Integrand size = 17, antiderivative size = 48 \[ \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx=\frac {2 e^{3 i a} \left (c x^{-\frac {i}{2}}\right )^{6 i} x}{\left (1+e^{2 i a} \left (c x^{-\frac {i}{2}}\right )^{4 i}\right )^2} \]
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Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4599, 4601, 270} \[ \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx=\frac {2 e^{3 i a} x \left (c x^{-\frac {i}{2}}\right )^{6 i}}{\left (1+e^{2 i a} \left (c x^{-\frac {i}{2}}\right )^{4 i}\right )^2} \]
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Rule 270
Rule 4599
Rule 4601
Rubi steps \begin{align*} \text {integral}& = \left (2 i \left (c x^{-\frac {i}{2}}\right )^{-2 i} x\right ) \text {Subst}\left (\int x^{-1+2 i} \sec ^3(a+2 \log (x)) \, dx,x,c x^{-\frac {i}{2}}\right ) \\ & = \left (16 i e^{3 i a} \left (c x^{-\frac {i}{2}}\right )^{-2 i} x\right ) \text {Subst}\left (\int \frac {x^{-1+8 i}}{\left (1+e^{2 i a} x^{4 i}\right )^3} \, dx,x,c x^{-\frac {i}{2}}\right ) \\ & = \frac {2 e^{3 i a} \left (c x^{-\frac {i}{2}}\right )^{6 i} x}{\left (1+e^{2 i a} \left (c x^{-\frac {i}{2}}\right )^{4 i}\right )^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(139\) vs. \(2(48)=96\).
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.90 \[ \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx=\frac {\sec ^2\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \left (\left (1+2 x^2\right ) \cos \left (a+2 \log \left (c x^{-\frac {i}{2}}\right )+i \log (x)\right )+i \left (-1+2 x^2\right ) \sin \left (a+2 \log \left (c x^{-\frac {i}{2}}\right )+i \log (x)\right )\right ) \left (-2 \cos \left (2 \left (a+2 \log \left (c x^{-\frac {i}{2}}\right )+i \log (x)\right )\right )+2 i \sin \left (2 \left (a+2 \log \left (c x^{-\frac {i}{2}}\right )+i \log (x)\right )\right )\right )}{4 x^2} \]
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Time = 272.79 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02
method | result | size |
parallelrisch | \(\frac {x \left (i \sin \left (a +2 \ln \left (c \,x^{-\frac {i}{2}}\right )\right )+\cos \left (a +2 \ln \left (c \,x^{-\frac {i}{2}}\right )\right )\right )}{\cos \left (2 a +4 \ln \left (c \,x^{-\frac {i}{2}}\right )\right )+1}\) | \(49\) |
risch | \(\frac {2 x \,c^{6 i} \left (x^{\frac {i}{2}}\right )^{-6 i} {\mathrm e}^{-3 \pi \,\operatorname {csgn}\left (i x^{-\frac {i}{2}}\right ) \operatorname {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{2}+3 \pi \,\operatorname {csgn}\left (i x^{-\frac {i}{2}}\right ) \operatorname {csgn}\left (i c \,x^{-\frac {i}{2}}\right ) \operatorname {csgn}\left (i c \right )+3 \pi \operatorname {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{3}-3 \pi \operatorname {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{2} \operatorname {csgn}\left (i c \right )+3 i a}}{\left (c^{4 i} \left (x^{\frac {i}{2}}\right )^{-4 i} {\mathrm e}^{-2 \pi \,\operatorname {csgn}\left (i x^{-\frac {i}{2}}\right ) \operatorname {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{2}} {\mathrm e}^{2 \pi \,\operatorname {csgn}\left (i x^{-\frac {i}{2}}\right ) \operatorname {csgn}\left (i c \,x^{-\frac {i}{2}}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 \pi \operatorname {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{3}} {\mathrm e}^{-2 \pi \operatorname {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )^{2}}\) | \(238\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx=-\frac {2 \, {\left (2 \, x^{2} e^{\left (2 i \, a + 4 i \, \log \left (c\right )\right )} + 1\right )}}{x^{4} e^{\left (5 i \, a + 10 i \, \log \left (c\right )\right )} + 2 \, x^{2} e^{\left (3 i \, a + 6 i \, \log \left (c\right )\right )} + e^{\left (i \, a + 2 i \, \log \left (c\right )\right )}} \]
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\[ \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx=\int \sec ^{3}{\left (a + 2 \log {\left (c x^{- \frac {i}{2}} \right )} \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 3.38 \[ \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx=\frac {2 \, {\left ({\left (\cos \left (3 \, a\right ) + i \, \sin \left (3 \, a\right )\right )} \cos \left (6 \, \log \left (c\right )\right ) + {\left (i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \sin \left (6 \, \log \left (c\right )\right )\right )} x e^{\left (6 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \left (x\right )\right ), \cos \left (\frac {1}{2} \, \log \left (x\right )\right )\right )\right )}}{{\left ({\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \left (c\right )\right ) - {\left (-i \, \cos \left (4 \, a\right ) + \sin \left (4 \, a\right )\right )} \sin \left (8 \, \log \left (c\right )\right )\right )} e^{\left (8 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \left (x\right )\right ), \cos \left (\frac {1}{2} \, \log \left (x\right )\right )\right )\right )} + 2 \, {\left ({\left (\cos \left (2 \, a\right ) + i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \left (c\right )\right ) + {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \sin \left (4 \, \log \left (c\right )\right )\right )} e^{\left (4 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \left (x\right )\right ), \cos \left (\frac {1}{2} \, \log \left (x\right )\right )\right )\right )} + 1} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).
Time = 1.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.73 \[ \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx=-\frac {4 \, c^{4 i} x^{2} e^{\left (2 i \, a\right )}}{c^{10 i} x^{4} e^{\left (5 i \, a\right )} + 2 \, c^{6 i} x^{2} e^{\left (3 i \, a\right )} + c^{2 i} e^{\left (i \, a\right )}} - \frac {2}{c^{10 i} x^{4} e^{\left (5 i \, a\right )} + 2 \, c^{6 i} x^{2} e^{\left (3 i \, a\right )} + c^{2 i} e^{\left (i \, a\right )}} \]
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Time = 31.77 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx=\frac {2\,x\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (\frac {c}{x^{\frac {1}{2}{}\mathrm {i}}}\right )}^{6{}\mathrm {i}}}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (\frac {c}{x^{\frac {1}{2}{}\mathrm {i}}}\right )}^{4{}\mathrm {i}}+1\right )}^2} \]
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