Optimal. Leaf size=79 \[ \frac{\cos (2 a) \text{CosIntegral}\left (2 b x^n\right )}{2 n}+\frac{\cos (4 a) \text{CosIntegral}\left (4 b x^n\right )}{8 n}-\frac{\sin (2 a) \text{Si}\left (2 b x^n\right )}{2 n}-\frac{\sin (4 a) \text{Si}\left (4 b x^n\right )}{8 n}+\frac{3 \log (x)}{8} \]
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Rubi [A] time = 0.0940672, antiderivative size = 79, normalized size of antiderivative = 1., 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} \[ \frac{\cos (2 a) \text{CosIntegral}\left (2 b x^n\right )}{2 n}+\frac{\cos (4 a) \text{CosIntegral}\left (4 b x^n\right )}{8 n}-\frac{\sin (2 a) \text{Si}\left (2 b x^n\right )}{2 n}-\frac{\sin (4 a) \text{Si}\left (4 b x^n\right )}{8 n}+\frac{3 \log (x)}{8} \]
Antiderivative was successfully verified.
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Rule 3426
Rule 3378
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\cos ^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*}
Mathematica [A] time = 0.14085, size = 66, normalized size = 0.84 \[ \frac{4 \cos (2 a) \text{CosIntegral}\left (2 b x^n\right )+\cos (4 a) \text{CosIntegral}\left (4 b x^n\right )-4 \sin (2 a) \text{Si}\left (2 b x^n\right )-\sin (4 a) \text{Si}\left (4 b x^n\right )}{8 n}+\frac{3 \log (x)}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 77, normalized size = 1. \begin{align*} -{\frac{{\it Si} \left ( 4\,b{x}^{n} \right ) \sin \left ( 4\,a \right ) }{8\,n}}+{\frac{{\it Ci} \left ( 4\,b{x}^{n} \right ) \cos \left ( 4\,a \right ) }{8\,n}}-{\frac{{\it Si} \left ( 2\,b{x}^{n} \right ) \sin \left ( 2\,a \right ) }{2\,n}}+{\frac{{\it Ci} \left ( 2\,b{x}^{n} \right ) \cos \left ( 2\,a \right ) }{2\,n}}+{\frac{3\,\ln \left ( b{x}^{n} \right ) }{8\,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 = 2.02452, size = 309, normalized size = 3.91 \begin{align*} \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 \left (x\right ) - 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} \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{\cos ^{4}{\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{\cos \left (b x^{n} + a\right )^{4}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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