Optimal. Leaf size=31 \[ a \log (x)+\frac{1}{3} b \sin (c) \text{CosIntegral}\left (d x^3\right )+\frac{1}{3} b \cos (c) \text{Si}\left (d x^3\right ) \]
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Rubi [A] time = 0.0387281, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 3377
Rule 3376
Rule 3375
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*}
Mathematica [A] time = 0.0511973, size = 29, normalized size = 0.94 \[ a \log (x)+\frac{1}{3} b \left (\sin (c) \text{CosIntegral}\left (d x^3\right )+\cos (c) \text{Si}\left (d x^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\sin \left ( d{x}^{3}+c \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.51283, size = 68, normalized size = 2.19 \begin{align*} -\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 \left (c\right ) -{\left ({\rm Ei}\left (i \, d x^{3}\right ) +{\rm Ei}\left (-i \, d x^{3}\right )\right )} \sin \left (c\right )\right )} b + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66128, size = 144, normalized size = 4.65 \begin{align*} \frac{1}{3} \, b \cos \left (c\right ) \operatorname{Si}\left (d x^{3}\right ) + a \log \left (x\right ) + \frac{1}{6} \,{\left (b \operatorname{Ci}\left (d x^{3}\right ) + b \operatorname{Ci}\left (-d x^{3}\right )\right )} \sin \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \sin{\left (c + d x^{3} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14297, size = 43, normalized size = 1.39 \begin{align*} \frac{1}{3} \, b \operatorname{Ci}\left (d x^{3}\right ) \sin \left (c\right ) + \frac{1}{3} \, b \cos \left (c\right ) \operatorname{Si}\left (d x^{3}\right ) + \frac{1}{3} \, a \log \left (d x^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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