\(\int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx\) [225]

Optimal result

Integrand size = 22, antiderivative size = 250 \[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {\sqrt {c+a^2 c x^2}}{2 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (\sqrt {1+a^2 x^2}-a x \sqrt {1+a^2 x^2} \arctan (a x)+\arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )-\arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{2 a^3 c \sqrt {1+a^2 x^2}} \] Input:

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

Output:

-1/2*(Sqrt[c*(1 + a^2*x^2)]*(Sqrt[1 + a^2*x^2] - a*x*Sqrt[1 + a^2*x^2]*Arc 
Tan[a*x] + 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[1 + a^2*x^2])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5487, 241, 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.

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5421

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[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 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 5487

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

Maple [A] (verified)

Time = 1.40 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.74

method	result	size
default	\(\frac {\left (\arctan \left (a x \right ) a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c \,a^{3}}+\frac {\left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a^{3} c}\)	\(184\)

Input:

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

Output:

1/2*(arctan(a*x)*a*x-1)*(c*(a*x-I)*(a*x+I))^(1/2)/c/a^3+1/2*(arctan(a*x)*l 
n(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
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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