Optimal. Leaf size=106 \[ -\frac {e^{\frac {2 a (m+1)}{\sqrt {-(m+1)^2}}} x^{m+1} \left (c x^2\right )^{\frac {m+1}{2}}}{8 (m+1)}-\frac {1}{4} e^{-\frac {2 a (m+1)}{\sqrt {-(m+1)^2}}} x^{m+1} \log (x) \left (c x^2\right )^{\frac {1}{2} (-m-1)}+\frac {x^{m+1}}{2 (m+1)} \]
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Rubi [A] time = 0.14, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4493, 4489} \[ -\frac {e^{\frac {2 a (m+1)}{\sqrt {-(m+1)^2}}} x^{m+1} \left (c x^2\right )^{\frac {m+1}{2}}}{8 (m+1)}-\frac {1}{4} e^{-\frac {2 a (m+1)}{\sqrt {-(m+1)^2}}} x^{m+1} \log (x) \left (c x^2\right )^{\frac {1}{2} (-m-1)}+\frac {x^{m+1}}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 4489
Rule 4493
Rubi steps
\begin {align*} \int x^m \sin ^2\left (a+\frac {1}{4} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx &=\frac {1}{2} \left (x^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1+m}{2}} \sin ^2\left (a+\frac {1}{4} \sqrt {-(1+m)^2} \log (x)\right ) \, dx,x,c x^2\right )\\ &=-\left (\frac {1}{8} \left (x^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \operatorname {Subst}\left (\int \left (\frac {e^{-\frac {2 a (1+m)}{\sqrt {-(1+m)^2}}}}{x}-2 x^{\frac {1}{2} (-1+m)}+e^{\frac {2 a (1+m)}{\sqrt {-(1+m)^2}}} x^m\right ) \, dx,x,c x^2\right )\right )\\ &=\frac {x^{1+m}}{2 (1+m)}-\frac {e^{\frac {2 a (1+m)}{\sqrt {-(1+m)^2}}} x^{1+m} \left (c x^2\right )^{\frac {1+m}{2}}}{8 (1+m)}-\frac {1}{4} e^{-\frac {2 a (1+m)}{\sqrt {-(1+m)^2}}} x^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)} \log (x)\\ \end {align*}
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Mathematica [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int x^m \sin ^2\left (a+\frac {1}{4} \sqrt {-(1+m)^2} \log \left (c x^2\right )\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.44, size = 75, normalized size = 0.71 \[ -\frac {{\left (2 \, {\left (m + 1\right )} e^{\left (-{\left (m + 1\right )} \log \relax (c) - 2 \, {\left (m + 1\right )} \log \relax (x) + 4 i \, a\right )} \log \relax (x) - 4 \, e^{\left (-\frac {1}{2} \, {\left (m + 1\right )} \log \relax (c) - {\left (m + 1\right )} \log \relax (x) + 2 i \, a\right )} + 1\right )} e^{\left (\frac {1}{2} \, {\left (m + 1\right )} \log \relax (c) + 2 \, {\left (m + 1\right )} \log \relax (x) - 2 i \, a\right )}}{8 \, {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 2.00, size = 350, normalized size = 3.30 \[ \frac {m^{2} x x^{m} e^{\left (\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) + {\left | m + 1 \right |} \log \relax (x) - 2 i \, a\right )} - m x x^{m} {\left | m + 1 \right |} e^{\left (\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) + {\left | m + 1 \right |} \log \relax (x) - 2 i \, a\right )} + m^{2} x x^{m} e^{\left (-\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) - {\left | m + 1 \right |} \log \relax (x) + 2 i \, a\right )} + m x x^{m} {\left | m + 1 \right |} e^{\left (-\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) - {\left | m + 1 \right |} \log \relax (x) + 2 i \, a\right )} + 2 \, {\left (m + 1\right )}^{2} x x^{m} - 2 \, m^{2} x x^{m} + 2 \, m x x^{m} e^{\left (\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) + {\left | m + 1 \right |} \log \relax (x) - 2 i \, a\right )} - x x^{m} {\left | m + 1 \right |} e^{\left (\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) + {\left | m + 1 \right |} \log \relax (x) - 2 i \, a\right )} + 2 \, m x x^{m} e^{\left (-\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) - {\left | m + 1 \right |} \log \relax (x) + 2 i \, a\right )} + x x^{m} {\left | m + 1 \right |} e^{\left (-\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) - {\left | m + 1 \right |} \log \relax (x) + 2 i \, a\right )} - 4 \, m x x^{m} + x x^{m} e^{\left (\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) + {\left | m + 1 \right |} \log \relax (x) - 2 i \, a\right )} + x x^{m} e^{\left (-\frac {1}{2} \, {\left | m + 1 \right |} \log \relax (c) - {\left | m + 1 \right |} \log \relax (x) + 2 i \, a\right )} - 2 \, x x^{m}}{4 \, {\left ({\left (m + 1\right )}^{2} m - m^{3} + {\left (m + 1\right )}^{2} - 3 \, m^{2} - 3 \, m - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\sin ^{2}\left (a +\frac {\ln \left (c \,x^{2}\right ) \sqrt {-\left (1+m \right )^{2}}}{4}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 134, normalized size = 1.26 \[ -\frac {c^{m + 1} x^{2} x^{2 \, m} \cos \left (2 \, a\right ) - 4 \, {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} c^{\frac {1}{2} \, m + \frac {1}{2}} x x^{m} + 2 \, {\left (\cos \left (2 \, a\right )^{3} + \cos \left (2 \, a\right ) \sin \left (2 \, a\right )^{2} + {\left (\cos \left (2 \, a\right )^{3} + \cos \left (2 \, a\right ) \sin \left (2 \, a\right )^{2}\right )} m\right )} \log \relax (x)}{8 \, {\left ({\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} c^{\frac {1}{2} \, m} m + {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} c^{\frac {1}{2} \, m}\right )} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.04, size = 149, normalized size = 1.41 \[ \frac {x\,x^m}{2\,m+2}-\frac {\frac {1}{c^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{2}}}\,x\,x^m\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (x^2\right )}^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{2}}}}{4\,m+4-\sqrt {-{\left (m+1\right )}^2}\,4{}\mathrm {i}}-\frac {c^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{2}}\,x\,x^m\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (x^2\right )}^{\frac {\sqrt {-m^2-2\,m-1}\,1{}\mathrm {i}}{2}}}{4\,m+4+\sqrt {-{\left (m+1\right )}^2}\,4{}\mathrm {i}} \]
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 \[ \int x^{m} \sin ^{2}{\left (a + \frac {\sqrt {- m^{2} - 2 m - 1} \log {\left (c x^{2} \right )}}{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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