3.238 \(\int \frac{\tan ^{-1}(a x)}{x^3 (c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=300 \[ -\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 c \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 c \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 c^2 x^2}+\frac{a^3 x}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{a^2 c x^2+c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.614256, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4966, 4962, 264, 4958, 4954, 4930, 191} \[ -\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 c \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 c \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 c^2 x^2}+\frac{a^3 x}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^(3/2)),x]

[Out]

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

Rule 4966

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

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 4930

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

Rule 191

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

Rubi steps

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

Mathematica [A]  time = 1.23818, size = 258, normalized size = 0.86 \[ -\frac{a^2 \left (12 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-12 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+2 \sqrt{a^2 x^2+1} \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right )+12 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-12 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )+\sqrt{a^2 x^2+1} \tan ^{-1}(a x) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-\sqrt{a^2 x^2+1} \tan ^{-1}(a x) \sec ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-8 a x+8 \tan ^{-1}(a x)+a x \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )}{8 c \sqrt{a^2 c x^2+c}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^(3/2)),x]

[Out]

-(a^2*(-8*a*x + 8*ArcTan[a*x] + a*x*Csc[ArcTan[a*x]/2]^2 + Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2
+ 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + E^(I*
ArcTan[a*x])] + (12*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (12*I)*Sqrt[1 + a^2*x^2]*PolyLog[2,
E^(I*ArcTan[a*x])] - Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Sec[ArcTan[a*x]/2]^2 + 2*Sqrt[1 + a^2*x^2]*Tan[ArcTan[a*x]/
2]))/(8*c*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.367, size = 273, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2} \left ( \arctan \left ( ax \right ) +i \right ) \left ( 1+iax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -1+iax \right ) \left ( \arctan \left ( ax \right ) -i \right ){a}^{2}}{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ax+\arctan \left ( ax \right ) }{2\,{c}^{2}{x}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{{\frac{3\,i}{2}}{a}^{2}}{{c}^{2}} \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)/x^3/(a^2*c*x^2+c)^(3/2),x)

[Out]

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

________________________________________________________________________________________

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)/x^3/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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 )}{a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x^3/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)/(a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atan}{\left (a x \right )}}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)/x**3/(a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(atan(a*x)/(x**3*(c*(a**2*x**2 + 1))**(3/2)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x^3/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(arctan(a*x)/((a^2*c*x^2 + c)^(3/2)*x^3), x)