Optimal. Leaf size=110 \[ \frac {x^{1+m} \csc \left (a+2 \log \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )\right )}{2 (1+m)}-\frac {x^{1+m} \cot \left (a+2 \log \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )\right ) \csc \left (a+2 \log \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )\right )}{2 \sqrt {-(1+m)^2}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.13, antiderivative size = 142, normalized size of antiderivative = 1.29, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {4606, 4602,
371} \begin {gather*} -\frac {8 e^{3 i a} x^{m+1} \left (c x^{\frac {1}{2} \sqrt {-(m+1)^2}}\right )^{6 i} \, _2F_1\left (3,\frac {1}{2} \left (3-\frac {i (m+1)}{\sqrt {-(m+1)^2}}\right );\frac {1}{2} \left (5-\frac {i (m+1)}{\sqrt {-(m+1)^2}}\right );e^{2 i a} \left (c x^{\frac {1}{2} \sqrt {-(m+1)^2}}\right )^{4 i}\right )}{i m-3 \sqrt {-(m+1)^2}+i} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 371
Rule 4602
Rule 4606
Rubi steps
\begin {align*} \int x^m \csc ^3\left (a+2 \log \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )\right ) \, dx &=\frac {\left (2 x^{1+m} \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )^{-\frac {2 (1+m)}{\sqrt {-(1+m)^2}}}\right ) \text {Subst}\left (\int x^{-1+\frac {2 (1+m)}{\sqrt {-(1+m)^2}}} \csc ^3(a+2 \log (x)) \, dx,x,c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )}{\sqrt {-(1+m)^2}}\\ &=\frac {\left (16 i e^{3 i a} x^{1+m} \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )^{-\frac {2 (1+m)}{\sqrt {-(1+m)^2}}}\right ) \text {Subst}\left (\int \frac {x^{(-1+6 i)+\frac {2 (1+m)}{\sqrt {-(1+m)^2}}}}{\left (1-e^{2 i a} x^{4 i}\right )^3} \, dx,x,c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )}{\sqrt {-(1+m)^2}}\\ &=-\frac {8 e^{3 i a} x^{1+m} \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )^{6 i} \, _2F_1\left (3,\frac {1}{2} \left (3-\frac {i (1+m)}{\sqrt {-(1+m)^2}}\right );\frac {1}{2} \left (5-\frac {i (1+m)}{\sqrt {-(1+m)^2}}\right );e^{2 i a} \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )^{4 i}\right )}{i+i m-3 \sqrt {-(1+m)^2}}\\ \end {align*}
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Mathematica [A]
time = 2.25, size = 79, normalized size = 0.72 \begin {gather*} \frac {x^{1+m} \left (1+m+\sqrt {-(1+m)^2} \cot \left (a+2 \log \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )\right )\right ) \csc \left (a+2 \log \left (c x^{\frac {1}{2} \sqrt {-(1+m)^2}}\right )\right )}{2 (1+m)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int x^{m} \left (\csc ^{3}\left (a +2 \ln \left (c \,x^{\frac {\sqrt {-\left (1+m \right )^{2}}}{2}}\right )\right )\right )\, dx\]
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. 974 vs.
\(2 (92) = 184\).
time = 0.38, size = 974, normalized size = 8.85 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.08, size = 83, normalized size = 0.75 \begin {gather*} -\frac {2 \, {\left (2 i \, x^{2} x^{2 \, m} e^{\left (3 i \, a + 6 i \, \log \left (c\right )\right )} - i \, e^{\left (5 i \, a + 10 i \, \log \left (c\right )\right )}\right )}}{{\left (m + 1\right )} x^{4} x^{4 \, m} - 2 \, {\left (m + 1\right )} x^{2} x^{2 \, m} e^{\left (2 i \, a + 4 i \, \log \left (c\right )\right )} + {\left (m + 1\right )} e^{\left (4 i \, a + 8 i \, \log \left (c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 5.82, size = 839, normalized size = 7.63 \begin {gather*} \frac {i \, c^{6 i} m x x^{m} x^{{\left | m + 1 \right |}} e^{\left (3 i \, a\right )}}{c^{8 i} m^{2} e^{\left (4 i \, a\right )} + 2 \, c^{8 i} m e^{\left (4 i \, a\right )} + c^{8 i} e^{\left (4 i \, a\right )} - 2 \, c^{4 i} m^{2} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 4 \, c^{4 i} m x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 2 \, c^{4 i} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} + m^{2} x^{4 \, {\left | m + 1 \right |}} + 2 \, m x^{4 \, {\left | m + 1 \right |}} + x^{4 \, {\left | m + 1 \right |}}} - \frac {i \, c^{6 i} x x^{m} x^{{\left | m + 1 \right |}} {\left | m + 1 \right |} e^{\left (3 i \, a\right )}}{c^{8 i} m^{2} e^{\left (4 i \, a\right )} + 2 \, c^{8 i} m e^{\left (4 i \, a\right )} + c^{8 i} e^{\left (4 i \, a\right )} - 2 \, c^{4 i} m^{2} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 4 \, c^{4 i} m x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 2 \, c^{4 i} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} + m^{2} x^{4 \, {\left | m + 1 \right |}} + 2 \, m x^{4 \, {\left | m + 1 \right |}} + x^{4 \, {\left | m + 1 \right |}}} + \frac {i \, c^{6 i} x x^{m} x^{{\left | m + 1 \right |}} e^{\left (3 i \, a\right )}}{c^{8 i} m^{2} e^{\left (4 i \, a\right )} + 2 \, c^{8 i} m e^{\left (4 i \, a\right )} + c^{8 i} e^{\left (4 i \, a\right )} - 2 \, c^{4 i} m^{2} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 4 \, c^{4 i} m x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 2 \, c^{4 i} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} + m^{2} x^{4 \, {\left | m + 1 \right |}} + 2 \, m x^{4 \, {\left | m + 1 \right |}} + x^{4 \, {\left | m + 1 \right |}}} - \frac {i \, c^{2 i} m x x^{m} x^{3 \, {\left | m + 1 \right |}} e^{\left (i \, a\right )}}{c^{8 i} m^{2} e^{\left (4 i \, a\right )} + 2 \, c^{8 i} m e^{\left (4 i \, a\right )} + c^{8 i} e^{\left (4 i \, a\right )} - 2 \, c^{4 i} m^{2} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 4 \, c^{4 i} m x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 2 \, c^{4 i} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} + m^{2} x^{4 \, {\left | m + 1 \right |}} + 2 \, m x^{4 \, {\left | m + 1 \right |}} + x^{4 \, {\left | m + 1 \right |}}} - \frac {i \, c^{2 i} x x^{m} x^{3 \, {\left | m + 1 \right |}} {\left | m + 1 \right |} e^{\left (i \, a\right )}}{c^{8 i} m^{2} e^{\left (4 i \, a\right )} + 2 \, c^{8 i} m e^{\left (4 i \, a\right )} + c^{8 i} e^{\left (4 i \, a\right )} - 2 \, c^{4 i} m^{2} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 4 \, c^{4 i} m x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 2 \, c^{4 i} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} + m^{2} x^{4 \, {\left | m + 1 \right |}} + 2 \, m x^{4 \, {\left | m + 1 \right |}} + x^{4 \, {\left | m + 1 \right |}}} - \frac {i \, c^{2 i} x x^{m} x^{3 \, {\left | m + 1 \right |}} e^{\left (i \, a\right )}}{c^{8 i} m^{2} e^{\left (4 i \, a\right )} + 2 \, c^{8 i} m e^{\left (4 i \, a\right )} + c^{8 i} e^{\left (4 i \, a\right )} - 2 \, c^{4 i} m^{2} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 4 \, c^{4 i} m x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} - 2 \, c^{4 i} x^{2 \, {\left | m + 1 \right |}} e^{\left (2 i \, a\right )} + m^{2} x^{4 \, {\left | m + 1 \right |}} + 2 \, m x^{4 \, {\left | m + 1 \right |}} + x^{4 \, {\left | m + 1 \right |}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.96, size = 171, normalized size = 1.55 \begin {gather*} \frac {\frac {x^{m+1}\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^{\frac {\sqrt {-m^2-2\,m-1}}{2}}\right )}^{6{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}}\,\sqrt {-{\left (m+1\right )}^2}\,1{}\mathrm {i}+m\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}{\sqrt {-{\left (m+1\right )}^2}}+\frac {x^{m+1}\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^{\frac {\sqrt {-m^2-2\,m-1}}{2}}\right )}^{2{}\mathrm {i}}\,\left (m+1-\sqrt {-{\left (m+1\right )}^2}\,1{}\mathrm {i}\right )}{\sqrt {-{\left (m+1\right )}^2}}}{\left (m+1\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^{\frac {\sqrt {-m^2-2\,m-1}}{2}}\right )}^{4{}\mathrm {i}}-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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