Optimal. Leaf size=43 \[ \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\cos \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2633} \[ \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\cos \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin {align*} \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \sin ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac {\cos \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 45, normalized size = 1.05 \[ \frac {\cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )}{12 b n}-\frac {3 \cos \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 37, normalized size = 0.86 \[ \frac {\cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - 3 \, \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{3 \, b n} \]
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 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 35, normalized size = 0.81 \[ -\frac {\left (2+\sin ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 233, normalized size = 5.42 \[ \frac {{\left (\cos \left (6 \, b \log \relax (c)\right ) \cos \left (3 \, b \log \relax (c)\right ) + \sin \left (6 \, b \log \relax (c)\right ) \sin \left (3 \, b \log \relax (c)\right ) + \cos \left (3 \, b \log \relax (c)\right )\right )} \cos \left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right ) - 9 \, {\left (\cos \left (4 \, b \log \relax (c)\right ) \cos \left (3 \, b \log \relax (c)\right ) + \cos \left (3 \, b \log \relax (c)\right ) \cos \left (2 \, b \log \relax (c)\right ) + \sin \left (4 \, b \log \relax (c)\right ) \sin \left (3 \, b \log \relax (c)\right ) + \sin \left (3 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right )\right )} \cos \left (b \log \left (x^{n}\right ) + a\right ) - {\left (\cos \left (3 \, b \log \relax (c)\right ) \sin \left (6 \, b \log \relax (c)\right ) - \cos \left (6 \, b \log \relax (c)\right ) \sin \left (3 \, b \log \relax (c)\right ) + \sin \left (3 \, b \log \relax (c)\right )\right )} \sin \left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right ) + 9 \, {\left (\cos \left (3 \, b \log \relax (c)\right ) \sin \left (4 \, b \log \relax (c)\right ) - \cos \left (4 \, b \log \relax (c)\right ) \sin \left (3 \, b \log \relax (c)\right ) + \cos \left (2 \, b \log \relax (c)\right ) \sin \left (3 \, b \log \relax (c)\right ) - \cos \left (3 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right )\right )} \sin \left (b \log \left (x^{n}\right ) + a\right )}{24 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.43, size = 37, normalized size = 0.86 \[ -\frac {3\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )-{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{3\,b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.95, size = 83, normalized size = 1.93 \[ \begin {cases} \log {\relax (x )} \sin ^{3}{\relax (a )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\relax (x )} \sin ^{3}{\left (a + b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\- \frac {\sin ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cos {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b n} - \frac {2 \cos ^{3}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{3 b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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