\(\int \frac {x^2 \arctan (a x)}{(c+a^2 c x^2)^{3/2}} \, dx\) [233]

Optimal result

Integrand size = 22, antiderivative size = 251 \[ \int \frac {x^2 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {1}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \arctan (a x)}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 c \sqrt {c+a^2 c x^2}} \] Output:

-1/a^3/c/(a^2*c*x^2+c)^(1/2)-x*arctan(a*x)/a^2/c/(a^2*c*x^2+c)^(1/2)-2*I*( 
a^2*x^2+1)^(1/2)*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^3/c 
/(a^2*c*x^2+c)^(1/2)+I*(a^2*x^2+1)^(1/2)*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I 
*a*x)^(1/2))/a^3/c/(a^2*c*x^2+c)^(1/2)-I*(a^2*x^2+1)^(1/2)*polylog(2,I*(1+ 
I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^3/c/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.62 \[ \int \frac {x^2 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1+a^2 x^2} \left (\frac {1}{\sqrt {1+a^2 x^2}}+\frac {a x \arctan (a x)}{\sqrt {1+a^2 x^2}}-\arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )+\arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{a^3 c \sqrt {c \left (1+a^2 x^2\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5469, 5425, 5421}

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.

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

\(\Big \downarrow \) 5469

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5421

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

Input:

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

Output:

-(1/(a^3*c*Sqrt[c + a^2*c*x^2])) - (x*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x 
^2]) + (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))/(a^2*c*Sqrt[c + 
 a^2*c*x^2])
 

Defintions of rubi rules used

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 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 5469

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^2*((d_) + (e_.)*(x_)^2)^(q_), x 
_Symbol] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c^3*d*(q + 1)^2)), x] + (Simp 
[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*c^2*d*(q + 1))), x] - Simp[1 
/(2*c^2*d*(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] && LtQ[q, -1] && NeQ[q, -5/2]
 

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {\left (\arctan \left (a x \right )+i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a^{3} c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )-i\right )}{2 \left (a^{2} x^{2}+1\right ) a^{3} c^{2}}-\frac {\left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{3} c^{2}}\)	\(247\)

Input:

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

Output:

-1/2*(arctan(a*x)+I)*(a*x-I)*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)/a^3/c^2 
-1/2*(c*(a*x-I)*(a*x+I))^(1/2)*(a*x+I)*(arctan(a*x)-I)/(a^2*x^2+1)/a^3/c^2 
-(arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)*ln(1-I*(1+I* 
a*x)/(a^2*x^2+1)^(1/2))+I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1 
+I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^(1/ 
2)/a^3/c^2
 

Fricas [F]

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

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

Output:

integral(sqrt(a^2*c*x^2 + c)*x^2*arctan(a*x)/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 
+ c^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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