Optimal. Leaf size=120 \[ \frac {2 b^2 n^2 x^{1+m}}{(1+m) \left ((1+m)^2+4 b^2 n^2\right )}+\frac {(1+m) x^{1+m} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+4 b^2 n^2}+\frac {2 b n x^{1+m} \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+4 b^2 n^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4576, 30}
\begin {gather*} \frac {(m+1) x^{m+1} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+(m+1)^2}+\frac {2 b n x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+(m+1)^2}+\frac {2 b^2 n^2 x^{m+1}}{(m+1) \left (4 b^2 n^2+(m+1)^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 4576
Rubi steps
\begin {align*} \int x^m \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {(1+m) x^{1+m} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+4 b^2 n^2}+\frac {2 b n x^{1+m} \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+4 b^2 n^2}+\frac {\left (2 b^2 n^2\right ) \int x^m \, dx}{(1+m)^2+4 b^2 n^2}\\ &=\frac {2 b^2 n^2 x^{1+m}}{(1+m) \left ((1+m)^2+4 b^2 n^2\right )}+\frac {(1+m) x^{1+m} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+4 b^2 n^2}+\frac {2 b n x^{1+m} \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+4 b^2 n^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.38, size = 91, normalized size = 0.76 \begin {gather*} \frac {x^{1+m} \left (1+2 m+m^2+4 b^2 n^2+(1+m)^2 \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+2 b (1+m) n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 (1+m) (1+m-2 i b n) (1+m+2 i b n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int x^{m} \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
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. 646 vs.
\(2 (120) = 240\).
time = 0.32, size = 646, normalized size = 5.38 \begin {gather*} \frac {{\left ({\left (\cos \left (4 \, 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 (2 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right )\right )} m^{2} + 2 \, {\left (\cos \left (4 \, 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 (2 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right )\right )} m + 2 \, {\left (b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - b \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + {\left (b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - b \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right )\right )} m + b \sin \left (2 \, b \log \left (c\right )\right )\right )} n + \cos \left (4 \, 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 (2 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right )\right )} x x^{m} \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) - {\left ({\left (\cos \left (2 \, 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 (2 \, b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right )\right )} m^{2} + 2 \, {\left (\cos \left (2 \, 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 (2 \, b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right )\right )} m - 2 \, {\left (b \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + b \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + {\left (b \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + b \sin \left (4 \, 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 )\right )} m + b \cos \left (2 \, b \log \left (c\right )\right )\right )} n + \cos \left (2 \, 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 (2 \, b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right )\right )} x x^{m} \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + 2 \, {\left ({\left (\cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} m^{2} + 4 \, {\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + 2 \, {\left (\cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} m + \cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} x x^{m}}{4 \, {\left ({\left (\cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} m^{3} + 3 \, {\left (\cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} m^{2} + 4 \, {\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2} + {\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} m\right )} n^{2} + 3 \, {\left (\cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} m + \cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, 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 = 2.76, size = 105, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (b m + b\right )} n x x^{m} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (2 \, b^{2} n^{2} x + {\left (m^{2} + 2 \, m + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} x^{m}}{m^{3} + 4 \, {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 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} \log {\left (x \right )} \cos ^{2}{\left (a \right )} & \text {for}\: b = 0 \wedge m = -1 \\\int x^{m} \cos ^{2}{\left (- a + \frac {i m \log {\left (c x^{n} \right )}}{2 n} + \frac {i \log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = \frac {i \left (- m - 1\right )}{2 n} \\\int x^{m} \cos ^{2}{\left (a + \frac {i m \log {\left (c x^{n} \right )}}{2 n} + \frac {i \log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = \frac {i \left (m + 1\right )}{2 n} \\\frac {\begin {cases} \log {\left (x \right )} \cos {\left (2 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (2 a + 2 b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (2 a + 2 b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases}}{2} + \frac {\log {\left (x \right )}}{2} & \text {for}\: m = -1 \\\frac {2 b^{2} n^{2} x x^{m} \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} + m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b^{2} n^{2} x x^{m} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} + m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b m n x x^{m} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} + m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 b n x x^{m} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} + m^{3} + 3 m^{2} + 3 m + 1} + \frac {m^{2} x x^{m} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} + m^{3} + 3 m^{2} + 3 m + 1} + \frac {2 m x x^{m} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} + m^{3} + 3 m^{2} + 3 m + 1} + \frac {x x^{m} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} + m^{3} + 3 m^{2} + 3 m + 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. 8742 vs.
\(2 (120) = 240\).
time = 0.71, size = 8742, normalized size = 72.85 \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 = 2.79, size = 82, normalized size = 0.68 \begin {gather*} \frac {x\,x^m}{2\,m+2}+\frac {x\,x^m\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}{4\,m+4+b\,n\,8{}\mathrm {i}}+\frac {x\,x^m\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{m\,4{}\mathrm {i}+8\,b\,n+4{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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