Optimal. Leaf size=101 \[ -\frac {a}{2 x^2}-\frac {b d e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{4 \sqrt [3]{-i d x^3}}-\frac {b d e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{4 \sqrt [3]{i d x^3}}-\frac {b \sin \left (c+d x^3\right )}{2 x^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {14, 3468, 3437,
2239} \begin {gather*} -\frac {b e^{i c} d x \text {Gamma}\left (\frac {1}{3},-i d x^3\right )}{4 \sqrt [3]{-i d x^3}}-\frac {b e^{-i c} d x \text {Gamma}\left (\frac {1}{3},i d x^3\right )}{4 \sqrt [3]{i d x^3}}-\frac {a}{2 x^2}-\frac {b \sin \left (c+d x^3\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2239
Rule 3437
Rule 3468
Rubi steps
\begin {align*} \int \frac {a+b \sin \left (c+d x^3\right )}{x^3} \, dx &=\int \left (\frac {a}{x^3}+\frac {b \sin \left (c+d x^3\right )}{x^3}\right ) \, dx\\ &=-\frac {a}{2 x^2}+b \int \frac {\sin \left (c+d x^3\right )}{x^3} \, dx\\ &=-\frac {a}{2 x^2}-\frac {b \sin \left (c+d x^3\right )}{2 x^2}+\frac {1}{2} (3 b d) \int \cos \left (c+d x^3\right ) \, dx\\ &=-\frac {a}{2 x^2}-\frac {b \sin \left (c+d x^3\right )}{2 x^2}+\frac {1}{4} (3 b d) \int e^{-i c-i d x^3} \, dx+\frac {1}{4} (3 b d) \int e^{i c+i d x^3} \, dx\\ &=-\frac {a}{2 x^2}-\frac {b d e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{4 \sqrt [3]{-i d x^3}}-\frac {b d e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{4 \sqrt [3]{i d x^3}}-\frac {b \sin \left (c+d x^3\right )}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 120, normalized size = 1.19 \begin {gather*} \frac {-i b \left (-i d x^3\right )^{4/3} \Gamma \left (\frac {1}{3},i d x^3\right ) (\cos (c)-i \sin (c))+i b \left (i d x^3\right )^{4/3} \Gamma \left (\frac {1}{3},-i d x^3\right ) (\cos (c)+i \sin (c))-2 \sqrt [3]{d^2 x^6} \left (a+b \sin \left (c+d x^3\right )\right )}{4 x^2 \sqrt [3]{d^2 x^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \sin \left (d \,x^{3}+c \right )}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 90, normalized size = 0.89 \begin {gather*} \frac {\left (d x^{3}\right )^{\frac {2}{3}} {\left ({\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 )} \cos \left (c\right ) - {\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 )} \sin \left (c\right )\right )} b}{12 \, x^{2}} - \frac {a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.11, size = 66, normalized size = 0.65 \begin {gather*} \frac {i \, 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 ) - i \, 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 ) - 2 \, b \sin \left (d x^{3} + c\right ) - 2 \, a}{4 \, x^{2}} \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 \frac {a + b \sin {\left (c + d x^{3} \right )}}{x^{3}}\, 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 \frac {a+b\,\sin \left (d\,x^3+c\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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