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

Optimal. Leaf size=240 \[ \frac{i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.348808, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4946, 4962, 264, 4958, 4954} \[ \frac{i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Sqrt[c + a^2*c*x^2])/(2*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) - (a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a
*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((I/2)*a^2*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/2)*a^2*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 4946

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 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]

Rubi steps

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

Mathematica [A]  time = 1.04801, size = 165, normalized size = 0.69 \[ \frac{a^2 \sqrt{c \left (a^2 x^2+1\right )} \left (4 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-4 i \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-2 \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right )+4 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-4 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )-2 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\tan ^{-1}(a x) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+\tan ^{-1}(a x) \sec ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )}{8 \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*Sqrt[c*(1 + a^2*x^2)]*(-2*Cot[ArcTan[a*x]/2] - ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 + 4*ArcTan[a*x]*Log[1 - E
^(I*ArcTan[a*x])] - 4*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (4*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (4*I)*Po
lyLog[2, E^(I*ArcTan[a*x])] + ArcTan[a*x]*Sec[ArcTan[a*x]/2]^2 - 2*Tan[ArcTan[a*x]/2]))/(8*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 0.418, size = 169, normalized size = 0.7 \begin{align*} -{\frac{ax+\arctan \left ( ax \right ) }{2\,{x}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{{\frac{i}{2}}{a}^{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \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 ){\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)*(a^2*c*x^2+c)^(1/2)/x^3,x)

[Out]

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

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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)*(a^2*c*x^2+c)^(1/2)/x^3,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 )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)*(a^2*c*x^2+c)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)*(a^2*c*x^2+c)^(1/2)/x^3,x, algorithm="giac")

[Out]

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