Optimal. Leaf size=42 \[ \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, 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 = {2713}
\begin {gather*} \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rubi steps
\begin {align*} \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \cos ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin ^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 = 42, normalized size = 1.00 \begin {gather*} \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 35, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\left (2+\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 n b}\) | \(35\) |
default | \(\frac {\left (2+\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 n b}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs.
\(2 (40) = 80\).
time = 0.30, size = 232, normalized size = 5.52 \begin {gather*} \frac {{\left (\cos \left (3 \, b \log \left (c\right )\right ) \sin \left (6 \, b \log \left (c\right )\right ) - \cos \left (6 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) + \sin \left (3 \, b \log \left (c\right )\right )\right )} \cos \left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right ) + 9 \, {\left (\cos \left (3 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) - \cos \left (3 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right )\right )} \cos \left (b \log \left (x^{n}\right ) + a\right ) + {\left (\cos \left (6 \, b \log \left (c\right )\right ) \cos \left (3 \, b \log \left (c\right )\right ) + \sin \left (6 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) + \cos \left (3 \, b \log \left (c\right )\right )\right )} \sin \left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right ) + 9 \, {\left (\cos \left (4 \, b \log \left (c\right )\right ) \cos \left (3 \, b \log \left (c\right )\right ) + \cos \left (3 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) + \sin \left (3 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right )\right )} \sin \left (b \log \left (x^{n}\right ) + a\right )}{24 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.34, size = 36, normalized size = 0.86 \begin {gather*} \frac {{\left (\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (32) = 64\).
time = 1.76, size = 71, normalized size = 1.69 \begin {gather*} \begin {cases} \log {\left (x \right )} \cos ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {2 \sin ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{3 b n} + \frac {\sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.35, size = 37, normalized size = 0.88 \begin {gather*} \frac {3\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )-{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{3\,b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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