3.246 \(\int \frac{\tan ^{-1}(a x)}{x (c+a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=279 \[ \frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{11 a x}{9 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{a x}{9 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\tan ^{-1}(a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

-(a*x)/(9*c*(c + a^2*c*x^2)^(3/2)) - (11*a*x)/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]/(3*c*(c + a^2*c*x^2)^(
3/2)) + ArcTan[a*x]/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[
1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/
(c^2*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^
2*c*x^2])

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Rubi [A]  time = 0.430169, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4966, 4958, 4954, 4930, 191, 192} \[ \frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{11 a x}{9 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{a x}{9 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{\tan ^{-1}(a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)^(5/2)),x]

[Out]

-(a*x)/(9*c*(c + a^2*c*x^2)^(3/2)) - (11*a*x)/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]/(3*c*(c + a^2*c*x^2)^(
3/2)) + ArcTan[a*x]/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[
1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/
(c^2*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*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 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]

Rule 192

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

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\left (a^2 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{\tan ^{-1}(a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{1}{3} a \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac{\int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx}{c^2}-\frac{a^2 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac{a x}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)}{c^2 \sqrt{c+a^2 c x^2}}-\frac{(2 a) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 c}-\frac{a \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}+\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{a x}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{11 a x}{9 c^2 \sqrt{c+a^2 c x^2}}+\frac{\tan ^{-1}(a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\tan ^{-1}(a x)}{c^2 \sqrt{c+a^2 c x^2}}-\frac{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^2 \sqrt{c+a^2 c x^2}}+\frac{i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.366334, size = 168, normalized size = 0.6 \[ \frac{\left (a^2 x^2+1\right )^{3/2} \left (36 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-36 i \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-\frac{45 a x}{\sqrt{a^2 x^2+1}}+\frac{45 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+36 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-36 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )-\sin \left (3 \tan ^{-1}(a x)\right )+3 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )\right )}{36 c \left (c \left (a^2 x^2+1\right )\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)^(5/2)),x]

[Out]

((1 + a^2*x^2)^(3/2)*((-45*a*x)/Sqrt[1 + a^2*x^2] + (45*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + 3*ArcTan[a*x]*Cos[3*A
rcTan[a*x]] + 36*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 36*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (36*I)*P
olyLog[2, -E^(I*ArcTan[a*x])] - (36*I)*PolyLog[2, E^(I*ArcTan[a*x])] - Sin[3*ArcTan[a*x]]))/(36*c*(c*(1 + a^2*
x^2))^(3/2))

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Maple [A]  time = 0.323, size = 370, normalized size = 1.3 \begin{align*} -{\frac{ \left ( i+3\,\arctan \left ( ax \right ) \right ) \left ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{72\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 5\,\arctan \left ( ax \right ) +5\,i \right ) \left ( 1+iax \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -5+5\,iax \right ) \left ( \arctan \left ( ax \right ) -i \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( i{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-3\,iax+1 \right ) \left ( -i+3\,\arctan \left ( ax \right ) \right ) }{72\,{c}^{3} \left ({a}^{4}{x}^{4}+2\,{a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{i}{{c}^{3}} \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/(a^2*c*x^2+c)^(5/2),x)

[Out]

-1/72*(I+3*arctan(a*x))*(I*x^3*a^3+3*a^2*x^2-3*I*a*x-1)*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^2/c^3+5/8*(arcta
n(a*x)+I)*(1+I*a*x)*(c*(a*x-I)*(a*x+I))^(1/2)/c^3/(a^2*x^2+1)-5/8*(c*(a*x-I)*(a*x+I))^(1/2)*(-1+I*a*x)*(arctan
(a*x)-I)/c^3/(a^2*x^2+1)+1/72*(c*(a*x-I)*(a*x+I))^(1/2)*(I*x^3*a^3-3*a^2*x^2-3*I*a*x+1)*(-I+3*arctan(a*x))/c^3
/(a^4*x^4+2*a^2*x^2+1)-I*(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)))*(c*(a*x-I)*(a*x+I)
)^(1/2)/(a^2*x^2+1)^(1/2)/c^3

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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)/x/(a^2*c*x^2+c)^(5/2),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 )}{a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)/(a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)/x/(a**2*c*x**2+c)**(5/2),x)

[Out]

Exception raised: TypeError

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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{5}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(arctan(a*x)/((a^2*c*x^2 + c)^(5/2)*x), x)