Optimal. Leaf size=58 \[ \frac {1}{2} x \csc \left (a+2 \log \left (c x^{\frac {i}{2}}\right )\right )+\frac {1}{2} i x \cot \left (a+2 \log \left (c x^{\frac {i}{2}}\right )\right ) \csc \left (a+2 \log \left (c x^{\frac {i}{2}}\right )\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 51, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4504, 4506, 261} \[ -\frac {2 i e^{i a} x \left (c x^{\frac {i}{2}}\right )^{2 i}}{\left (1-e^{2 i a} \left (c x^{\frac {i}{2}}\right )^{4 i}\right )^2} \]
Warning: Unable to verify antiderivative.
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Rule 261
Rule 4504
Rule 4506
Rubi steps
\begin {align*} \int \csc ^3\left (a+2 \log \left (c x^{\frac {i}{2}}\right )\right ) \, dx &=-\left (\left (2 i \left (c x^{\frac {i}{2}}\right )^{2 i} x\right ) \operatorname {Subst}\left (\int x^{-1-2 i} \csc ^3(a+2 \log (x)) \, dx,x,c x^{\frac {i}{2}}\right )\right )\\ &=\left (16 e^{3 i a} \left (c x^{\frac {i}{2}}\right )^{2 i} x\right ) \operatorname {Subst}\left (\int \frac {x^{-1+4 i}}{\left (1-e^{2 i a} x^{4 i}\right )^3} \, dx,x,c x^{\frac {i}{2}}\right )\\ &=-\frac {2 i e^{i a} \left (c x^{\frac {i}{2}}\right )^{2 i} x}{\left (1-e^{2 i a} \left (c x^{\frac {i}{2}}\right )^{4 i}\right )^2}\\ \end {align*}
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Mathematica [B] time = 0.17, size = 137, normalized size = 2.36 \[ \frac {\csc ^2\left (a+2 \log \left (c x^{\frac {i}{2}}\right )\right ) \left (\left (2 x^2+1\right ) \sin \left (a+2 \log \left (c x^{\frac {i}{2}}\right )-i \log (x)\right )+i \left (2 x^2-1\right ) \cos \left (a+2 \log \left (c x^{\frac {i}{2}}\right )-i \log (x)\right )\right ) \left (i \sin \left (2 \left (a+2 \log \left (c x^{\frac {i}{2}}\right )-i \log (x)\right )\right )+\cos \left (2 \left (a+2 \log \left (c x^{\frac {i}{2}}\right )-i \log (x)\right )\right )\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 56, normalized size = 0.97 \[ \frac {-4 i \, x^{2} e^{\left (3 i \, a + 6 i \, \log \relax (c)\right )} + 2 i \, e^{\left (5 i \, a + 10 i \, \log \relax (c)\right )}}{x^{4} - 2 \, x^{2} e^{\left (2 i \, a + 4 i \, \log \relax (c)\right )} + e^{\left (4 i \, a + 8 i \, \log \relax (c)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.66, size = 74, normalized size = 1.28 \[ \frac {2 i \, c^{10 i} e^{\left (5 i \, a\right )}}{c^{8 i} e^{\left (4 i \, a\right )} - 2 \, c^{4 i} x^{2} e^{\left (2 i \, a\right )} + x^{4}} - \frac {4 i \, c^{6 i} x^{2} e^{\left (3 i \, a\right )}}{c^{8 i} e^{\left (4 i \, a\right )} - 2 \, c^{4 i} x^{2} e^{\left (2 i \, a\right )} + x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 209, normalized size = 3.60 \[ -\frac {2 i x \,c^{2 i} \left (x^{\frac {i}{2}}\right )^{2 i} {\mathrm e}^{\pi \mathrm {csgn}\left (i c \,x^{\frac {i}{2}}\right )^{3}-\pi \mathrm {csgn}\left (i c \,x^{\frac {i}{2}}\right )^{2} \mathrm {csgn}\left (i c \right )-\pi \mathrm {csgn}\left (i c \,x^{\frac {i}{2}}\right )^{2} \mathrm {csgn}\left (i x^{\frac {i}{2}}\right )+\pi \,\mathrm {csgn}\left (i c \,x^{\frac {i}{2}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{\frac {i}{2}}\right )+i a}}{\left (\left (x^{\frac {i}{2}}\right )^{4 i} c^{4 i} {\mathrm e}^{2 \pi \mathrm {csgn}\left (i c \,x^{\frac {i}{2}}\right )^{3}} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{\frac {i}{2}}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{\frac {i}{2}}\right )^{2} \mathrm {csgn}\left (i x^{\frac {i}{2}}\right )} {\mathrm e}^{2 \pi \,\mathrm {csgn}\left (i c \,x^{\frac {i}{2}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{\frac {i}{2}}\right )} {\mathrm e}^{2 i a}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 159, normalized size = 2.74 \[ -\frac {{\left (2 \, {\left (i \, \cos \relax (a) - \sin \relax (a)\right )} \cos \left (2 \, \log \relax (c)\right ) - {\left (2 \, \cos \relax (a) + 2 i \, \sin \relax (a)\right )} \sin \left (2 \, \log \relax (c)\right )\right )} x e^{\left (6 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \relax (x)\right ), \cos \left (\frac {1}{2} \, \log \relax (x)\right )\right )\right )}}{{\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \relax (c)\right ) - {\left ({\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \relax (c)\right ) + 2 \, {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \sin \left (4 \, \log \relax (c)\right )\right )} e^{\left (4 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \relax (x)\right ), \cos \left (\frac {1}{2} \, \log \relax (x)\right )\right )\right )} + {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \sin \left (8 \, \log \relax (c)\right ) + e^{\left (8 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \relax (x)\right ), \cos \left (\frac {1}{2} \, \log \relax (x)\right )\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 55, normalized size = 0.95 \[ -\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^{\frac {1}{2}{}\mathrm {i}}\right )}^{2{}\mathrm {i}}\,2{}\mathrm {i}}{1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^{\frac {1}{2}{}\mathrm {i}}\right )}^{8{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^{\frac {1}{2}{}\mathrm {i}}\right )}^{4{}\mathrm {i}}} \]
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 \[ \int \csc ^{3}{\left (a + 2 \log {\left (c x^{\frac {i}{2}} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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