Optimal. Leaf size=43 \[ -\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right )}{2 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2} \]
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Rubi [A] time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3425, 3378, 3376, 3375} \[ -\frac {\cos (2 a) \text {CosIntegral}\left (2 b x^n\right )}{2 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 3375
Rule 3376
Rule 3378
Rule 3425
Rubi steps
\begin {align*} \int \frac {\sin ^2\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac {1}{2 x}-\frac {\cos \left (2 a+2 b x^n\right )}{2 x}\right ) \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \int \frac {\cos \left (2 a+2 b x^n\right )}{x} \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \cos (2 a) \int \frac {\cos \left (2 b x^n\right )}{x} \, dx+\frac {1}{2} \sin (2 a) \int \frac {\sin \left (2 b x^n\right )}{x} \, dx\\ &=-\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 37, normalized size = 0.86 \[ \frac {-\cos (2 a) \text {Ci}\left (2 b x^n\right )+\sin (2 a) \text {Si}\left (2 b x^n\right )+n \log (x)}{2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 48, normalized size = 1.12 \[ -\frac {\cos \left (2 \, a\right ) \operatorname {Ci}\left (2 \, b x^{n}\right ) + \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, b x^{n}\right ) - 2 \, n \log \relax (x) - 2 \, \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right )}{4 \, n} \]
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 \[ \int \frac {\sin \left (b x^{n} + a\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 1.05 \[ \frac {\ln \left (b \,x^{n}\right )}{2 n}+\frac {\Si \left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2 n}-\frac {\Ci \left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.10, size = 100, normalized size = 2.33 \[ -\frac {{\left ({\rm Ei}\left (2 i \, b x^{n}\right ) + {\rm Ei}\left (-2 i \, b x^{n}\right ) + {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \left (2 \, a\right ) - 4 \, n \log \relax (x) - {\left (-i \, {\rm Ei}\left (2 i \, b x^{n}\right ) + i \, {\rm Ei}\left (-2 i \, b x^{n}\right ) - i \, {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + i \, {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \left (2 \, a\right )}{8 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\sin \left (a+b\,x^n\right )}^2}{x} \,d x \]
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 \[ \int \frac {\sin ^{2}{\left (a + b x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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