3.338 \(\int \frac{\tan ^{-1}(a x)^2}{x^4 \sqrt{c+a^2 c x^2}} \, dx\)

Optimal. Leaf size=311 \[ -\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c x}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{10 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.627991, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4962, 264, 4958, 4954, 4944} \[ -\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c x}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{10 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4962

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]

Rule 264

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 4958

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 4954

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

Rule 4944

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]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^4 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{1}{3} (2 a) \int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (2 a^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(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{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{3} \left (4 a^3\right ) \int \frac{\tan ^{-1}(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} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x}-\frac{\left (a^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(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{\tan ^{-1}(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} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c x}+\frac{10 a^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\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} \text{Li}_2\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} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 2.60985, size = 228, normalized size = 0.73 \[ \frac{a^3 \sqrt{a^2 c x^2+c} \left (\frac{\left (a^2 x^2+1\right )^{3/2} \left (\frac{20 i a^3 x^3 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\tan ^{-1}(a x) \left (-2 \sin \left (2 \tan ^{-1}(a x)\right )+\frac{5 \left (\log \left (1-e^{i \tan ^{-1}(a x)}\right )-\log \left (1+e^{i \tan ^{-1}(a x)}\right )\right ) \left (\sqrt{a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )-3 a x\right )}{\sqrt{a^2 x^2+1}}\right )+\tan ^{-1}(a x)^2 \left (2-6 \cos \left (2 \tan ^{-1}(a x)\right )\right )+2 \left (\cos \left (2 \tan ^{-1}(a x)\right )-1\right )\right )}{a^3 x^3}-20 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )\right )}{12 c \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[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])

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Maple [A]  time = 0.798, size = 206, normalized size = 0.7 \begin{align*}{\frac{2\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}-{a}^{2}{x}^{2}-\arctan \left ( ax \right ) xa- \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3\,c{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{{\frac{5\,i}{3}}{a}^{3}}{c} \left ( i\arctan \left ( ax \right ) \ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i\arctan \left ( ax \right ) \ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +{\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -{\it polylog} \left ( 2,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{a^{2} c x^{6} + c x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x^{4} \sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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