Optimal. Leaf size=101 \[ -\frac {a}{x}-\frac {b \sin \left (c+d x^3\right )}{x}-\frac {b e^{i c} d x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{2 \left (-i d x^3\right )^{2/3}}-\frac {b e^{-i c} d x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{2 \left (i d x^3\right )^{2/3}} \]
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Rubi [A] time = 0.08, 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, 3387, 3390, 2218} \[ -\frac {b e^{i c} d x^2 \text {Gamma}\left (\frac {2}{3},-i d x^3\right )}{2 \left (-i d x^3\right )^{2/3}}-\frac {b e^{-i c} d x^2 \text {Gamma}\left (\frac {2}{3},i d x^3\right )}{2 \left (i d x^3\right )^{2/3}}-\frac {a}{x}-\frac {b \sin \left (c+d x^3\right )}{x} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2218
Rule 3387
Rule 3390
Rubi steps
\begin {align*} \int \frac {a+b \sin \left (c+d x^3\right )}{x^2} \, dx &=\int \left (\frac {a}{x^2}+\frac {b \sin \left (c+d x^3\right )}{x^2}\right ) \, dx\\ &=-\frac {a}{x}+b \int \frac {\sin \left (c+d x^3\right )}{x^2} \, dx\\ &=-\frac {a}{x}-\frac {b \sin \left (c+d x^3\right )}{x}+(3 b d) \int x \cos \left (c+d x^3\right ) \, dx\\ &=-\frac {a}{x}-\frac {b \sin \left (c+d x^3\right )}{x}+\frac {1}{2} (3 b d) \int e^{-i c-i d x^3} x \, dx+\frac {1}{2} (3 b d) \int e^{i c+i d x^3} x \, dx\\ &=-\frac {a}{x}-\frac {b d e^{i c} x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{2 \left (-i d x^3\right )^{2/3}}-\frac {b d e^{-i c} x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{2 \left (i d x^3\right )^{2/3}}-\frac {b \sin \left (c+d x^3\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 120, normalized size = 1.19 \[ \frac {-2 \left (d^2 x^6\right )^{2/3} \left (a+b \sin \left (c+d x^3\right )\right )-i b \left (-i d x^3\right )^{5/3} (\cos (c)-i \sin (c)) \Gamma \left (\frac {2}{3},i d x^3\right )+i b \left (i d x^3\right )^{5/3} (\cos (c)+i \sin (c)) \Gamma \left (\frac {2}{3},-i d x^3\right )}{2 x \left (d^2 x^6\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 62, normalized size = 0.61 \[ \frac {i \, b \left (i \, d\right )^{\frac {1}{3}} x e^{\left (-i \, c\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) - i \, b \left (-i \, d\right )^{\frac {1}{3}} x e^{\left (i \, c\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right ) - 2 \, b \sin \left (d x^{3} + c\right ) - 2 \, a}{2 \, x} \]
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 \[ \int \frac {b \sin \left (d x^{3} + c\right ) + a}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {a +b \sin \left (d \,x^{3}+c \right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 89, normalized size = 0.88 \[ -\frac {\left (d x^{3}\right )^{\frac {1}{3}} {\left ({\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {1}{3}, i \, d x^{3}\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {1}{3}, -i \, d x^{3}\right )\right )} \cos \relax (c) + {\left ({\left (\sqrt {3} + i\right )} \Gamma \left (-\frac {1}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (-\frac {1}{3}, -i \, d x^{3}\right )\right )} \sin \relax (c)\right )} b}{12 \, x} - \frac {a}{x} \]
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 \[ \int \frac {a+b\,\sin \left (d\,x^3+c\right )}{x^2} \,d x \]
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 \[ \int \frac {a + b \sin {\left (c + d x^{3} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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