Optimal. Leaf size=67 \[ \frac {3 \cos (a) \text {Ci}\left (b x^n\right )}{4 n}+\frac {\cos (3 a) \text {Ci}\left (3 b x^n\right )}{4 n}-\frac {3 \sin (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {\sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \]
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Rubi [A] time = 0.09, antiderivative size = 67, 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 = {3426, 3378, 3376, 3375} \[ \frac {3 \cos (a) \text {CosIntegral}\left (b x^n\right )}{4 n}+\frac {\cos (3 a) \text {CosIntegral}\left (3 b x^n\right )}{4 n}-\frac {3 \sin (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {\sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \]
Antiderivative was successfully verified.
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Rule 3375
Rule 3376
Rule 3378
Rule 3426
Rubi steps
\begin {align*} \int \frac {\cos ^3\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac {3 \cos \left (a+b x^n\right )}{4 x}+\frac {\cos \left (3 a+3 b x^n\right )}{4 x}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\cos \left (3 a+3 b x^n\right )}{x} \, dx+\frac {3}{4} \int \frac {\cos \left (a+b x^n\right )}{x} \, dx\\ &=\frac {1}{4} (3 \cos (a)) \int \frac {\cos \left (b x^n\right )}{x} \, dx+\frac {1}{4} \cos (3 a) \int \frac {\cos \left (3 b x^n\right )}{x} \, dx-\frac {1}{4} (3 \sin (a)) \int \frac {\sin \left (b x^n\right )}{x} \, dx-\frac {1}{4} \sin (3 a) \int \frac {\sin \left (3 b x^n\right )}{x} \, dx\\ &=\frac {3 \cos (a) \text {Ci}\left (b x^n\right )}{4 n}+\frac {\cos (3 a) \text {Ci}\left (3 b x^n\right )}{4 n}-\frac {3 \sin (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {\sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 53, normalized size = 0.79 \[ \frac {3 \cos (a) \text {Ci}\left (b x^n\right )+\cos (3 a) \text {Ci}\left (3 b x^n\right )-3 \sin (a) \text {Si}\left (b x^n\right )-\sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 74, normalized size = 1.10 \[ \frac {\cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x^{n}\right ) + 3 \, \cos \relax (a) \operatorname {Ci}\left (b x^{n}\right ) + 3 \, \cos \relax (a) \operatorname {Ci}\left (-b x^{n}\right ) + \cos \left (3 \, a\right ) \operatorname {Ci}\left (-3 \, b x^{n}\right ) - 2 \, \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) - 6 \, \sin \relax (a) \operatorname {Si}\left (b x^{n}\right )}{8 \, n} \]
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 \[ \int \frac {\cos \left (b x^{n} + a\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 0.78 \[ \frac {-\frac {\Si \left (3 b \,x^{n}\right ) \sin \left (3 a \right )}{4}+\frac {\Ci \left (3 b \,x^{n}\right ) \cos \left (3 a \right )}{4}-\frac {3 \Si \left (b \,x^{n}\right ) \sin \relax (a )}{4}+\frac {3 \Ci \left (b \,x^{n}\right ) \cos \relax (a )}{4}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.86, size = 179, normalized size = 2.67 \[ \frac {{\left ({\rm Ei}\left (3 i \, b x^{n}\right ) + {\rm Ei}\left (-3 i \, b x^{n}\right ) + {\rm Ei}\left (3 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-3 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \left (3 \, a\right ) + 3 \, {\left ({\rm Ei}\left (i \, b x^{n}\right ) + {\rm Ei}\left (-i \, b x^{n}\right ) + {\rm Ei}\left (i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \relax (a) + {\left (i \, {\rm Ei}\left (3 i \, b x^{n}\right ) - i \, {\rm Ei}\left (-3 i \, b x^{n}\right ) + i \, {\rm Ei}\left (3 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) - i \, {\rm Ei}\left (-3 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \left (3 \, a\right ) + {\left (3 i \, {\rm Ei}\left (i \, b x^{n}\right ) - 3 i \, {\rm Ei}\left (-i \, b x^{n}\right ) + 3 i \, {\rm Ei}\left (i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) - 3 i \, {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \relax (a)}{16 \, n} \]
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 \[ \int \frac {{\cos \left (a+b\,x^n\right )}^3}{x} \,d x \]
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 \[ \int \frac {\cos ^{3}{\left (a + b x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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