Optimal. Leaf size=275 \[ -\frac{3 i a b e^{i c} d^2 x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{10 \sqrt [3]{-i d x^3}}+\frac{3 i a b e^{-i c} d^2 x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{10 \sqrt [3]{i d x^3}}-\frac{3 b^2 e^{2 i c} d^2 x \text{Gamma}\left (\frac{1}{3},-2 i d x^3\right )}{10 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac{3 b^2 e^{-2 i c} d^2 x \text{Gamma}\left (\frac{1}{3},2 i d x^3\right )}{10 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac{2 a^2+b^2}{10 x^5}-\frac{2 a b \sin \left (c+d x^3\right )}{5 x^5}-\frac{3 a b d \cos \left (c+d x^3\right )}{5 x^2}-\frac{3 b^2 d \sin \left (2 c+2 d x^3\right )}{10 x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{10 x^5} \]
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Rubi [A] time = 0.182991, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {3403, 6, 3388, 3387, 3356, 2208, 3355} \[ -\frac{3 i a b e^{i c} d^2 x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{10 \sqrt [3]{-i d x^3}}+\frac{3 i a b e^{-i c} d^2 x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{10 \sqrt [3]{i d x^3}}-\frac{3 b^2 e^{2 i c} d^2 x \text{Gamma}\left (\frac{1}{3},-2 i d x^3\right )}{10 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac{3 b^2 e^{-2 i c} d^2 x \text{Gamma}\left (\frac{1}{3},2 i d x^3\right )}{10 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac{2 a^2+b^2}{10 x^5}-\frac{2 a b \sin \left (c+d x^3\right )}{5 x^5}-\frac{3 a b d \cos \left (c+d x^3\right )}{5 x^2}-\frac{3 b^2 d \sin \left (2 c+2 d x^3\right )}{10 x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{10 x^5} \]
Antiderivative was successfully verified.
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Rule 3403
Rule 6
Rule 3388
Rule 3387
Rule 3356
Rule 2208
Rule 3355
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin \left (c+d x^3\right )\right )^2}{x^6} \, dx &=\int \left (\frac{a^2}{x^6}+\frac{b^2}{2 x^6}-\frac{b^2 \cos \left (2 c+2 d x^3\right )}{2 x^6}+\frac{2 a b \sin \left (c+d x^3\right )}{x^6}\right ) \, dx\\ &=\int \left (\frac{a^2+\frac{b^2}{2}}{x^6}-\frac{b^2 \cos \left (2 c+2 d x^3\right )}{2 x^6}+\frac{2 a b \sin \left (c+d x^3\right )}{x^6}\right ) \, dx\\ &=-\frac{2 a^2+b^2}{10 x^5}+(2 a b) \int \frac{\sin \left (c+d x^3\right )}{x^6} \, dx-\frac{1}{2} b^2 \int \frac{\cos \left (2 c+2 d x^3\right )}{x^6} \, dx\\ &=-\frac{2 a^2+b^2}{10 x^5}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{10 x^5}-\frac{2 a b \sin \left (c+d x^3\right )}{5 x^5}+\frac{1}{5} (6 a b d) \int \frac{\cos \left (c+d x^3\right )}{x^3} \, dx+\frac{1}{5} \left (3 b^2 d\right ) \int \frac{\sin \left (2 c+2 d x^3\right )}{x^3} \, dx\\ &=-\frac{2 a^2+b^2}{10 x^5}-\frac{3 a b d \cos \left (c+d x^3\right )}{5 x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{10 x^5}-\frac{2 a b \sin \left (c+d x^3\right )}{5 x^5}-\frac{3 b^2 d \sin \left (2 c+2 d x^3\right )}{10 x^2}-\frac{1}{5} \left (9 a b d^2\right ) \int \sin \left (c+d x^3\right ) \, dx+\frac{1}{5} \left (9 b^2 d^2\right ) \int \cos \left (2 c+2 d x^3\right ) \, dx\\ &=-\frac{2 a^2+b^2}{10 x^5}-\frac{3 a b d \cos \left (c+d x^3\right )}{5 x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{10 x^5}-\frac{2 a b \sin \left (c+d x^3\right )}{5 x^5}-\frac{3 b^2 d \sin \left (2 c+2 d x^3\right )}{10 x^2}-\frac{1}{10} \left (9 i a b d^2\right ) \int e^{-i c-i d x^3} \, dx+\frac{1}{10} \left (9 i a b d^2\right ) \int e^{i c+i d x^3} \, dx+\frac{1}{10} \left (9 b^2 d^2\right ) \int e^{-2 i c-2 i d x^3} \, dx+\frac{1}{10} \left (9 b^2 d^2\right ) \int e^{2 i c+2 i d x^3} \, dx\\ &=-\frac{2 a^2+b^2}{10 x^5}-\frac{3 a b d \cos \left (c+d x^3\right )}{5 x^2}+\frac{b^2 \cos \left (2 c+2 d x^3\right )}{10 x^5}-\frac{3 i a b d^2 e^{i c} x \Gamma \left (\frac{1}{3},-i d x^3\right )}{10 \sqrt [3]{-i d x^3}}+\frac{3 i a b d^2 e^{-i c} x \Gamma \left (\frac{1}{3},i d x^3\right )}{10 \sqrt [3]{i d x^3}}-\frac{3 b^2 d^2 e^{2 i c} x \Gamma \left (\frac{1}{3},-2 i d x^3\right )}{10 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac{3 b^2 d^2 e^{-2 i c} x \Gamma \left (\frac{1}{3},2 i d x^3\right )}{10 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac{2 a b \sin \left (c+d x^3\right )}{5 x^5}-\frac{3 b^2 d \sin \left (2 c+2 d x^3\right )}{10 x^2}\\ \end{align*}
Mathematica [A] time = 2.5157, size = 294, normalized size = 1.07 \[ -\frac{6 i a b \sqrt [3]{i d x^3} \left (d^2 x^6\right )^{2/3} (\cos (c)+i \sin (c)) \text{Gamma}\left (\frac{1}{3},-i d x^3\right )+6 i a b \left (i d x^3\right )^{5/3} (\cos (c)-i \sin (c)) \text{Gamma}\left (\frac{1}{3},i d x^3\right )-3\ 2^{2/3} b^2 \cos (2 c) \left (i d x^3\right )^{5/3} \text{Gamma}\left (\frac{1}{3},2 i d x^3\right )+3 i 2^{2/3} b^2 \sin (2 c) \left (i d x^3\right )^{5/3} \text{Gamma}\left (\frac{1}{3},2 i d x^3\right )-3\ 2^{2/3} b^2 \left (-i d x^3\right )^{5/3} (\cos (2 c)+i \sin (2 c)) \text{Gamma}\left (\frac{1}{3},-2 i d x^3\right )+4 a^2+8 a b \sin \left (c+d x^3\right )+12 a b d x^3 \cos \left (c+d x^3\right )+6 b^2 d x^3 \sin \left (2 \left (c+d x^3\right )\right )-2 b^2 \cos \left (2 \left (c+d x^3\right )\right )+2 b^2}{20 x^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.22, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\sin \left ( d{x}^{3}+c \right ) \right ) ^{2}}{{x}^{6}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.27423, size = 756, normalized size = 2.75 \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 = 1.91319, size = 506, normalized size = 1.84 \begin{align*} \frac{3 i \, b^{2} \left (2 i \, d\right )^{\frac{2}{3}} d x^{5} e^{\left (-2 i \, c\right )} \Gamma \left (\frac{1}{3}, 2 i \, d x^{3}\right ) + 6 \, a b \left (i \, d\right )^{\frac{2}{3}} d x^{5} e^{\left (-i \, c\right )} \Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + 6 \, a b \left (-i \, d\right )^{\frac{2}{3}} d x^{5} e^{\left (i \, c\right )} \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right ) - 3 i \, b^{2} \left (-2 i \, d\right )^{\frac{2}{3}} d x^{5} e^{\left (2 i \, c\right )} \Gamma \left (\frac{1}{3}, -2 i \, d x^{3}\right ) - 12 \, a b d x^{3} \cos \left (d x^{3} + c\right ) + 4 \, b^{2} \cos \left (d x^{3} + c\right )^{2} - 4 \, a^{2} - 4 \, b^{2} - 4 \,{\left (3 \, b^{2} d x^{3} \cos \left (d x^{3} + c\right ) + 2 \, a b\right )} \sin \left (d x^{3} + c\right )}{20 \, x^{5}} \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^{6}}\, 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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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