Optimal. Leaf size=226 \[ \frac {4 n \sqrt {-\frac {(m+1)^2}{n^2}} x^{m+1} \sin \left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)^2}-\frac {4 x^{m+1} \cos ^3\left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)}+\frac {8 x^{m+1} \cos \left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)}-\frac {6 n \sqrt {-\frac {(m+1)^2}{n^2}} x^{m+1} \sin \left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right ) \cos ^2\left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)^2} \]
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Rubi [A] time = 0.08, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4488, 4486} \[ \frac {4 n \sqrt {-\frac {(m+1)^2}{n^2}} x^{m+1} \sin \left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)^2}-\frac {4 x^{m+1} \cos ^3\left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)}+\frac {8 x^{m+1} \cos \left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)}-\frac {6 n \sqrt {-\frac {(m+1)^2}{n^2}} x^{m+1} \sin \left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right ) \cos ^2\left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)^2} \]
Antiderivative was successfully verified.
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Rule 4486
Rule 4488
Rubi steps
\begin {align*} \int x^m \cos ^3\left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx &=-\frac {4 x^{1+m} \cos ^3\left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)}-\frac {6 \sqrt {-\frac {(1+m)^2}{n^2}} n x^{1+m} \cos ^2\left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \sin \left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)^2}+\frac {6}{5} \int x^m \cos \left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx\\ &=\frac {8 x^{1+m} \cos \left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)}-\frac {4 x^{1+m} \cos ^3\left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)}+\frac {4 \sqrt {-\frac {(1+m)^2}{n^2}} n x^{1+m} \sin \left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)^2}-\frac {6 \sqrt {-\frac {(1+m)^2}{n^2}} n x^{1+m} \cos ^2\left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \sin \left (a+\frac {1}{2} \sqrt {-\frac {(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)^2}\\ \end {align*}
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Mathematica [A] time = 1.69, size = 158, normalized size = 0.70 \[ \frac {x^{m+1} \left (n \sqrt {-\frac {(m+1)^2}{n^2}} \left (5 \sin \left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )-3 \sin \left (3 a+\frac {3}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )\right )+10 (m+1) \cos \left (a+\frac {1}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )-2 (m+1) \cos \left (3 a+\frac {3}{2} \sqrt {-\frac {(m+1)^2}{n^2}} \log \left (c x^n\right )\right )\right )}{10 (m+1)^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.48, size = 128, normalized size = 0.57 \[ \frac {{\left (5 \, e^{\left (-\frac {{\left (m + 1\right )} n \log \relax (x) - 2 i \, a n + {\left (m + 1\right )} \log \relax (c)}{n}\right )} + 15 \, e^{\left (-\frac {2 \, {\left ({\left (m + 1\right )} n \log \relax (x) - 2 i \, a n + {\left (m + 1\right )} \log \relax (c)\right )}}{n}\right )} - 5 \, e^{\left (-\frac {3 \, {\left ({\left (m + 1\right )} n \log \relax (x) - 2 i \, a n + {\left (m + 1\right )} \log \relax (c)\right )}}{n}\right )} + 1\right )} e^{\left (\frac {5 \, {\left ({\left (m + 1\right )} n \log \relax (x) - 2 i \, a n + {\left (m + 1\right )} \log \relax (c)\right )}}{2 \, n} + \frac {2 i \, a n - {\left (m + 1\right )} \log \relax (c)}{n}\right )}}{20 \, {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 14.21, size = 1870, normalized size = 8.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cos ^{3}\left (a +\frac {\ln \left (c \,x^{n}\right ) \sqrt {-\frac {\left (1+m \right )^{2}}{n^{2}}}}{2}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 195, normalized size = 0.86 \[ \frac {{\left (c^{\frac {3 \, m}{n} + \frac {3}{n}} x \cos \left (3 \, a\right ) e^{\left (m \log \relax (x) + \frac {3 \, m \log \left (x^{n}\right )}{n} + \frac {3 \, \log \left (x^{n}\right )}{n}\right )} + 5 \, c^{\frac {2 \, m}{n} + \frac {2}{n}} x \cos \relax (a) e^{\left (m \log \relax (x) + \frac {2 \, m \log \left (x^{n}\right )}{n} + \frac {2 \, \log \left (x^{n}\right )}{n}\right )} + 15 \, c^{\frac {m}{n} + \frac {1}{n}} x \cos \relax (a) e^{\left (m \log \relax (x) + \frac {m \log \left (x^{n}\right )}{n} + \frac {\log \left (x^{n}\right )}{n}\right )} - 5 \, x x^{m} \cos \left (3 \, a\right )\right )} e^{\left (-\frac {3 \, m \log \left (x^{n}\right )}{2 \, n} - \frac {3 \, \log \left (x^{n}\right )}{2 \, n}\right )}}{20 \, {\left (c^{\frac {3 \, m}{2 \, n} + \frac {3}{2 \, n}} m + c^{\frac {3 \, m}{2 \, n} + \frac {3}{2 \, n}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.71, size = 277, normalized size = 1.23 \[ \frac {x\,x^m\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {2\,m}{n^2}-\frac {1}{n^2}-\frac {m^2}{n^2}}\,1{}\mathrm {i}}{2}}}\,\left (2\,m+2+n\,\sqrt {-\frac {{\left (m+1\right )}^2}{n^2}}\,1{}\mathrm {i}\right )}{4\,{\left (m+1\right )}^2}+\frac {x\,x^m\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {2\,m}{n^2}-\frac {1}{n^2}-\frac {m^2}{n^2}}\,1{}\mathrm {i}}{2}}\,\left (2\,m+2-n\,\sqrt {-\frac {{\left (m+1\right )}^2}{n^2}}\,1{}\mathrm {i}\right )}{4\,{\left (m+1\right )}^2}-\frac {x\,x^m\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {2\,m}{n^2}-\frac {1}{n^2}-\frac {m^2}{n^2}}\,3{}\mathrm {i}}{2}}}\,\left (2\,m+2+n\,\sqrt {-\frac {{\left (m+1\right )}^2}{n^2}}\,3{}\mathrm {i}\right )}{20\,{\left (m+1\right )}^2}-\frac {x\,x^m\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {2\,m}{n^2}-\frac {1}{n^2}-\frac {m^2}{n^2}}\,3{}\mathrm {i}}{2}}\,\left (2\,m+2-n\,\sqrt {-\frac {{\left (m+1\right )}^2}{n^2}}\,3{}\mathrm {i}\right )}{20\,{\left (m+1\right )}^2} \]
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 \[ \int x^{m} \cos ^{3}{\left (a + \frac {\sqrt {- \frac {m^{2}}{n^{2}} - \frac {2 m}{n^{2}} - \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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