\(\int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx\) [338]

Optimal result

Integrand size = 24, antiderivative size = 311 \[ \int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx=-\frac {a^2 \sqrt {c+a^2 c x^2}}{3 c x}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x}+\frac {10 a^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {5 i a^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {5 i a^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}} \]

[Out]

10/3*a^3*arctan(a*x)*arctanh((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5/3*I*a^3*
polylog(2,-(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+5/3*I*a^3*polylog(2,(1+I*a*x
)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-1/3*a^2*(a^2*c*x^2+c)^(1/2)/c/x-1/3*a*arctan(a*
x)*(a^2*c*x^2+c)^(1/2)/c/x^2-1/3*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/c/x^3+2/3*a^2*arctan(a*x)^2*(a^2*c*x^2+c)^(
1/2)/c/x

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5082, 270, 5078, 5074, 5064} \[ \int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\frac {2 a^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x}-\frac {a \arctan (a x) \sqrt {a^2 c x^2+c}}{3 c x^2}-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}-\frac {a^2 \sqrt {a^2 c x^2+c}}{3 c x}+\frac {10 a^3 \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}-\frac {5 i a^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}+\frac {5 i a^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}} \]

[In]

Int[ArcTan[a*x]^2/(x^4*Sqrt[c + a^2*c*x^2]),x]

[Out]

-1/3*(a^2*Sqrt[c + a^2*c*x^2])/(c*x) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c*x^2) - (Sqrt[c + a^2*c*x^2]*Ar
cTan[a*x]^2)/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c*x) + (10*a^3*Sqrt[1 + a^2*x^2]*ArcTan[
a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) - (((5*I)/3)*a^3*Sqrt[1 + a^2*x^2]*Poly
Log[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] + (((5*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2,
Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 5064

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 5074

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[(-2/Sqrt[d])*(a + b
*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/S
qrt[1 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]], x]) /; FreeQ[{a, b, c, d
, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rule 5078

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[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 5082

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] + (-Dist[b*c*(p/(f*(m + 1))), Int[(f*x
)^(m + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[c^2*((m + 2)/(f^2*(m + 1))), 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] && G
tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x^3}+\frac {1}{3} (2 a) \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {\arctan (a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x}+\frac {1}{3} a^2 \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} a^3 \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (4 a^3\right ) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {a^2 \sqrt {c+a^2 c x^2}}{3 c x}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x}-\frac {\left (a^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 a^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 \sqrt {c+a^2 c x^2}}{3 c x}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c x}+\frac {10 a^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {5 i a^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {5 i a^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.24 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.73 \[ \int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\frac {a^3 \sqrt {c+a^2 c x^2} \left (-20 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+\frac {\left (1+a^2 x^2\right )^{3/2} \left (\arctan (a x)^2 (2-6 \cos (2 \arctan (a x)))+2 (-1+\cos (2 \arctan (a x)))+\frac {20 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}+\arctan (a x) \left (-2 \sin (2 \arctan (a x))+\frac {5 \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right ) \left (-3 a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )}{\sqrt {1+a^2 x^2}}\right )\right )}{a^3 x^3}\right )}{12 c \sqrt {1+a^2 x^2}} \]

[In]

Integrate[ArcTan[a*x]^2/(x^4*Sqrt[c + a^2*c*x^2]),x]

[Out]

(a^3*Sqrt[c + a^2*c*x^2]*((-20*I)*PolyLog[2, -E^(I*ArcTan[a*x])] + ((1 + a^2*x^2)^(3/2)*(ArcTan[a*x]^2*(2 - 6*
Cos[2*ArcTan[a*x]]) + 2*(-1 + Cos[2*ArcTan[a*x]]) + ((20*I)*a^3*x^3*PolyLog[2, E^(I*ArcTan[a*x])])/(1 + a^2*x^
2)^(3/2) + ArcTan[a*x]*(-2*Sin[2*ArcTan[a*x]] + (5*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])])*(
-3*a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]))/Sqrt[1 + a^2*x^2])))/(a^3*x^3)))/(12*c*Sqrt[1 + a^2*x^2])

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\left (2 x^{2} \arctan \left (a x \right )^{2} a^{2}-a^{2} x^{2}-x \arctan \left (a x \right ) a -\arctan \left (a x \right )^{2}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 c \,x^{3}}-\frac {5 i a^{3} \left (i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 \sqrt {a^{2} x^{2}+1}\, c}\)	\(206\)

[In]

int(arctan(a*x)^2/x^4/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(2*x^2*arctan(a*x)^2*a^2-a^2*x^2-x*arctan(a*x)*a-arctan(a*x)^2)*(c*(a*x-I)*(I+a*x))^(1/2)/c/x^3-5/3*I*a^3*
(I*arctan(a*x)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-I*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+polylog(2,-(1
+I*a*x)/(a^2*x^2+1)^(1/2))-polylog(2,(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)
/c

Fricas [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c} x^{4}} \,d x } \]

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^2/(a^2*c*x^6 + c*x^4), x)

Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{4} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(atan(a*x)**2/x**4/(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(atan(a*x)**2/(x**4*sqrt(c*(a**2*x**2 + 1))), x)

Maxima [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c} x^{4}} \,d x } \]

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c} x^{4}} \,d x } \]

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

int(atan(a*x)^2/(x^4*(c + a^2*c*x^2)^(1/2)),x)

[Out]

int(atan(a*x)^2/(x^4*(c + a^2*c*x^2)^(1/2)), x)