Optimal. Leaf size=48 \[ \frac {2 e^{3 i a} x \left (c x^{-\frac {i}{2}}\right )^{6 i}}{\left (1+e^{2 i a} \left (c x^{-\frac {i}{2}}\right )^{4 i}\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 48, 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 = {4503, 4505, 264} \[ \frac {2 e^{3 i a} x \left (c x^{-\frac {i}{2}}\right )^{6 i}}{\left (1+e^{2 i a} \left (c x^{-\frac {i}{2}}\right )^{4 i}\right )^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 4503
Rule 4505
Rubi steps
\begin {align*} \int \sec ^3\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \, dx &=\left (2 i \left (c x^{-\frac {i}{2}}\right )^{-2 i} x\right ) \operatorname {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 ) \operatorname {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*}
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Mathematica [B] time = 0.17, size = 139, normalized size = 2.90 \[ \frac {\sec ^2\left (a+2 \log \left (c x^{-\frac {i}{2}}\right )\right ) \left (i \left (2 x^2-1\right ) \sin \left (a+2 \log \left (c x^{-\frac {i}{2}}\right )+i \log (x)\right )+\left (2 x^2+1\right ) \cos \left (a+2 \log \left (c x^{-\frac {i}{2}}\right )+i \log (x)\right )\right ) \left (2 i \sin \left (2 \left (a+2 \log \left (c x^{-\frac {i}{2}}\right )+i \log (x)\right )\right )-2 \cos \left (2 \left (a+2 \log \left (c x^{-\frac {i}{2}}\right )+i \log (x)\right )\right )\right )}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 57, normalized size = 1.19 \[ -\frac {2 \, {\left (2 \, x^{2} e^{\left (2 i \, a + 4 i \, \log \relax (c)\right )} + 1\right )}}{x^{4} e^{\left (5 i \, a + 10 i \, \log \relax (c)\right )} + 2 \, x^{2} e^{\left (3 i \, a + 6 i \, \log \relax (c)\right )} + e^{\left (i \, a + 2 i \, \log \relax (c)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.16, size = 83, normalized size = 1.73 \[ -\frac {4 \, c^{4 i} x^{2} e^{\left (2 i \, a\right )}}{c^{10 i} x^{4} e^{\left (5 i \, a\right )} + 2 \, c^{6 i} x^{2} e^{\left (3 i \, a\right )} + c^{2 i} e^{\left (i \, a\right )}} - \frac {2}{c^{10 i} x^{4} e^{\left (5 i \, a\right )} + 2 \, c^{6 i} x^{2} e^{\left (3 i \, a\right )} + c^{2 i} e^{\left (i \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 238, normalized size = 4.96 \[ \frac {2 x \left (x^{\frac {i}{2}}\right )^{-6 i} c^{6 i} {\mathrm e}^{3 \pi \mathrm {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{3}-3 \pi \mathrm {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{2} \mathrm {csgn}\left (i c \right )-3 \pi \mathrm {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{2} \mathrm {csgn}\left (i x^{-\frac {i}{2}}\right )+3 \pi \,\mathrm {csgn}\left (i c \,x^{-\frac {i}{2}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{-\frac {i}{2}}\right )+3 i a}}{\left (\left (x^{\frac {i}{2}}\right )^{-4 i} c^{4 i} {\mathrm e}^{2 \pi \mathrm {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{3}} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{-\frac {i}{2}}\right )^{2} \mathrm {csgn}\left (i x^{-\frac {i}{2}}\right )} {\mathrm e}^{2 \pi \,\mathrm {csgn}\left (i c \,x^{-\frac {i}{2}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{-\frac {i}{2}}\right )} {\mathrm e}^{2 i a}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 166, normalized size = 3.46 \[ \frac {{\left ({\left (2 \, \cos \left (3 \, a\right ) + 2 i \, \sin \left (3 \, a\right )\right )} \cos \left (6 \, \log \relax (c)\right ) + 2 \, {\left (i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \sin \left (6 \, \log \relax (c)\right )\right )} x e^{\left (6 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \relax (x)\right ), \cos \left (\frac {1}{2} \, \log \relax (x)\right )\right )\right )}}{{\left ({\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \relax (c)\right ) - {\left (-i \, \cos \left (4 \, a\right ) + \sin \left (4 \, a\right )\right )} \sin \left (8 \, \log \relax (c)\right )\right )} e^{\left (8 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \relax (x)\right ), \cos \left (\frac {1}{2} \, \log \relax (x)\right )\right )\right )} + {\left ({\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \relax (c)\right ) + 2 \, {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \sin \left (4 \, \log \relax (c)\right )\right )} e^{\left (4 \, \arctan \left (\sin \left (\frac {1}{2} \, \log \relax (x)\right ), \cos \left (\frac {1}{2} \, \log \relax (x)\right )\right )\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.28, size = 39, normalized size = 0.81 \[ \frac {2\,x\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (\frac {c}{x^{\frac {1}{2}{}\mathrm {i}}}\right )}^{6{}\mathrm {i}}}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (\frac {c}{x^{\frac {1}{2}{}\mathrm {i}}}\right )}^{4{}\mathrm {i}}+1\right )}^2} \]
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 \[ \int \sec ^{3}{\left (a + 2 \log {\left (c x^{- \frac {i}{2}} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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