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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5072, 267, 5010, 5006} \[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}+\frac {i \sqrt {a^2 x^2+1} \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{a^3 \sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c} \]

[In]

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

[Out]

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

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5006

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]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rule 5010

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[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 5072

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] + (-Dist[b*f*(p/(c*m)), Int[(f*x)^(m - 1
)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[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] && Gt
Q[m, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {\int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{2 a} \\ & = -\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 {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (\sqrt {1+a^2 x^2}-a x \sqrt {1+a^2 x^2} \arctan (a x)+\arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )-\arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{2 a^3 c \sqrt {1+a^2 x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.74

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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