Optimal. Leaf size=52 \[ -\frac {b n x \cos \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2}+\frac {x \sin \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4563}
\begin {gather*} \frac {x \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1}-\frac {b n x \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 4563
Rubi steps
\begin {align*} \int \sin \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {b n x \cos \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2}+\frac {x \sin \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 40, normalized size = 0.77 \begin {gather*} \frac {x \left (-b n \cos \left (a+b \log \left (c x^n\right )\right )+\sin \left (a+b \log \left (c x^n\right )\right )\right )}{1+b^2 n^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sin \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
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. 206 vs.
\(2 (52) = 104\).
time = 0.29, size = 206, normalized size = 3.96 \begin {gather*} -\frac {{\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \cos \left (b \log \left (c\right )\right )\right )} n - \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) - \sin \left (b \log \left (c\right )\right )\right )} x \cos \left (b \log \left (x^{n}\right ) + a\right ) - {\left ({\left (b \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \sin \left (b \log \left (c\right )\right )\right )} n + \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + \cos \left (b \log \left (c\right )\right )\right )} x \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \left (c\right )\right )^{2} + b^{2} \sin \left (b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (b \log \left (c\right )\right )^{2} + \sin \left (b \log \left (c\right )\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.26, size = 45, normalized size = 0.87 \begin {gather*} -\frac {b n x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - x \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} + 1} \end {gather*}
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 \begin {gather*} \begin {cases} \int \sin {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i}{n} \\\int \sin {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i}{n} \\- \frac {b n x \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + 1} + \frac {x \sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 882 vs.
\(2 (52) = 104\).
time = 0.40, size = 882, normalized size = 16.96 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.33, size = 40, normalized size = 0.77 \begin {gather*} \frac {x\,\left (\sin \left (a+b\,\ln \left (c\,x^n\right )\right )-b\,n\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b^2\,n^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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