3.5.23 \(\int (c+a^2 c x^2)^{3/2} \arctan (a x)^3 \, dx\) [423]

3.5.23.1 Optimal result

Integrand size = 21, antiderivative size = 760 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {9 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \]

output

-1/4*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2/a+1/4*x*(a^2*c*x^2+c)^(3/2)*arctan( 
a*x)^3-5*I*c^2*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^ 
2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)+9/4*I*c^2*polylog(4,I*(1+I*a*x)/(a^2*x^2+ 
1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-5/2*I*c^2*polylog(2,I*(1 
+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-9/4 
*I*c^2*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2* 
c*x^2+c)^(1/2)-9/8*I*c^2*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/(a^2*c*x^2+c)^(1/2)+5/2*I*c^2*polylog(2,-I*(1+I* 
a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)-9/4*c^ 
2*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/(a^2*c*x^2+c)^(1/2)+9/4*c^2*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/(a^2*c*x^2+c)^(1/2)-3/4*I*c^2*arctan((1+I*a*x 
)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2) 
+9/8*I*c^2*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/(a^2*c*x^2+c)^(1/2)-1/4*c*(a^2*c*x^2+c)^(1/2)/a+1/4*c*x*arcta 
n(a*x)*(a^2*c*x^2+c)^(1/2)-9/8*c*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a+3/8*c 
*x*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)
 

3.5.23.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2105\) vs. \(2(760)=1520\).

Time = 12.73 (sec) , antiderivative size = 2105, normalized size of antiderivative = 2.77 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\text {Result too large to show} \]

input

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

output

((-1/2*I)*c*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x 
] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan 
[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]^2 
)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I* 
E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] 
+ (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^ 
(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(a*Sqrt[1 + a^2*x^2 
]) + (c*((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2] 
) + (Sqrt[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - 
Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - P 
olyLog[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2 
*x^2)]*(-1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - Ar 
cTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - Arc 
Tan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^( 
I*(Pi/2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^( 
I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(P 
i/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E 
^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] 
+ PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a 
*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a...
 

3.5.23.3 Rubi [A] (verified)

Time = 2.30 (sec) , antiderivative size = 652, normalized size of antiderivative = 0.86, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5415, 5413, 5415, 5425, 5421, 5423, 3042, 4669, 3011, 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.

\(\displaystyle \int \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2} \, dx\)

\(\Big \downarrow \) 5415

\(\displaystyle \frac {1}{2} c \int \sqrt {a^2 c x^2+c} \arctan (a x)dx+\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^3dx+\frac {1}{4} x \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{4 a}\)

\(\Big \downarrow \) 5413

\(\displaystyle \frac {1}{2} c \left (\frac {1}{2} c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^3dx+\frac {1}{4} x \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{4 a}\)

\(\Big \downarrow \) 5415

\(\displaystyle \frac {1}{2} c \left (\frac {1}{2} c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {3}{4} c \left (3 c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} c \int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{4 a}\)

\(\Big \downarrow \) 5425

\(\displaystyle \frac {1}{2} c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {3}{4} c \left (\frac {3 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{4 a}\)

\(\Big \downarrow \) 5421

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {3 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^3 \sqrt {a^2 c x^2+c}-\frac {3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{2} c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )+\frac {1}{4} x \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{4 a}\)

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.5.23.3.1 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 5413

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

rule 5415

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

rule 5421

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

3.5.23.4 Maple [A] (verified)

Time = 3.75 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right )^{3} a^{3} x^{3}-2 x^{2} \arctan \left (a x \right )^{2} a^{2}+5 \arctan \left (a x \right )^{3} a x +2 x \arctan \left (a x \right ) a -11 \arctan \left (a x \right )^{2}-2\right )}{8 a}-\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-9 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+9 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+20 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-20 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-20 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+20 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 a \sqrt {a^{2} x^{2}+1}}\)	\(466\)

input

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

output

1/8*c/a*(c*(a*x-I)*(I+a*x))^(1/2)*(2*arctan(a*x)^3*a^3*x^3-2*x^2*arctan(a* 
x)^2*a^2+5*arctan(a*x)^3*a*x+2*x*arctan(a*x)*a-11*arctan(a*x)^2-2)-1/8*c*( 
c*(a*x-I)*(I+a*x))^(1/2)*(3*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/ 
2))-3*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-9*I*arctan(a*x)^2* 
polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+9*I*arctan(a*x)^2*polylog(2,I*(1 
+I*a*x)/(a^2*x^2+1)^(1/2))+20*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/ 
2))+18*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-20*arctan(a*x 
)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-18*arctan(a*x)*polylog(3,I*(1+I*a*x) 
/(a^2*x^2+1)^(1/2))+18*I*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-18*I*po 
lylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-20*I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1 
)^(1/2))+20*I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a/(a^2*x^2+1)^(1/2)
 

3.5.23.5 Fricas [F]

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

input

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

output

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

3.5.23.6 Sympy [F]

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

input

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

output

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

3.5.23.7 Maxima [F]

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

input

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

output

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

3.5.23.8 Giac [F(-2)]

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

input

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

3.5.23.9 Mupad [F(-1)]

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

input

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

output

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