Optimal. Leaf size=109 \[ -\frac{i x \sec ^2(x) \text{PolyLog}\left (2,e^{2 i x}\right )}{\sqrt{a \sec ^4(x)}}+\frac{\sec ^2(x) \text{PolyLog}\left (3,e^{2 i x}\right )}{2 \sqrt{a \sec ^4(x)}}-\frac{i x^3 \sec ^2(x)}{3 \sqrt{a \sec ^4(x)}}+\frac{x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.57421, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6720, 3717, 2190, 2531, 2282, 6589} \[ -\frac{i x \sec ^2(x) \text{PolyLog}\left (2,e^{2 i x}\right )}{\sqrt{a \sec ^4(x)}}+\frac{\sec ^2(x) \text{PolyLog}\left (3,e^{2 i x}\right )}{2 \sqrt{a \sec ^4(x)}}-\frac{i x^3 \sec ^2(x)}{3 \sqrt{a \sec ^4(x)}}+\frac{x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6720
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^2 \csc (x) \sec (x)}{\sqrt{a \sec ^4(x)}} \, dx &=\frac{\sec ^2(x) \int x^2 \cot (x) \, dx}{\sqrt{a \sec ^4(x)}}\\ &=-\frac{i x^3 \sec ^2(x)}{3 \sqrt{a \sec ^4(x)}}-\frac{\left (2 i \sec ^2(x)\right ) \int \frac{e^{2 i x} x^2}{1-e^{2 i x}} \, dx}{\sqrt{a \sec ^4(x)}}\\ &=-\frac{i x^3 \sec ^2(x)}{3 \sqrt{a \sec ^4(x)}}+\frac{x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}-\frac{\left (2 \sec ^2(x)\right ) \int x \log \left (1-e^{2 i x}\right ) \, dx}{\sqrt{a \sec ^4(x)}}\\ &=-\frac{i x^3 \sec ^2(x)}{3 \sqrt{a \sec ^4(x)}}+\frac{x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}-\frac{i x \text{Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}+\frac{\left (i \sec ^2(x)\right ) \int \text{Li}_2\left (e^{2 i x}\right ) \, dx}{\sqrt{a \sec ^4(x)}}\\ &=-\frac{i x^3 \sec ^2(x)}{3 \sqrt{a \sec ^4(x)}}+\frac{x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}-\frac{i x \text{Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}+\frac{\sec ^2(x) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i x}\right )}{2 \sqrt{a \sec ^4(x)}}\\ &=-\frac{i x^3 \sec ^2(x)}{3 \sqrt{a \sec ^4(x)}}+\frac{x^2 \log \left (1-e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}-\frac{i x \text{Li}_2\left (e^{2 i x}\right ) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}+\frac{\text{Li}_3\left (e^{2 i x}\right ) \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0595842, size = 75, normalized size = 0.69 \[ \frac{\sec ^2(x) \left (24 i x \text{PolyLog}\left (2,e^{-2 i x}\right )+12 \text{PolyLog}\left (3,e^{-2 i x}\right )+8 i x^3+24 x^2 \log \left (1-e^{-2 i x}\right )-i \pi ^3\right )}{24 \sqrt{a \sec ^4(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.067, size = 183, normalized size = 1.7 \begin{align*}{\frac{{\frac{i}{3}}{{\rm e}^{2\,ix}}{x}^{3}}{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,ix}}}{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{4}}}}}}}-2\,{\frac{i/3{{\rm e}^{2\,ix}}{x}^{3}-1/2\,{{\rm e}^{2\,ix}}{x}^{2}\ln \left ({{\rm e}^{ix}}+1 \right ) +i{{\rm e}^{2\,ix}}x{\it polylog} \left ( 2,-{{\rm e}^{ix}} \right ) -{{\rm e}^{2\,ix}}{\it polylog} \left ( 3,-{{\rm e}^{ix}} \right ) -1/2\,{{\rm e}^{2\,ix}}{x}^{2}\ln \left ( 1-{{\rm e}^{ix}} \right ) +i{{\rm e}^{2\,ix}}x{\it polylog} \left ( 2,{{\rm e}^{ix}} \right ) -{{\rm e}^{2\,ix}}{\it polylog} \left ( 3,{{\rm e}^{ix}} \right ) }{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,ix}}}{ \left ( 1+{{\rm e}^{2\,ix}} \right ) ^{4}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.53651, size = 153, normalized size = 1.4 \begin{align*} \frac{-2 i \, x^{3} + 6 i \, x^{2} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 6 i \, x^{2} \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + 3 \, x^{2} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + 3 \, x^{2} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 12 i \, x{\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 12 i \, x{\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + 12 \,{\rm Li}_{3}(-e^{\left (i \, x\right )}) + 12 \,{\rm Li}_{3}(e^{\left (i \, x\right )})}{6 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 2.68834, size = 821, normalized size = 7.53 \begin{align*} \frac{2 \, \sqrt{\frac{a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2}{\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 2 \, \sqrt{\frac{a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2}{\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, \sqrt{\frac{a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2}{\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 2 \, \sqrt{\frac{a}{\cos \left (x\right )^{4}}} \cos \left (x\right )^{2}{\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) +{\left (x^{2} \cos \left (x\right )^{2} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x^{2} \cos \left (x\right )^{2} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + x^{2} \cos \left (x\right )^{2} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x^{2} \cos \left (x\right )^{2} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - 2 i \, x \cos \left (x\right )^{2}{\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 2 i \, x \cos \left (x\right )^{2}{\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 i \, x \cos \left (x\right )^{2}{\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 2 i \, x \cos \left (x\right )^{2}{\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right )\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4}}}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \csc{\left (x \right )} \sec{\left (x \right )}}{\sqrt{a \sec ^{4}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \csc \left (x\right ) \sec \left (x\right )}{\sqrt{a \sec \left (x\right )^{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]