Optimal. Leaf size=25 \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right )}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right )}{n} \]
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Rubi [A] time = 0.038134, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3377, 3376, 3375} \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right )}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 3377
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\sin \left (a+b x^n\right )}{x} \, dx &=\cos (a) \int \frac{\sin \left (b x^n\right )}{x} \, dx+\sin (a) \int \frac{\cos \left (b x^n\right )}{x} \, dx\\ &=\frac{\text{Ci}\left (b x^n\right ) \sin (a)}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0612245, size = 23, normalized size = 0.92 \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right )+\cos (a) \text{Si}\left (b x^n\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 24, normalized size = 1. \begin{align*}{\frac{{\it Si} \left ( b{x}^{n} \right ) \cos \left ( a \right ) +{\it Ci} \left ( b{x}^{n} \right ) \sin \left ( a \right ) }{n}} \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 = 1.7626, size = 131, normalized size = 5.24 \begin{align*} \frac{\operatorname{Ci}\left (b x^{n}\right ) \sin \left (a\right ) + \operatorname{Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 2 \, \cos \left (a\right ) \operatorname{Si}\left (b x^{n}\right )}{2 \, 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{\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 )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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