Optimal. Leaf size=31 \[ a \log (x)+\frac {1}{3} b \sin (c) \text {Ci}\left (d x^3\right )+\frac {1}{3} b \cos (c) \text {Si}\left (d x^3\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {14, 3377, 3376, 3375} \[ a \log (x)+\frac {1}{3} b \sin (c) \text {CosIntegral}\left (d x^3\right )+\frac {1}{3} b \cos (c) \text {Si}\left (d x^3\right ) \]
Antiderivative was successfully verified.
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Rule 14
Rule 3375
Rule 3376
Rule 3377
Rubi steps
\begin {align*} \int \frac {a+b \sin \left (c+d x^3\right )}{x} \, dx &=\int \left (\frac {a}{x}+\frac {b \sin \left (c+d x^3\right )}{x}\right ) \, dx\\ &=a \log (x)+b \int \frac {\sin \left (c+d x^3\right )}{x} \, dx\\ &=a \log (x)+(b \cos (c)) \int \frac {\sin \left (d x^3\right )}{x} \, dx+(b \sin (c)) \int \frac {\cos \left (d x^3\right )}{x} \, dx\\ &=a \log (x)+\frac {1}{3} b \text {Ci}\left (d x^3\right ) \sin (c)+\frac {1}{3} b \cos (c) \text {Si}\left (d x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 29, normalized size = 0.94 \[ a \log (x)+\frac {1}{3} b \left (\sin (c) \text {Ci}\left (d x^3\right )+\cos (c) \text {Si}\left (d x^3\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 38, normalized size = 1.23 \[ \frac {1}{3} \, b \cos \relax (c) \operatorname {Si}\left (d x^{3}\right ) + a \log \relax (x) + \frac {1}{6} \, {\left (b \operatorname {Ci}\left (d x^{3}\right ) + b \operatorname {Ci}\left (-d x^{3}\right )\right )} \sin \relax (c) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 32, normalized size = 1.03 \[ \frac {1}{3} \, b \operatorname {Ci}\left (d x^{3}\right ) \sin \relax (c) + \frac {1}{3} \, b \cos \relax (c) \operatorname {Si}\left (d x^{3}\right ) + \frac {1}{3} \, a \log \left (d x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {a +b \sin \left (d \,x^{3}+c \right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.43, size = 50, normalized size = 1.61 \[ -\frac {1}{6} \, {\left ({\left (i \, {\rm Ei}\left (i \, d x^{3}\right ) - i \, {\rm Ei}\left (-i \, d x^{3}\right )\right )} \cos \relax (c) - {\left ({\rm Ei}\left (i \, d x^{3}\right ) + {\rm Ei}\left (-i \, d x^{3}\right )\right )} \sin \relax (c)\right )} b + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ a\,\ln \relax (x)+\frac {b\,\sin \relax (c)\,\mathrm {cosint}\left (d\,x^3\right )}{3}+\frac {b\,\cos \relax (c)\,\mathrm {sinint}\left (d\,x^3\right )}{3} \]
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 \[ \int \frac {a + b \sin {\left (c + d x^{3} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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