Optimal. Leaf size=45 \[ \frac {e^{i a} x^2 \left (c x^i\right )^{2 i}}{\left (1+e^{2 i a} \left (c x^i\right )^{4 i}\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 45, 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 = {4509, 4505, 261} \[ \frac {e^{i a} x^2 \left (c x^i\right )^{2 i}}{\left (1+e^{2 i a} \left (c x^i\right )^{4 i}\right )^2} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4505
Rule 4509
Rubi steps
\begin {align*} \int x \sec ^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 ) \operatorname {Subst}\left (\int x^{-1-2 i} \sec ^3(a+2 \log (x)) \, dx,x,c x^i\right )\right )\\ &=-\left (\left (8 i e^{3 i a} \left (c x^i\right )^{2 i} x^2\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^i\right )\right )\\ &=\frac {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] time = 0.17, size = 127, normalized size = 2.82 \[ -\frac {\sec ^2\left (a+2 \log \left (c x^i\right )\right ) \left (i \left (1-2 x^4\right ) \sin \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )+\left (2 x^4+1\right ) \cos \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right ) \left (i \sin \left (2 \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right )+\cos \left (2 \left (a+2 \log \left (c x^i\right )-2 i \log (x)\right )\right )\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 55, normalized size = 1.22 \[ -\frac {2 \, x^{4} e^{\left (3 i \, a + 6 i \, \log \relax (c)\right )} + e^{\left (5 i \, a + 10 i \, \log \relax (c)\right )}}{x^{8} + 2 \, x^{4} e^{\left (2 i \, a + 4 i \, \log \relax (c)\right )} + e^{\left (4 i \, a + 8 i \, \log \relax (c)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sec \left (a + 2 \, \log \left (c x^{i}\right )\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 209, normalized size = 4.64 \[ \frac {x^{2} c^{2 i} \left (x^{i}\right )^{2 i} {\mathrm e}^{\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{3}-\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i c \right )-\pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i x^{i}\right )+\pi \,\mathrm {csgn}\left (i c \,x^{i}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{i}\right )+i a}}{\left (\left (x^{i}\right )^{4 i} c^{4 i} {\mathrm e}^{2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{3}} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i c \right )} {\mathrm e}^{-2 \pi \mathrm {csgn}\left (i c \,x^{i}\right )^{2} \mathrm {csgn}\left (i x^{i}\right )} {\mathrm e}^{2 \pi \,\mathrm {csgn}\left (i c \,x^{i}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{i}\right )} {\mathrm e}^{2 i a}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 140, normalized size = 3.11 \[ \frac {{\left ({\left (\cos \relax (a) + i \, \sin \relax (a)\right )} \cos \left (2 \, \log \relax (c)\right ) - {\left (-i \, \cos \relax (a) + \sin \relax (a)\right )} \sin \left (2 \, \log \relax (c)\right )\right )} x^{2} e^{\left (6 \, \arctan \left (\sin \left (\log \relax (x)\right ), \cos \left (\log \relax (x)\right )\right )\right )}}{{\left (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \relax (c)\right ) + {\left ({\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \relax (c)\right ) - 2 \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \sin \left (4 \, \log \relax (c)\right )\right )} e^{\left (4 \, \arctan \left (\sin \left (\log \relax (x)\right ), \cos \left (\log \relax (x)\right )\right )\right )} + {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \sin \left (8 \, \log \relax (c)\right ) + e^{\left (8 \, \arctan \left (\sin \left (\log \relax (x)\right ), \cos \left (\log \relax (x)\right )\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.43, size = 46, normalized size = 1.02 \[ \frac {x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{2{}\mathrm {i}}}{2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{4{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^{1{}\mathrm {i}}\right )}^{8{}\mathrm {i}}+1} \]
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 \[ \int x \sec ^{3}{\left (a + 2 \log {\left (c x^{i} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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