Optimal. Leaf size=49 \[ -\frac {i e^{i a} \left (c x^i\right )^{2 i} x^2}{\left (1-e^{2 i a} \left (c x^i\right )^{4 i}\right )^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {4606, 4602,
267} \begin {gather*} -\frac {i e^{i a} x^2 \left (c x^i\right )^{2 i}}{\left (1-e^{2 i a} \left (c x^i\right )^{4 i}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4602
Rule 4606
Rubi steps
\begin {align*} \int x \csc ^3\left (a+2 \log \left (c x^i\right )\right ) \, dx &=-\left (\left (i \left (c x^i\right )^{2 i} x^2\right ) \text {Subst}\left (\int x^{-1-2 i} \csc ^3(a+2 \log (x)) \, dx,x,c x^i\right )\right )\\ &=\left (8 e^{3 i a} \left (c x^i\right )^{2 i} x^2\right ) \text {Subst}\left (\int \frac {x^{-1+4 i}}{\left (1-e^{2 i a} x^{4 i}\right )^3} \, dx,x,c x^i\right )\\ &=-\frac {i e^{i a} \left (c x^i\right )^{2 i} x^2}{\left (1-e^{2 i a} \left (c x^i\right )^{4 i}\right )^2}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(127\) vs. \(2(49)=98\).
time = 0.20, size = 127, normalized size = 2.59 \begin {gather*} \frac {\csc ^2\left (a+2 \log \left (c x^i\right )\right ) \left (i \left (-1+2 x^4\right ) \cos \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )+\left (1+2 x^4\right ) \sin \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right ) \left (\cos \left (2 \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right )+i \sin \left (2 \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right )\right )}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.23, size = 211, normalized size = 4.31
method | result | size |
risch | \(-\frac {i x^{2} \left (x^{i}\right )^{2 i} c^{2 i} {\mathrm e}^{\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{3}-\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i c \right )-\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i x^{i}\right )+\pi \,\mathrm {csgn}\left (i c \,x^{i}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{i}\right )+i a}}{\left (\left (x^{i}\right )^{4 i} c^{4 i} {\mathrm e}^{2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{3}} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i x^{i}\right )} {\mathrm e}^{2 \pi \,\mathrm {csgn}\left (i c \,x^{i}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{i}\right )} {\mathrm e}^{2 i a}-1\right )^{2}}\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 139 vs. \(2 (32) = 64\).
time = 0.31, size = 139, normalized size = 2.84 \begin {gather*} \frac {{\left ({\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \cos \left (2 \, \log \left (c\right )\right ) + {\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \sin \left (2 \, \log \left (c\right )\right )\right )} x^{2} e^{\left (6 \, \arctan \left (\sin \left (\log \left (x\right )\right ), \cos \left (\log \left (x\right )\right )\right )\right )}}{{\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \left (c\right )\right ) - 2 \, {\left ({\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \left (c\right )\right ) + {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \sin \left (4 \, \log \left (c\right )\right )\right )} e^{\left (4 \, \arctan \left (\sin \left (\log \left (x\right )\right ), \cos \left (\log \left (x\right )\right )\right )\right )} + {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \sin \left (8 \, \log \left (c\right )\right ) + e^{\left (8 \, \arctan \left (\sin \left (\log \left (x\right )\right ), \cos \left (\log \left (x\right )\right )\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.83, size = 56, normalized size = 1.14 \begin {gather*} \frac {-2 i \, x^{4} 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 )}}{x^{8} - 2 \, x^{4} e^{\left (2 i \, a + 4 i \, \log \left (c\right )\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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \csc ^{3}{\left (a + 2 \log {\left (c x^{i} \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.41, size = 45, normalized size = 0.92 \begin {gather*} -\frac {x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{2{}\mathrm {i}}\,1{}\mathrm {i}}{1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{8{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{4{}\mathrm {i}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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