\(\int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx\) [416]

Optimal result

Integrand size = 24, antiderivative size = 600 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\frac {6 i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+\sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \] Output:

6*I*c*(a^2*x^2+1)^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2/ 
(a^2*c*x^2+c)^(1/2)+(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3-2*c*(a^2*x^2+1)^(1/2 
)*arctan(a*x)^3*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+3 
*I*c*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2 
))/(a^2*c*x^2+c)^(1/2)-6*I*c*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,-I*(1 
+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+6*I*c*(a^2*x^2+1)^(1/2)*arc 
tan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-3*I* 
c*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))/( 
a^2*c*x^2+c)^(1/2)-6*c*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(3,-(1+I*a*x)/ 
(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+6*c*(a^2*x^2+1)^(1/2)*polylog(3,-I* 
(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-6*c*(a^2*x^2+1)^(1/2)*pol 
ylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+6*c*(a^2*x^2+1)^ 
(1/2)*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/ 
2)-6*I*c*(a^2*x^2+1)^(1/2)*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c* 
x^2+c)^(1/2)+6*I*c*(a^2*x^2+1)^(1/2)*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2) 
)/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-i \pi ^4+8 \sqrt {1+a^2 x^2} \arctan (a x)^3+2 i \arctan (a x)^4+8 \arctan (a x)^3 \log \left (1-e^{-i \arctan (a x)}\right )-24 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+24 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-8 \arctan (a x)^3 \log \left (1+e^{i \arctan (a x)}\right )+24 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+24 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-48 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+48 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+48 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-48 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-48 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )\right )}{8 \sqrt {1+a^2 x^2}} \] Input:

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

Output:

(Sqrt[c + a^2*c*x^2]*((-I)*Pi^4 + 8*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + (2*I 
)*ArcTan[a*x]^4 + 8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 24*ArcTa 
n[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + 24*ArcTan[a*x]^2*Log[1 + I*E^(I*Ar 
cTan[a*x])] - 8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (24*I)*ArcTan[a 
*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, - 
E^(I*ArcTan[a*x])] - (48*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] 
 + (48*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 48*ArcTan[a*x]*Pol 
yLog[3, E^((-I)*ArcTan[a*x])] - 48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x 
])] + 48*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 48*PolyLog[3, I*E^(I*ArcTan[ 
a*x])] - (48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (48*I)*PolyLog[4, -E^(I 
*ArcTan[a*x])]))/(8*Sqrt[1 + a^2*x^2])
 

Rubi [A] (verified)

Time = 2.99 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.60, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {5485, 5465, 5425, 5423, 3042, 4669, 3011, 2720, 5493, 5491, 3042, 4671, 3011, 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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5491

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

a^2*c*((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^2*c) - (3*Sqrt[1 + a^2*x^2]* 
((-2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog 
[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*A 
rcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x] 
)])))/(a^2*Sqrt[c + a^2*c*x^2])) + (c*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^3* 
ArcTanh[E^(I*ArcTan[a*x])] + 3*(I*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a* 
x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + PolyLog[4, 
 -E^(I*ArcTan[a*x])])) - 3*(I*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])] 
- (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])] + PolyLog[4, E^(I* 
ArcTan[a*x])]))))/Sqrt[c + a^2*c*x^2]
 

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 5465

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

Maple [A] (verified)

Time = 6.24 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.76

method	result	size
default	\(\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{3}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{\sqrt {a^{2} x^{2}+1}}\)	\(454\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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