\(\int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^4} \, dx\) [314]

Optimal result

Integrand size = 24, antiderivative size = 275 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^4} \, dx=-\frac {a^2 \sqrt {c+a^2 c x^2}}{3 x}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 c x^3}-\frac {2 a^3 c \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {i a^3 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {i a^3 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5064, 5066, 5082, 270, 5078, 5074} \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^4} \, dx=-\frac {a \arctan (a x) \sqrt {a^2 c x^2+c}}{3 x^2}-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}-\frac {a^2 \sqrt {a^2 c x^2+c}}{3 x}-\frac {2 a^3 c \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}+\frac {i a^3 c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}-\frac {i a^3 c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

-1/3*(a^2*Sqrt[c + a^2*c*x^2])/x - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - ((c + a^2*c*x^2)^(3/2)*ArcTan
[a*x]^2)/(3*c*x^3) - (2*a^3*c*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]) + ((I/3)*a^3*c*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] - ((I/3)*a^3*c*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 5066

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((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])/(f*(m + 2))), x] + (Dist[d/(m + 2), Int[(f*x)^m*((a + b*ArcTan[c*x]
)/Sqrt[d + e*x^2]), x], x] - Dist[b*c*(d/(f*(m + 2))), Int[(f*x)^(m + 1)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]

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 {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 c x^3}+\frac {1}{3} (2 a) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^3} \, dx \\ & = -\frac {2 a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 c x^3}-\frac {1}{3} (2 a c) \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{3} \left (2 a^2 c\right ) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {2 a^2 \sqrt {c+a^2 c x^2}}{3 x}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 c x^3}-\frac {1}{3} \left (a^2 c\right ) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{3} \left (a^3 c\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 x}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 c x^3}+\frac {\left (a^3 c \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 x}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 c x^3}-\frac {2 a^3 c \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 {i a^3 c \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 {i a^3 c \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 = 1.48 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^4} \, dx=-\frac {c \sqrt {1+a^2 x^2} \left (-4 i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+4 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+\sqrt {1+a^2 x^2} \left (4 a^2 x^2+4 \left (1+a^2 x^2\right ) \arctan (a x)^2+\arctan (a x) \left (a x \left (4-3 \sqrt {1+a^2 x^2} \log \left (1-e^{i \arctan (a x)}\right )+3 \sqrt {1+a^2 x^2} \log \left (1+e^{i \arctan (a x)}\right )\right )+\left (1+a^2 x^2\right ) \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right ) \sin (3 \arctan (a x))\right )\right )\right )}{12 x^3 \sqrt {c+a^2 c x^2}} \]

[In]

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

[Out]

-1/12*(c*Sqrt[1 + a^2*x^2]*((-4*I)*a^3*x^3*PolyLog[2, -E^(I*ArcTan[a*x])] + (4*I)*a^3*x^3*PolyLog[2, E^(I*ArcT
an[a*x])] + Sqrt[1 + a^2*x^2]*(4*a^2*x^2 + 4*(1 + a^2*x^2)*ArcTan[a*x]^2 + ArcTan[a*x]*(a*x*(4 - 3*Sqrt[1 + a^
2*x^2]*Log[1 - E^(I*ArcTan[a*x])] + 3*Sqrt[1 + a^2*x^2]*Log[1 + E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)*(Log[1 - E
^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])])*Sin[3*ArcTan[a*x]]))))/(x^3*Sqrt[c + a^2*c*x^2])

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.71

method	result	size
default	\(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (x^{2} \arctan \left (a x \right )^{2} a^{2}+a^{2} x^{2}+x \arctan \left (a x \right ) a +\arctan \left (a x \right )^{2}\right )}{3 x^{3}}+\frac {i a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{3 \sqrt {a^{2} x^{2}+1}}\)	\(195\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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