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

Optimal. Leaf size=304 \[ \frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{a c \sqrt{a^2 c x^2+c}}{2 x}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2} \]

[Out]

-(a*c*Sqrt[c + a^2*c*x^2])/(2*x) + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])
/(2*x^2) - (3*a^2*c^2*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] - a^2*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (((3*I)/2)*a^2*c^2*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] - (((3*I)/2)*a^2*c^2*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 = 0.641778, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4950, 4946, 4962, 264, 4958, 4954, 217, 206} \[ \frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{a c \sqrt{a^2 c x^2+c}}{2 x}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*c*Sqrt[c + a^2*c*x^2])/(2*x) + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])
/(2*x^2) - (3*a^2*c^2*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] - a^2*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (((3*I)/2)*a^2*c^2*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] - (((3*I)/2)*a^2*c^2*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 4950

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

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]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x^3} \, dx &=c \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx+\left (a^2 c\right ) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx\\ &=a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^2}-c^2 \int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx+\left (a c^2\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\left (a^3 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{a c \sqrt{c+a^2 c x^2}}{x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}-\frac{1}{2} \left (a c^2\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{2} \left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (a^2 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{a c \sqrt{c+a^2 c x^2}}{2 x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}-\frac{2 a^2 c^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 )}{\sqrt{c+a^2 c x^2}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^2 c^2 \sqrt{1+a^2 x^2} \text{Li}_2\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^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (a^2 c^2 \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 c \sqrt{c+a^2 c x^2}}{2 x}+a^2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}-\frac{3 a^2 c^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 )}{\sqrt{c+a^2 c x^2}}-a^2 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{3 i a^2 c^2 \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{3 i a^2 c^2 \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 = 1.60369, size = 301, normalized size = 0.99 \[ \frac{a^2 c \sqrt{a^2 c x^2+c} \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \left (12 i \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-12 i \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )-2 \cot ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+4 a x \tan ^{-1}(a x) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+12 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-12 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\tan ^{-1}(a x) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+\tan ^{-1}(a x) \csc \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \sec \left (\frac{1}{2} \tan ^{-1}(a x)\right )+8 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \log \left (\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-8 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \log \left (\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )+\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-2\right )}{8 \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*c*Sqrt[c + a^2*c*x^2]*(-2 - 2*Cot[ArcTan[a*x]/2]^2 + 4*a*x*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - ArcTan[a*x]
*Cot[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 12*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 12*A
rcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + 8*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] - Sin[A
rcTan[a*x]/2]] - 8*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (12*I)*Cot[ArcTan[a*x]/2]
*PolyLog[2, -E^(I*ArcTan[a*x])] - (12*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] + ArcTan[a*x]*Csc[Ar
cTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 0.326, size = 180, normalized size = 0.6 \begin{align*}{\frac{c \left ( 2\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-ax-\arctan \left ( ax \right ) \right ) }{2\,{x}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{a}^{2}c}{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 3\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -4\,i\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\,i{\it dilog} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\,i{\it dilog} \left ( 1+{(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((a^2*c*x^2+c)^(3/2)*arctan(a*x)/x^3,x)

[Out]

1/2*c*(c*(a*x-I)*(a*x+I))^(1/2)*(2*arctan(a*x)*a^2*x^2-a*x-arctan(a*x))/x^2-1/2*a^2*c*(c*(a*x-I)*(a*x+I))^(1/2
)*(3*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*dilog((1+I*a*x)
/(a^2*x^2+1)^(1/2))-3*I*dilog(1+(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((a^2*c*x^2+c)^(3/2)*arctan(a*x)/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{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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