Optimal. Leaf size=625 \[ \frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {3 i \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}-\frac {6 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt {a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.49, antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589} \[ \frac {3 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {a^2 c x^2+c}}+\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac {3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}-\frac {6 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2531
Rule 4181
Rule 4886
Rule 4888
Rule 4890
Rule 4930
Rule 4952
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac {\int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {3 \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}-\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 7.87, size = 812, normalized size = 1.30 \[ \frac {\sqrt {c \left (a^2 x^2+1\right )} \left (-\frac {1}{2} i \tan ^{-1}(a x)^4-2 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3+2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3+\frac {\tan ^{-1}(a x)^3}{\left (\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )^2}-\frac {\tan ^{-1}(a x)^3}{\left (\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )+\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )^2}+i \pi \tan ^{-1}(a x)^3+3 \pi \log \left (1-i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-3 \pi \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-6 i \text {Li}_2\left (-i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-6 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-\frac {6 \sin \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2}{\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )}+\frac {6 \sin \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2}{\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )+\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )}-\frac {3}{4} i \pi ^2 \tan ^{-1}(a x)^2-6 \tan ^{-1}(a x)^2-\frac {3}{2} \pi ^2 \log \left (1-i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+12 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac {3}{2} \pi ^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-12 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+6 i \pi \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-12 \text {Li}_3\left (-i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+12 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac {1}{4} i \pi ^3 \tan ^{-1}(a x)+\frac {1}{4} \pi ^3 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right )-\frac {1}{4} \pi ^3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-\frac {1}{4} \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (2 \tan ^{-1}(a x)+\pi \right )\right )\right )-\frac {3}{2} i \pi \left (\pi -4 \tan ^{-1}(a x)\right ) \text {Li}_2\left (i e^{-i \tan ^{-1}(a x)}\right )-\frac {3}{2} i \pi ^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+12 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-12 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+6 \pi \text {Li}_3\left (i e^{-i \tan ^{-1}(a x)}\right )-6 \pi \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+12 i \text {Li}_4\left (-i e^{-i \tan ^{-1}(a x)}\right )+12 i \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )+\frac {7 i \pi ^4}{32}\right )}{4 a^3 c \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 3.42, size = 430, normalized size = 0.69 \[ \frac {\left (\arctan \left (a x \right ) x a -3\right ) \arctan \left (a x \right )^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c \,a^{3}}+\frac {i \left (i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \polylog \left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^3}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________