Optimal. Leaf size=130 \[ -\frac {a}{4 x^4}-\frac {3 i b e^{i c} d^2 x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{8 \left (-i d x^3\right )^{2/3}}+\frac {3 i b e^{-i c} d^2 x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{8 \left (i d x^3\right )^{2/3}}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {b \sin \left (c+d x^3\right )}{4 x^4} \]
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Rubi [A] time = 0.10, antiderivative size = 130, 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, 3389, 2218} \[ -\frac {3 i b e^{i c} d^2 x^2 \text {Gamma}\left (\frac {2}{3},-i d x^3\right )}{8 \left (-i d x^3\right )^{2/3}}+\frac {3 i b e^{-i c} d^2 x^2 \text {Gamma}\left (\frac {2}{3},i d x^3\right )}{8 \left (i d x^3\right )^{2/3}}-\frac {a}{4 x^4}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2218
Rule 3387
Rule 3388
Rule 3389
Rubi steps
\begin {align*} \int \frac {a+b \sin \left (c+d x^3\right )}{x^5} \, dx &=\int \left (\frac {a}{x^5}+\frac {b \sin \left (c+d x^3\right )}{x^5}\right ) \, dx\\ &=-\frac {a}{4 x^4}+b \int \frac {\sin \left (c+d x^3\right )}{x^5} \, dx\\ &=-\frac {a}{4 x^4}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}+\frac {1}{4} (3 b d) \int \frac {\cos \left (c+d x^3\right )}{x^2} \, dx\\ &=-\frac {a}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}-\frac {1}{4} \left (9 b d^2\right ) \int x \sin \left (c+d x^3\right ) \, dx\\ &=-\frac {a}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}-\frac {1}{8} \left (9 i b d^2\right ) \int e^{-i c-i d x^3} x \, dx+\frac {1}{8} \left (9 i b d^2\right ) \int e^{i c+i d x^3} x \, dx\\ &=-\frac {a}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {3 i b d^2 e^{i c} x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{8 \left (-i d x^3\right )^{2/3}}+\frac {3 i b d^2 e^{-i c} x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{8 \left (i d x^3\right )^{2/3}}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 143, normalized size = 1.10 \[ \frac {-2 \left (d^2 x^6\right )^{2/3} \left (a+b \sin \left (c+d x^3\right )+3 b d x^3 \cos \left (c+d x^3\right )\right )+3 b d^2 x^6 \left (i d x^3\right )^{2/3} (\sin (c)-i \cos (c)) \Gamma \left (\frac {2}{3},-i d x^3\right )+3 b d^2 x^6 \left (-i d x^3\right )^{2/3} (\sin (c)+i \cos (c)) \Gamma \left (\frac {2}{3},i d x^3\right )}{8 x^4 \left (d^2 x^6\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 83, normalized size = 0.64 \[ \frac {3 \, b \left (i \, d\right )^{\frac {1}{3}} d x^{4} e^{\left (-i \, c\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) + 3 \, b \left (-i \, d\right )^{\frac {1}{3}} d x^{4} e^{\left (i \, c\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right ) - 6 \, b d x^{3} \cos \left (d x^{3} + c\right ) - 2 \, b \sin \left (d x^{3} + c\right ) - 2 \, a}{8 \, x^{4}} \]
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 \[ \int \frac {b \sin \left (d x^{3} + c\right ) + a}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {a +b \sin \left (d \,x^{3}+c \right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 91, normalized size = 0.70 \[ \frac {\left (d x^{3}\right )^{\frac {1}{3}} {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (-\frac {4}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (-\frac {4}{3}, -i \, d x^{3}\right )\right )} \cos \relax (c) - {\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {4}{3}, i \, d x^{3}\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {4}{3}, -i \, d x^{3}\right )\right )} \sin \relax (c)\right )} b d}{12 \, x} - \frac {a}{4 \, x^{4}} \]
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 \[ \int \frac {a+b\,\sin \left (d\,x^3+c\right )}{x^5} \,d x \]
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 \[ \int \frac {a + b \sin {\left (c + d x^{3} \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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