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 = 0.0427162, antiderivative size = 45, normalized size of antiderivative = 1., 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 4509
Rule 4505
Rule 261
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*}
Mathematica [B] time = 0.151096, 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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Maple [C] time = 0.199, size = 215, normalized size = 4.8 \begin{align*}{\frac{{x}^{2}{{\rm e}^{-i \left ( i\pi \, \left ({\it csgn} \left ( ic{x}^{i} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{i} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{i} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{i} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{i} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{i} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{i} \right ) -a \right ) }}}{ \left ({{\rm e}^{-2\,i \left ( i\pi \, \left ({\it csgn} \left ( ic{x}^{i} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( ic{x}^{i} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{i} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{i} \right ) +i\pi \,{\it csgn} \left ( ic{x}^{i} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{i} \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{i} \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 = 1.11996, size = 189, normalized size = 4.2 \begin{align*} \frac{{\left ({\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \cos \left (2 \, \log \left (c\right )\right ) -{\left (-i \, \cos \left (a\right ) + \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 ) +{\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 (\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.449068, size = 127, normalized size = 2.82 \begin{align*} \frac{x^{2} e^{\left (i \, a + 2 i \, \log \left (c x^{i}\right )\right )}}{e^{\left (4 i \, a + 8 i \, \log \left (c x^{i}\right )\right )} + 2 \, e^{\left (2 i \, a + 4 i \, \log \left (c x^{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 x \sec ^{3}{\left (a + 2 \log{\left (c x^{i} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sec \left (a + 2 \, \log \left (c x^{i}\right )\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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