Optimal. Leaf size=25 \[ \frac {\sin (a) \text {Ci}\left (b x^n\right )}{n}+\frac {\cos (a) \text {Si}\left (b x^n\right )}{n} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 3375
Rule 3376
Rule 3377
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*}
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Mathematica [A] time = 0.07, size = 23, normalized size = 0.92 \[ \frac {\sin (a) \text {Ci}\left (b x^n\right )+\cos (a) \text {Si}\left (b x^n\right )}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 35, normalized size = 1.40 \[ \frac {\operatorname {Ci}\left (b x^{n}\right ) \sin \relax (a) + \operatorname {Ci}\left (-b x^{n}\right ) \sin \relax (a) + 2 \, \cos \relax (a) \operatorname {Si}\left (b x^{n}\right )}{2 \, n} \]
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 \[ \int \frac {\sin \left (b x^{n} + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 24, normalized size = 0.96 \[ \frac {\Si \left (b \,x^{n}\right ) \cos \relax (a )+\Ci \left (b \,x^{n}\right ) \sin \relax (a )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.02, size = 91, normalized size = 3.64 \[ -\frac {{\left (i \, {\rm Ei}\left (i \, b x^{n}\right ) - i \, {\rm Ei}\left (-i \, b x^{n}\right ) + i \, {\rm Ei}\left (i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) - i \, {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \relax (a) - {\left ({\rm Ei}\left (i \, b x^{n}\right ) + {\rm Ei}\left (-i \, b x^{n}\right ) + {\rm Ei}\left (i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \relax (a)}{4 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sin \left (a+b\,x^n\right )}{x} \,d x \]
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 \[ \int \frac {\sin {\left (a + b x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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