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},2 i d x^3\right )-5 \sqrt [3]{2} b^2 \sin (2 c) \left (i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},-2 i d x^3\right )-5 \sqrt [3]{2} b^2 \sin (2 c) \left (-i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},2 i d x^3\right )+36 b^2 d x^3 \left (d^2 x^6\right )^{2/3}\right )}{360 \left (d^2 x^6\right )^{5/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 150, normalized size = 0.60 \[ \frac {36 \, {\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{5} - 60 \, b^{2} d x^{2} \cos \left (d x^{3} + c\right ) \sin \left (d x^{3} + c\right ) - 240 \, a b d x^{2} \cos \left (d x^{3} + c\right ) - 5 \, b^{2} \left (2 i \, d\right )^{\frac {1}{3}} e^{\left (-2 i \, c\right )} \Gamma \left (\frac {2}{3}, 2 i \, d x^{3}\right ) + 80 i \, a b \left (i \, d\right )^{\frac {1}{3}} e^{\left (-i \, c\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) - 80 i \, a b \left (-i \, d\right )^{\frac {1}{3}} e^{\left (i \, c\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right ) - 5 \, b^{2} \left (-2 i \, d\right )^{\frac {1}{3}} e^{\left (2 i \, c\right )} \Gamma \left (\frac {2}{3}, -2 i \, d x^{3}\right )}{360 \, d^{2}} \]
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 \[ \int {\left (b \sin \left (d x^{3} + c\right ) + a\right )}^{2} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int x^{4} \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 239, normalized size = 0.96 \[ \frac {1}{5} \, a^{2} x^{5} - \frac {{\left (6 \, d x^{3} \cos \left (d x^{3} + c\right ) - \left (d x^{3}\right )^{\frac {1}{3}} {\left ({\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right )\right )} \cos \relax (c) + {\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right )\right )} \sin \relax (c)\right )}\right )} a b}{9 \, d^{2} x} + \frac {{\left (72 \, d^{2} x^{6} - 60 \, d x^{3} \sin \left (2 \, d x^{3} + 2 \, c\right ) - 2^{\frac {1}{3}} \left (d x^{3}\right )^{\frac {1}{3}} {\left ({\left ({\left (5 \, \sqrt {3} + 5 i\right )} \Gamma \left (\frac {2}{3}, 2 i \, d x^{3}\right ) + {\left (5 \, \sqrt {3} - 5 i\right )} \Gamma \left (\frac {2}{3}, -2 i \, d x^{3}\right )\right )} \cos \left (2 \, c\right ) + 5 \, {\left ({\left (-i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, 2 i \, d x^{3}\right ) + {\left (i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, -2 i \, d x^{3}\right )\right )} \sin \left (2 \, c\right )\right )}\right )} b^{2}}{720 \, d^{2} x} \]
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 \[ \int x^4\,{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2 \,d x \]
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 \[ \int x^{4} \left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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