Optimal. Leaf size=225 \[ -\frac{a b e^{i c} d x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{2 \sqrt [3]{-i d x^3}}-\frac{a b e^{-i c} d x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{2 \sqrt [3]{i d x^3}}+\frac{i b^2 e^{2 i c} d x \text{Gamma}\left (\frac{1}{3},-2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac{i b^2 e^{-2 i c} d x \text{Gamma}\left (\frac{1}{3},2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac{2 a^2+b^2}{4 x^2}-\frac{a b \sin \left (c+d x^3\right )}{x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2} \]
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Rubi [A] time = 0.133969, antiderivative size = 225, normalized size of antiderivative = 1., 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, 3388, 3355, 2208, 3387, 3356} \[ -\frac{a b e^{i c} d x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{2 \sqrt [3]{-i d x^3}}-\frac{a b e^{-i c} d x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{2 \sqrt [3]{i d x^3}}+\frac{i b^2 e^{2 i c} d x \text{Gamma}\left (\frac{1}{3},-2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac{i b^2 e^{-2 i c} d x \text{Gamma}\left (\frac{1}{3},2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac{2 a^2+b^2}{4 x^2}-\frac{a b \sin \left (c+d x^3\right )}{x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 3403
Rule 6
Rule 3388
Rule 3355
Rule 2208
Rule 3387
Rule 3356
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin \left (c+d x^3\right )\right )^2}{x^3} \, dx &=\int \left (\frac{a^2}{x^3}+\frac{b^2}{2 x^3}-\frac{b^2 \cos \left (2 c+2 d x^3\right )}{2 x^3}+\frac{2 a b \sin \left (c+d x^3\right )}{x^3}\right ) \, dx\\ &=\int \left (\frac{a^2+\frac{b^2}{2}}{x^3}-\frac{b^2 \cos \left (2 c+2 d x^3\right )}{2 x^3}+\frac{2 a b \sin \left (c+d x^3\right )}{x^3}\right ) \, dx\\ &=-\frac{2 a^2+b^2}{4 x^2}+(2 a b) \int \frac{\sin \left (c+d x^3\right )}{x^3} \, dx-\frac{1}{2} b^2 \int \frac{\cos \left (2 c+2 d x^3\right )}{x^3} \, dx\\ &=-\frac{2 a^2+b^2}{4 x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2}-\frac{a b \sin \left (c+d x^3\right )}{x^2}+(3 a b d) \int \cos \left (c+d x^3\right ) \, dx+\frac{1}{2} \left (3 b^2 d\right ) \int \sin \left (2 c+2 d x^3\right ) \, dx\\ &=-\frac{2 a^2+b^2}{4 x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2}-\frac{a b \sin \left (c+d x^3\right )}{x^2}+\frac{1}{2} (3 a b d) \int e^{-i c-i d x^3} \, dx+\frac{1}{2} (3 a b d) \int e^{i c+i d x^3} \, dx+\frac{1}{4} \left (3 i b^2 d\right ) \int e^{-2 i c-2 i d x^3} \, dx-\frac{1}{4} \left (3 i b^2 d\right ) \int e^{2 i c+2 i d x^3} \, dx\\ &=-\frac{2 a^2+b^2}{4 x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2}-\frac{a b d e^{i c} x \Gamma \left (\frac{1}{3},-i d x^3\right )}{2 \sqrt [3]{-i d x^3}}-\frac{a b d e^{-i c} x \Gamma \left (\frac{1}{3},i d x^3\right )}{2 \sqrt [3]{i d x^3}}+\frac{i b^2 d e^{2 i c} x \Gamma \left (\frac{1}{3},-2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac{i b^2 d e^{-2 i c} x \Gamma \left (\frac{1}{3},2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac{a b \sin \left (c+d x^3\right )}{x^2}\\ \end{align*}
Mathematica [A] time = 0.562294, size = 332, normalized size = 1.48 \[ \frac{-4 i a b \left (-i d x^3\right )^{4/3} (\cos (c)-i \sin (c)) \text{Gamma}\left (\frac{1}{3},i d x^3\right )+4 i a b \left (i d x^3\right )^{4/3} (\cos (c)+i \sin (c)) \text{Gamma}\left (\frac{1}{3},-i d x^3\right )+2^{2/3} b^2 \cos (2 c) \left (i d x^3\right )^{4/3} \text{Gamma}\left (\frac{1}{3},-2 i d x^3\right )+2^{2/3} b^2 \cos (2 c) \left (-i d x^3\right )^{4/3} \text{Gamma}\left (\frac{1}{3},2 i d x^3\right )+i 2^{2/3} b^2 \sin (2 c) \left (i d x^3\right )^{4/3} \text{Gamma}\left (\frac{1}{3},-2 i d x^3\right )-i 2^{2/3} b^2 \sin (2 c) \left (-i d x^3\right )^{4/3} \text{Gamma}\left (\frac{1}{3},2 i d x^3\right )-4 a^2 \sqrt [3]{d^2 x^6}-8 a b \sqrt [3]{d^2 x^6} \sin \left (c+d x^3\right )+2 b^2 \sqrt [3]{d^2 x^6} \cos \left (2 \left (c+d x^3\right )\right )-2 b^2 \sqrt [3]{d^2 x^6}}{8 x^2 \sqrt [3]{d^2 x^6}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.208, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\sin \left ( d{x}^{3}+c \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.27861, size = 743, normalized size = 3.3 \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 = 1.92269, size = 405, normalized size = 1.8 \begin{align*} -\frac{b^{2} \left (2 i \, d\right )^{\frac{2}{3}} x^{2} e^{\left (-2 i \, c\right )} \Gamma \left (\frac{1}{3}, 2 i \, d x^{3}\right ) - 4 i \, a b \left (i \, d\right )^{\frac{2}{3}} x^{2} e^{\left (-i \, c\right )} \Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + 4 i \, a b \left (-i \, d\right )^{\frac{2}{3}} x^{2} 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}} x^{2} e^{\left (2 i \, c\right )} \Gamma \left (\frac{1}{3}, -2 i \, d x^{3}\right ) - 4 \, b^{2} \cos \left (d x^{3} + c\right )^{2} + 8 \, a b \sin \left (d x^{3} + c\right ) + 4 \, a^{2} + 4 \, b^{2}}{8 \, x^{2}} \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 \frac{\left (a + b \sin{\left (c + d x^{3} \right )}\right )^{2}}{x^{3}}\, 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 \frac{{\left (b \sin \left (d x^{3} + c\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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