Optimal. Leaf size=183 \[ \frac {1}{2} \left (2 a^2+b^2\right ) x+\frac {i a b e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{3 \sqrt [3]{-i d x^3}}-\frac {i a b e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{3 \sqrt [3]{i d x^3}}+\frac {b^2 e^{2 i c} x \Gamma \left (\frac {1}{3},-2 i d x^3\right )}{12 \sqrt [3]{2} \sqrt [3]{-i d x^3}}+\frac {b^2 e^{-2 i c} x \Gamma \left (\frac {1}{3},2 i d x^3\right )}{12 \sqrt [3]{2} \sqrt [3]{i d x^3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3438, 3437,
2239, 3436} \begin {gather*} \frac {i a b e^{i c} x \text {Gamma}\left (\frac {1}{3},-i d x^3\right )}{3 \sqrt [3]{-i d x^3}}-\frac {i a b e^{-i c} x \text {Gamma}\left (\frac {1}{3},i d x^3\right )}{3 \sqrt [3]{i d x^3}}+\frac {b^2 e^{2 i c} x \text {Gamma}\left (\frac {1}{3},-2 i d x^3\right )}{12 \sqrt [3]{2} \sqrt [3]{-i d x^3}}+\frac {b^2 e^{-2 i c} x \text {Gamma}\left (\frac {1}{3},2 i d x^3\right )}{12 \sqrt [3]{2} \sqrt [3]{i d x^3}}+\frac {1}{2} x \left (2 a^2+b^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 3436
Rule 3437
Rule 3438
Rubi steps
\begin {align*} \int \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\int \left (a^2+\frac {b^2}{2}-\frac {1}{2} b^2 \cos \left (2 c+2 d x^3\right )+2 a b \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac {1}{2} \left (2 a^2+b^2\right ) x+(2 a b) \int \sin \left (c+d x^3\right ) \, dx-\frac {1}{2} b^2 \int \cos \left (2 c+2 d x^3\right ) \, dx\\ &=\frac {1}{2} \left (2 a^2+b^2\right ) x+(i a b) \int e^{-i c-i d x^3} \, dx-(i a b) \int e^{i c+i d x^3} \, dx-\frac {1}{4} b^2 \int e^{-2 i c-2 i d x^3} \, dx-\frac {1}{4} b^2 \int e^{2 i c+2 i d x^3} \, dx\\ &=\frac {1}{2} \left (2 a^2+b^2\right ) x+\frac {i a b e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{3 \sqrt [3]{-i d x^3}}-\frac {i a b e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{3 \sqrt [3]{i d x^3}}+\frac {b^2 e^{2 i c} x \Gamma \left (\frac {1}{3},-2 i d x^3\right )}{12 \sqrt [3]{2} \sqrt [3]{-i d x^3}}+\frac {b^2 e^{-2 i c} x \Gamma \left (\frac {1}{3},2 i d x^3\right )}{12 \sqrt [3]{2} \sqrt [3]{i d x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 281, normalized size = 1.54 \begin {gather*} \frac {x \left (24 a^2 \sqrt [3]{d^2 x^6}+12 b^2 \sqrt [3]{d^2 x^6}+2^{2/3} b^2 \sqrt [3]{i d x^3} \cos (2 c) \Gamma \left (\frac {1}{3},-2 i d x^3\right )+2^{2/3} b^2 \sqrt [3]{-i d x^3} \cos (2 c) \Gamma \left (\frac {1}{3},2 i d x^3\right )-8 i a b \sqrt [3]{-i d x^3} \Gamma \left (\frac {1}{3},i d x^3\right ) (\cos (c)-i \sin (c))+8 i a b \sqrt [3]{i d x^3} \Gamma \left (\frac {1}{3},-i d x^3\right ) (\cos (c)+i \sin (c))+i 2^{2/3} b^2 \sqrt [3]{i d x^3} \Gamma \left (\frac {1}{3},-2 i d x^3\right ) \sin (2 c)-i 2^{2/3} b^2 \sqrt [3]{-i d x^3} \Gamma \left (\frac {1}{3},2 i d x^3\right ) \sin (2 c)\right )}{24 \sqrt [3]{d^2 x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 192, normalized size = 1.05 \begin {gather*} \frac {{\left ({\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 )} \cos \left (c\right ) - {\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 )} \sin \left (c\right )\right )} a b x}{6 \, \left (d x^{3}\right )^{\frac {1}{3}}} + \frac {2^{\frac {2}{3}} {\left ({\left ({\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 )} \cos \left (2 \, c\right ) + {\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 )} \sin \left (2 \, c\right )\right )} x + 12 \cdot 2^{\frac {1}{3}} \left (d x^{3}\right )^{\frac {1}{3}} x\right )} b^{2}}{48 \, \left (d x^{3}\right )^{\frac {1}{3}}} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.12, size = 105, normalized size = 0.57 \begin {gather*} \frac {-i \, 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 ) - 8 \, a b \left (i \, d\right )^{\frac {2}{3}} e^{\left (-i \, c\right )} \Gamma \left (\frac {1}{3}, i \, d x^{3}\right ) - 8 \, a b \left (-i \, d\right )^{\frac {2}{3}} e^{\left (i \, c\right )} \Gamma \left (\frac {1}{3}, -i \, d x^{3}\right ) + i \, 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 ) + 12 \, {\left (2 \, a^{2} + b^{2}\right )} d x}{24 \, d} \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 (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 = -0.01 \begin {gather*} \int {\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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