3.5.17 \(\int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx\) [417]

3.5.17.1 Optimal result

Integrand size = 24, antiderivative size = 622 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx=-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {2 i a c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {6 a c \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]

output

-2*I*a*c*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a^2*x^2+1)^(1/ 
2)/(a^2*c*x^2+c)^(1/2)-6*a*c*arctan(a*x)^2*arctanh((1+I*a*x)/(a^2*x^2+1)^( 
1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*I*a*c*arctan(a*x)*polylog(2, 
-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+3*I*a* 
c*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2 
)/(a^2*c*x^2+c)^(1/2)-3*I*a*c*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2 
+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*I*a*c*arctan(a*x)*polyl 
og(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6* 
a*c*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c 
)^(1/2)-6*a*c*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x 
^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*a*c*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a 
^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*a*c*polylog(3,(1+ 
I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*I*a*c*po 
lylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1 
/2)+6*I*a*c*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^ 
2*c*x^2+c)^(1/2)-arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x
 

3.5.17.2 Mathematica [A] (warning: unable to verify)

Time = 2.67 (sec) , antiderivative size = 768, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx=\frac {a \sqrt {c+a^2 c x^2} \csc \left (\frac {1}{2} \arctan (a x)\right ) \left (-7 i a \pi ^4 x-8 i a \pi ^3 x \arctan (a x)+24 i a \pi ^2 x \arctan (a x)^2-32 i a \pi x \arctan (a x)^3-64 \sqrt {1+a^2 x^2} \arctan (a x)^3+16 i a x \arctan (a x)^4+48 a \pi ^2 x \arctan (a x) \log \left (1-i e^{-i \arctan (a x)}\right )-96 a \pi x \arctan (a x)^2 \log \left (1-i e^{-i \arctan (a x)}\right )-8 a \pi ^3 x \log \left (1+i e^{-i \arctan (a x)}\right )+64 a x \arctan (a x)^3 \log \left (1+i e^{-i \arctan (a x)}\right )+192 a x \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )+8 a \pi ^3 x \log \left (1+i e^{i \arctan (a x)}\right )-48 a \pi ^2 x \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )+96 a \pi x \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-64 a x \arctan (a x)^3 \log \left (1+i e^{i \arctan (a x)}\right )-192 a x \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )+8 a \pi ^3 x \log \left (\tan \left (\frac {1}{4} (\pi +2 \arctan (a x))\right )\right )+192 i a x \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{-i \arctan (a x)}\right )+48 i a \pi x (\pi -4 \arctan (a x)) \operatorname {PolyLog}\left (2,i e^{-i \arctan (a x)}\right )+384 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+48 i a \pi ^2 x \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-192 i a \pi x \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+192 i a x \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-384 i a x \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+384 a x \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{-i \arctan (a x)}\right )-192 a \pi x \operatorname {PolyLog}\left (3,i e^{-i \arctan (a x)}\right )-384 a x \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+192 a \pi x \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-384 a x \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+384 a x \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )-384 i a x \operatorname {PolyLog}\left (4,-i e^{-i \arctan (a x)}\right )-384 i a x \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )}{128 \left (1+a^2 x^2\right )} \]

input

Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^2,x]
 

output

(a*Sqrt[c + a^2*c*x^2]*Csc[ArcTan[a*x]/2]*((-7*I)*a*Pi^4*x - (8*I)*a*Pi^3* 
x*ArcTan[a*x] + (24*I)*a*Pi^2*x*ArcTan[a*x]^2 - (32*I)*a*Pi*x*ArcTan[a*x]^ 
3 - 64*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + (16*I)*a*x*ArcTan[a*x]^4 + 48*a*P 
i^2*x*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])] - 96*a*Pi*x*ArcTan[a*x]^2*L 
og[1 - I/E^(I*ArcTan[a*x])] - 8*a*Pi^3*x*Log[1 + I/E^(I*ArcTan[a*x])] + 64 
*a*x*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] + 192*a*x*ArcTan[a*x]^2*Lo 
g[1 - E^(I*ArcTan[a*x])] + 8*a*Pi^3*x*Log[1 + I*E^(I*ArcTan[a*x])] - 48*a* 
Pi^2*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + 96*a*Pi*x*ArcTan[a*x]^2* 
Log[1 + I*E^(I*ArcTan[a*x])] - 64*a*x*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[ 
a*x])] - 192*a*x*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*a*Pi^3*x*Log 
[Tan[(Pi + 2*ArcTan[a*x])/4]] + (192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)/ 
E^(I*ArcTan[a*x])] + (48*I)*a*Pi*x*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I* 
ArcTan[a*x])] + (384*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + ( 
48*I)*a*Pi^2*x*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*a*Pi*x*ArcTan[ 
a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (192*I)*a*x*ArcTan[a*x]^2*PolyLo 
g[2, (-I)*E^(I*ArcTan[a*x])] - (384*I)*a*x*ArcTan[a*x]*PolyLog[2, E^(I*Arc 
Tan[a*x])] + 384*a*x*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])] - 192* 
a*Pi*x*PolyLog[3, I/E^(I*ArcTan[a*x])] - 384*a*x*PolyLog[3, -E^(I*ArcTan[a 
*x])] + 192*a*Pi*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 384*a*x*ArcTan[a*x 
]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 384*a*x*PolyLog[3, E^(I*ArcTan[a...
 

3.5.17.3 Rubi [A] (verified)

Time = 2.54 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.58, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {5485, 5425, 5423, 3042, 4669, 3011, 5479, 5493, 5491, 3042, 4671, 3011, 2720, 7143, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{x^2} \, dx\)

\(\Big \downarrow \) 5485

\(\displaystyle a^2 c \int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx\)

\(\Big \downarrow \) 5425

\(\displaystyle \frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx\)

\(\Big \downarrow \) 5423

\(\displaystyle \frac {a c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle c \int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 4669

\(\displaystyle c \int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3011

\(\displaystyle c \int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5479

\(\displaystyle c \left (3 a \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5493

\(\displaystyle c \left (\frac {3 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5491

\(\displaystyle c \left (\frac {3 a \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3042

\(\displaystyle c \left (\frac {3 a \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 4671

\(\displaystyle c \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 3011

\(\displaystyle c \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 2720

\(\displaystyle c \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle c \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{\sqrt {a^2 c x^2+c}}\)

input

Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^2,x]
 

output

c*(-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c*x)) + (3*a*Sqrt[1 + a^2*x^2]*( 
-2*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])] + 2*(I*ArcTan[a*x]*PolyLog[2, 
-E^(I*ArcTan[a*x])] - PolyLog[3, -E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*P 
olyLog[2, E^(I*ArcTan[a*x])] - PolyLog[3, E^(I*ArcTan[a*x])])))/Sqrt[c + a 
^2*c*x^2]) + (a*c*Sqrt[1 + a^2*x^2]*((-2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcT 
an[a*x]^3 + 3*(I*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (2*I)* 
((-I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + PolyLog[4, (-I)*E^( 
I*ArcTan[a*x])])) - 3*(I*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])] - ( 
2*I)*((-I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + PolyLog[4, I*E^(I 
*ArcTan[a*x])]))))/Sqrt[c + a^2*c*x^2]
 

3.5.17.3.1 Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5423

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[1/(c*Sqrt[d])   Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ 
c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt 
Q[d, 0]
 

rule 5425

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]   Int[(a + b*ArcTan[c*x])^ 
p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] & 
& IGtQ[p, 0] &&  !GtQ[d, 0]
 

rule 5479

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + 
 b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1)))   Int[(f*x) 
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, 
c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
&& NeQ[m, -1]
 

rule 5485

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(q_.), x_Symbol] :> Simp[d   Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + 
 b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2)   Int[(f*x)^(m + 2)*(d + e*x^2 
)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] 
&& EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] 
&& IntegerQ[q]))
 

rule 5491

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2] 
), x_Symbol] :> Simp[1/Sqrt[d]   Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcTa 
n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && 
GtQ[d, 0]
 

rule 5493

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 
]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]   Int[(a + b*ArcTan 
[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[ 
e, c^2*d] && IGtQ[p, 0] &&  !GtQ[d, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

3.5.17.4 Maple [A] (verified)

Time = 5.53 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{3}}{x}+\frac {i a \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3 i \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{\sqrt {a^{2} x^{2}+1}}\)	\(466\)

input

int(arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x^2,x,method=_RETURNVERBOSE)
 

output

-(c*(a*x-I)*(I+a*x))^(1/2)*arctan(a*x)^3/x+I*a*(c*(a*x-I)*(I+a*x))^(1/2)*( 
I*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*arctan(a*x)^3*ln(1-I 
*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^( 
1/2)+1)-3*I*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*arctan(a*x)^ 
2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*polylog(2,I*(1 
+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2 
+1)^(1/2))-6*I*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*arct 
an(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*arctan(a*x)*polylog(2,(1 
+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I 
*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(4,-I*(1+I*a*x)/(a^2*x^2+ 
1)^(1/2))+6*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)
 

3.5.17.5 Fricas [F]

\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]

input

integrate(arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x^2,x, algorithm="fricas")
 

output

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^3/x^2, x)
 

3.5.17.6 Sympy [F]

\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx \]

input

integrate(atan(a*x)**3*(a**2*c*x**2+c)**(1/2)/x**2,x)
 

output

Integral(sqrt(c*(a**2*x**2 + 1))*atan(a*x)**3/x**2, x)
 

3.5.17.7 Maxima [F]

\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \]

input

integrate(arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x^2,x, algorithm="maxima")
 

output

integrate(sqrt(a^2*c*x^2 + c)*arctan(a*x)^3/x^2, x)
 

3.5.17.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx=\text {Exception raised: TypeError} \]

input

integrate(arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x^2,x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.5.17.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c}}{x^2} \,d x \]

input

int((atan(a*x)^3*(c + a^2*c*x^2)^(1/2))/x^2,x)
 

output

int((atan(a*x)^3*(c + a^2*c*x^2)^(1/2))/x^2, x)