Optimal. Leaf size=67 \[ \frac{3 \sin (a) \text{CosIntegral}\left (b x^n\right )}{4 n}-\frac{\sin (3 a) \text{CosIntegral}\left (3 b x^n\right )}{4 n}+\frac{3 \cos (a) \text{Si}\left (b x^n\right )}{4 n}-\frac{\cos (3 a) \text{Si}\left (3 b x^n\right )}{4 n} \]
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Rubi [A] time = 0.0922735, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3425, 3377, 3376, 3375} \[ \frac{3 \sin (a) \text{CosIntegral}\left (b x^n\right )}{4 n}-\frac{\sin (3 a) \text{CosIntegral}\left (3 b x^n\right )}{4 n}+\frac{3 \cos (a) \text{Si}\left (b x^n\right )}{4 n}-\frac{\cos (3 a) \text{Si}\left (3 b x^n\right )}{4 n} \]
Antiderivative was successfully verified.
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Rule 3425
Rule 3377
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\sin ^3\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac{3 \sin \left (a+b x^n\right )}{4 x}-\frac{\sin \left (3 a+3 b x^n\right )}{4 x}\right ) \, dx\\ &=-\left (\frac{1}{4} \int \frac{\sin \left (3 a+3 b x^n\right )}{x} \, dx\right )+\frac{3}{4} \int \frac{\sin \left (a+b x^n\right )}{x} \, dx\\ &=\frac{1}{4} (3 \cos (a)) \int \frac{\sin \left (b x^n\right )}{x} \, dx-\frac{1}{4} \cos (3 a) \int \frac{\sin \left (3 b x^n\right )}{x} \, dx+\frac{1}{4} (3 \sin (a)) \int \frac{\cos \left (b x^n\right )}{x} \, dx-\frac{1}{4} \sin (3 a) \int \frac{\cos \left (3 b x^n\right )}{x} \, dx\\ &=\frac{3 \text{Ci}\left (b x^n\right ) \sin (a)}{4 n}-\frac{\text{Ci}\left (3 b x^n\right ) \sin (3 a)}{4 n}+\frac{3 \cos (a) \text{Si}\left (b x^n\right )}{4 n}-\frac{\cos (3 a) \text{Si}\left (3 b x^n\right )}{4 n}\\ \end{align*}
Mathematica [A] time = 0.105854, size = 54, normalized size = 0.81 \[ \frac{3 \sin (a) \text{CosIntegral}\left (b x^n\right )-\sin (3 a) \text{CosIntegral}\left (3 b x^n\right )+3 \cos (a) \text{Si}\left (b x^n\right )-\cos (3 a) \text{Si}\left (3 b x^n\right )}{4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 52, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ( -{\frac{{\it Si} \left ( 3\,b{x}^{n} \right ) \cos \left ( 3\,a \right ) }{4}}-{\frac{{\it Ci} \left ( 3\,b{x}^{n} \right ) \sin \left ( 3\,a \right ) }{4}}+{\frac{3\,{\it Si} \left ( b{x}^{n} \right ) \cos \left ( a \right ) }{4}}+{\frac{3\,{\it Ci} \left ( b{x}^{n} \right ) \sin \left ( a \right ) }{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72273, size = 275, normalized size = 4.1 \begin{align*} -\frac{\operatorname{Ci}\left (3 \, b x^{n}\right ) \sin \left (3 \, a\right ) + \operatorname{Ci}\left (-3 \, b x^{n}\right ) \sin \left (3 \, a\right ) - 3 \, \operatorname{Ci}\left (b x^{n}\right ) \sin \left (a\right ) - 3 \, \operatorname{Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 2 \, \cos \left (3 \, a\right ) \operatorname{Si}\left (3 \, b x^{n}\right ) - 6 \, \cos \left (a\right ) \operatorname{Si}\left (b x^{n}\right )}{8 \, n} \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{\sin ^{3}{\left (a + b x^{n} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x^{n} + a\right )^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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