Optimal. Leaf size=477 \[ -\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}-\frac{9 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^2 \sqrt{a^2 c x^2+c}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{2 a^2}-\frac{c x \sqrt{a^2 c x^2+c}}{20 a}+\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{3 x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}-\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{40 a}+\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}+\frac{9 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.416141, antiderivative size = 477, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4930, 4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195} \[ -\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}-\frac{9 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^2 \sqrt{a^2 c x^2+c}}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{2 a^2}-\frac{c x \sqrt{a^2 c x^2+c}}{20 a}+\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{3 x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}-\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{40 a}+\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}+\frac{9 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4930
Rule 4880
Rule 4890
Rule 4888
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 217
Rule 206
Rule 195
Rubi steps
\begin{align*} \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx &=\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{3 \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx}{5 a}\\ &=\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}-\frac{3 x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{c \int \sqrt{c+a^2 c x^2} \, dx}{10 a}-\frac{(9 c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx}{20 a}\\ &=-\frac{c x \sqrt{c+a^2 c x^2}}{20 a}+\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}-\frac{9 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{40 a}-\frac{3 x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{20 a}-\frac{\left (9 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{40 a}-\frac{\left (9 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{20 a}\\ &=-\frac{c x \sqrt{c+a^2 c x^2}}{20 a}+\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}-\frac{9 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{40 a}-\frac{3 x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{20 a}-\frac{\left (9 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{20 a}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{40 a \sqrt{c+a^2 c x^2}}\\ &=-\frac{c x \sqrt{c+a^2 c x^2}}{20 a}+\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}-\frac{9 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{40 a}-\frac{3 x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^2}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{40 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{c x \sqrt{c+a^2 c x^2}}{20 a}+\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}-\frac{9 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{40 a}-\frac{3 x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^2}+\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 a^2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{c x \sqrt{c+a^2 c x^2}}{20 a}+\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}-\frac{9 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{40 a}-\frac{3 x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^2}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 a^2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{c x \sqrt{c+a^2 c x^2}}{20 a}+\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}-\frac{9 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{40 a}-\frac{3 x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^2}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{c x \sqrt{c+a^2 c x^2}}{20 a}+\frac{9 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a^2}-\frac{9 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{40 a}-\frac{3 x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{20 a}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^2}-\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{9 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{9 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}-\frac{9 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 3.85049, size = 441, normalized size = 0.92 \[ \frac{c \sqrt{a^2 c x^2+c} \left (960 \left (-i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+\text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-\text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-\tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+48 \left (11 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-11 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-11 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+11 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+10 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-11 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+\left (a^2 x^2+1\right )^{5/2} \left (-\left (\frac{48 a x}{\left (a^2 x^2+1\right )^2}+\tan ^{-1}(a x)^2 \left (6 \sin \left (2 \tan ^{-1}(a x)\right )-33 \sin \left (4 \tan ^{-1}(a x)\right )\right )+32 \tan ^{-1}(a x)^3 \left (5 \cos \left (2 \tan ^{-1}(a x)\right )-1\right )+6 \tan ^{-1}(a x) \left (36 \cos \left (2 \tan ^{-1}(a x)\right )+11 \cos \left (4 \tan ^{-1}(a x)\right )+25\right )\right )\right )+80 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \left (4 \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+6 \cos \left (2 \tan ^{-1}(a x)\right )+6\right )\right )}{960 a^2 \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.543, size = 421, normalized size = 0.9 \begin{align*}{\frac{c \left ( 8\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{4}{a}^{4}-6\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+16\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}+4\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-15\, \left ( \arctan \left ( ax \right ) \right ) ^{2}xa+8\, \left ( \arctan \left ( ax \right ) \right ) ^{3}-2\,ax+22\,\arctan \left ( ax \right ) \right ) }{40\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{3\,c}{40\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}-3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{3\,c}{40\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}+6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{ic}{{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x \arctan \left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{3} + c x\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}, x\right ) \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 x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}^{3}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]