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

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

[Out]

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

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Rubi [A]  time = 0.175936, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4930, 4878, 4890, 4886} \[ -\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 a^2 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 a^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c}}{3 a^2}+\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac{2 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{3 a^2 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]

[Out]

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

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 4878

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

Rule 4890

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/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 4886

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

Rubi steps

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

Mathematica [A]  time = 0.580792, size = 260, normalized size = 0.93 \[ \frac{\left (a^2 x^2+1\right ) \sqrt{c \left (a^2 x^2+1\right )} \left (-\frac{4 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\frac{4 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}-\frac{3 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 x^2+1}}+\frac{3 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 x^2+1}}+4 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+2 \cos \left (2 \tan ^{-1}(a x)\right )-\tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+\tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+2\right )}{12 a^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]

[Out]

((1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^
(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTan[
a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*
x])] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])]
)/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/(12*a^2)

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Maple [A]  time = 0.404, size = 198, normalized size = 0.7 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}-\arctan \left ( ax \right ) xa+ \left ( \arctan \left ( ax \right ) \right ) ^{2}+1}{3\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{1}{3\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( \arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\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(x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError