Optimal. Leaf size=79 \[ -\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right )}{2 n}+\frac {\cos (4 a) \text {Ci}\left (4 b x^n\right )}{8 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}-\frac {\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8} \]
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Rubi [A] time = 0.10, antiderivative size = 79, normalized size of antiderivative = 1.00, 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, 3378, 3376, 3375} \[ -\frac {\cos (2 a) \text {CosIntegral}\left (2 b x^n\right )}{2 n}+\frac {\cos (4 a) \text {CosIntegral}\left (4 b x^n\right )}{8 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}-\frac {\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8} \]
Antiderivative was successfully verified.
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Rule 3375
Rule 3376
Rule 3378
Rule 3425
Rubi steps
\begin {align*} \int \frac {\sin ^4\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac {3}{8 x}-\frac {\cos \left (2 a+2 b x^n\right )}{2 x}+\frac {\cos \left (4 a+4 b x^n\right )}{8 x}\right ) \, dx\\ &=\frac {3 \log (x)}{8}+\frac {1}{8} \int \frac {\cos \left (4 a+4 b x^n\right )}{x} \, dx-\frac {1}{2} \int \frac {\cos \left (2 a+2 b x^n\right )}{x} \, dx\\ &=\frac {3 \log (x)}{8}-\frac {1}{2} \cos (2 a) \int \frac {\cos \left (2 b x^n\right )}{x} \, dx+\frac {1}{8} \cos (4 a) \int \frac {\cos \left (4 b x^n\right )}{x} \, dx+\frac {1}{2} \sin (2 a) \int \frac {\sin \left (2 b x^n\right )}{x} \, dx-\frac {1}{8} \sin (4 a) \int \frac {\sin \left (4 b x^n\right )}{x} \, dx\\ &=-\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right )}{2 n}+\frac {\cos (4 a) \text {Ci}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}-\frac {\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 66, normalized size = 0.84 \[ \frac {-4 \cos (2 a) \text {Ci}\left (2 b x^n\right )+\cos (4 a) \text {Ci}\left (4 b x^n\right )+4 \sin (2 a) \text {Si}\left (2 b x^n\right )-\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 87, normalized size = 1.10 \[ \frac {\cos \left (4 \, a\right ) \operatorname {Ci}\left (4 \, b x^{n}\right ) - 4 \, \cos \left (2 \, a\right ) \operatorname {Ci}\left (2 \, b x^{n}\right ) - 4 \, \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, b x^{n}\right ) + \cos \left (4 \, a\right ) \operatorname {Ci}\left (-4 \, b x^{n}\right ) + 6 \, n \log \relax (x) - 2 \, \sin \left (4 \, a\right ) \operatorname {Si}\left (4 \, b x^{n}\right ) + 8 \, \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right )}{16 \, n} \]
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 \[ \int \frac {\sin \left (b x^{n} + a\right )^{4}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 77, normalized size = 0.97 \[ \frac {3 \ln \left (b \,x^{n}\right )}{8 n}-\frac {\Si \left (4 b \,x^{n}\right ) \sin \left (4 a \right )}{8 n}+\frac {\Ci \left (4 b \,x^{n}\right ) \cos \left (4 a \right )}{8 n}+\frac {\Si \left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2 n}-\frac {\Ci \left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.32, size = 188, normalized size = 2.38 \[ \frac {{\left ({\rm Ei}\left (4 i \, b x^{n}\right ) + {\rm Ei}\left (-4 i \, b x^{n}\right ) + {\rm Ei}\left (4 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-4 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \left (4 \, a\right ) - 4 \, {\left ({\rm Ei}\left (2 i \, b x^{n}\right ) + {\rm Ei}\left (-2 i \, b x^{n}\right ) + {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \left (2 \, a\right ) + 12 \, n \log \relax (x) + {\left (i \, {\rm Ei}\left (4 i \, b x^{n}\right ) - i \, {\rm Ei}\left (-4 i \, b x^{n}\right ) + i \, {\rm Ei}\left (4 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) - i \, {\rm Ei}\left (-4 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \left (4 \, a\right ) + {\left (-4 i \, {\rm Ei}\left (2 i \, b x^{n}\right ) + 4 i \, {\rm Ei}\left (-2 i \, b x^{n}\right ) - 4 i \, {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + 4 i \, {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \left (2 \, a\right )}{32 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+b\,x^n\right )}^4}{x} \,d x \]
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 \[ \int \frac {\sin ^{4}{\left (a + b x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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