Optimal. Leaf size=58 \[ \frac{1}{2} x \sec \left (a+2 \log \left (c x^{\frac{i}{2}}\right )\right )-\frac{1}{2} i x \tan \left (a+2 \log \left (c x^{\frac{i}{2}}\right )\right ) \sec \left (a+2 \log \left (c x^{\frac{i}{2}}\right )\right ) \]
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Rubi [A] time = 0.0349121, antiderivative size = 48, normalized size of antiderivative = 0.83, 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, 261} \[ \frac{2 e^{i a} x \left (c x^{\frac{i}{2}}\right )^{2 i}}{\left (1+e^{2 i a} \left (c x^{\frac{i}{2}}\right )^{4 i}\right )^2} \]
Warning: Unable to verify antiderivative.
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Rule 4503
Rule 4505
Rule 261
Rubi steps
\begin{align*} \int \sec ^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} \sec ^3(a+2 \log (x)) \, dx,x,c x^{\frac{i}{2}}\right )\right )\\ &=-\left (\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+4 i}}{\left (1+e^{2 i a} x^{4 i}\right )^3} \, dx,x,c x^{\frac{i}{2}}\right )\right )\\ &=\frac{2 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*}
Mathematica [B] time = 0.113546, size = 137, normalized size = 2.36 \[ -\frac{\sec ^2\left (a+2 \log \left (c x^{\frac{i}{2}}\right )\right ) \left (i \left (1-2 x^2\right ) \sin \left (a+2 \log \left (c x^{\frac{i}{2}}\right )-i \log (x)\right )+\left (2 x^2+1\right ) \cos \left (a+2 \log \left (c x^{\frac{i}{2}}\right )-i \log (x)\right )\right ) \left (i \sin \left (2 \left (a+2 \log \left (c x^{\frac{i}{2}}\right )-i \log (x)\right )\right )+\cos \left (2 \left (a+2 \log \left (c x^{\frac{i}{2}}\right )-i \log (x)\right )\right )\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.204, size = 214, normalized size = 3.7 \begin{align*} 2\,{\frac{x{{\rm e}^{-i \left ( i\pi \, \left ({\it csgn} \left ( ic{x}^{i/2} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{i/2} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{i/2} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{i/2} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{i/2} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{i/2} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{i/2} \right ) -a \right ) }}}{ \left ({{\rm e}^{-2\,i \left ( i\pi \, \left ({\it csgn} \left ( ic{x}^{i/2} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{i/2} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{i/2} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{i/2} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{i/2} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{i/2} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{i/2} \right ) -a \right ) }}+1 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.24468, size = 208, normalized size = 3.59 \begin{align*} \frac{{\left ({\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} \cos \left (2 \, \log \left (c\right )\right ) + 2 \,{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \sin \left (2 \, \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 (\cos \left (4 \, a\right ) + i \, \sin \left (4 \, a\right )\right )} \cos \left (8 \, \log \left (c\right )\right ) +{\left ({\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \cos \left (4 \, \log \left (c\right )\right ) - 2 \,{\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 )} +{\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 (\frac{1}{2} \, \log \left (x\right )\right ), \cos \left (\frac{1}{2} \, \log \left (x\right )\right )\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.449053, size = 151, normalized size = 2.6 \begin{align*} \frac{2 \, x e^{\left (i \, a + 2 i \, \log \left (c x^{\frac{1}{2} i}\right )\right )}}{e^{\left (4 i \, a + 8 i \, \log \left (c x^{\frac{1}{2} i}\right )\right )} + 2 \, e^{\left (2 i \, a + 4 i \, \log \left (c x^{\frac{1}{2} i}\right )\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec ^{3}{\left (a + 2 \log{\left (c x^{\frac{i}{2}} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.75822, size = 100, normalized size = 1.72 \begin{align*} -\frac{2 \, 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 \, 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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