Optimal. Leaf size=126 \[ -\frac {a}{5 x^5}-\frac {3 i b e^{i c} d^2 x \Gamma \left (\frac {1}{3},-i d x^3\right )}{20 \sqrt [3]{-i d x^3}}+\frac {3 i b e^{-i c} d^2 x \Gamma \left (\frac {1}{3},i d x^3\right )}{20 \sqrt [3]{i d x^3}}-\frac {b \sin \left (c+d x^3\right )}{5 x^5}-\frac {3 b d \cos \left (c+d x^3\right )}{10 x^2} \]
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Rubi [A] time = 0.07, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {14, 3387, 3388, 3355, 2208} \[ -\frac {3 i b e^{i c} d^2 x \text {Gamma}\left (\frac {1}{3},-i d x^3\right )}{20 \sqrt [3]{-i d x^3}}+\frac {3 i b e^{-i c} d^2 x \text {Gamma}\left (\frac {1}{3},i d x^3\right )}{20 \sqrt [3]{i d x^3}}-\frac {a}{5 x^5}-\frac {b \sin \left (c+d x^3\right )}{5 x^5}-\frac {3 b d \cos \left (c+d x^3\right )}{10 x^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2208
Rule 3355
Rule 3387
Rule 3388
Rubi steps
\begin {align*} \int \frac {a+b \sin \left (c+d x^3\right )}{x^6} \, dx &=\int \left (\frac {a}{x^6}+\frac {b \sin \left (c+d x^3\right )}{x^6}\right ) \, dx\\ &=-\frac {a}{5 x^5}+b \int \frac {\sin \left (c+d x^3\right )}{x^6} \, dx\\ &=-\frac {a}{5 x^5}-\frac {b \sin \left (c+d x^3\right )}{5 x^5}+\frac {1}{5} (3 b d) \int \frac {\cos \left (c+d x^3\right )}{x^3} \, dx\\ &=-\frac {a}{5 x^5}-\frac {3 b d \cos \left (c+d x^3\right )}{10 x^2}-\frac {b \sin \left (c+d x^3\right )}{5 x^5}-\frac {1}{10} \left (9 b d^2\right ) \int \sin \left (c+d x^3\right ) \, dx\\ &=-\frac {a}{5 x^5}-\frac {3 b d \cos \left (c+d x^3\right )}{10 x^2}-\frac {b \sin \left (c+d x^3\right )}{5 x^5}-\frac {1}{20} \left (9 i b d^2\right ) \int e^{-i c-i d x^3} \, dx+\frac {1}{20} \left (9 i b d^2\right ) \int e^{i c+i d x^3} \, dx\\ &=-\frac {a}{5 x^5}-\frac {3 b d \cos \left (c+d x^3\right )}{10 x^2}-\frac {3 i b d^2 e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{20 \sqrt [3]{-i d x^3}}+\frac {3 i b d^2 e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{20 \sqrt [3]{i d x^3}}-\frac {b \sin \left (c+d x^3\right )}{5 x^5}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 146, normalized size = 1.16 \[ \frac {-2 \sqrt [3]{d^2 x^6} \left (2 a+2 b \sin \left (c+d x^3\right )+3 b d x^3 \cos \left (c+d x^3\right )\right )+3 b d^2 x^6 \sqrt [3]{i d x^3} (\sin (c)-i \cos (c)) \Gamma \left (\frac {1}{3},-i d x^3\right )+3 b d^2 x^6 \sqrt [3]{-i d x^3} (\sin (c)+i \cos (c)) \Gamma \left (\frac {1}{3},i d x^3\right )}{20 x^5 \sqrt [3]{d^2 x^6}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 83, normalized size = 0.66 \[ \frac {3 \, b \left (i \, d\right )^{\frac {2}{3}} d x^{5} e^{\left (-i \, c\right )} \Gamma \left (\frac {1}{3}, i \, d x^{3}\right ) + 3 \, b \left (-i \, d\right )^{\frac {2}{3}} d x^{5} e^{\left (i \, c\right )} \Gamma \left (\frac {1}{3}, -i \, d x^{3}\right ) - 6 \, b d x^{3} \cos \left (d x^{3} + c\right ) - 4 \, b \sin \left (d x^{3} + c\right ) - 4 \, a}{20 \, x^{5}} \]
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 \[ \int \frac {b \sin \left (d x^{3} + c\right ) + a}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {a +b \sin \left (d \,x^{3}+c \right )}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 91, normalized size = 0.72 \[ -\frac {\left (d x^{3}\right )^{\frac {2}{3}} {\left ({\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {5}{3}, i \, d x^{3}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {5}{3}, -i \, d x^{3}\right )\right )} \cos \relax (c) - {\left ({\left (\sqrt {3} - i\right )} \Gamma \left (-\frac {5}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (-\frac {5}{3}, -i \, d x^{3}\right )\right )} \sin \relax (c)\right )} b d}{12 \, x^{2}} - \frac {a}{5 \, x^{5}} \]
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 \[ \int \frac {a+b\,\sin \left (d\,x^3+c\right )}{x^6} \,d x \]
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 \[ \int \frac {a + b \sin {\left (c + d x^{3} \right )}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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