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

Optimal. Leaf size=334 \[ -\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{10 a^2 \sqrt{a^2 c x^2+c}}+\frac{\left (a^2 c x^2+c\right )^{3/2}}{30 a^2}+\frac{3 c \sqrt{a^2 c x^2+c}}{20 a^2}+\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac{x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{10 a}-\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a} \]

[Out]

(3*c*Sqrt[c + a^2*c*x^2])/(20*a^2) + (c + a^2*c*x^2)^(3/2)/(30*a^2) - (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/
(20*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(10*a) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(5*a^2*c) + (((3
*I)/10)*c^2*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]) -
(((3*I)/20)*c^2*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
]) + (((3*I)/20)*c^2*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 = 0.232196, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4930, 4878, 4890, 4886} \[ -\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{10 a^2 \sqrt{a^2 c x^2+c}}+\frac{\left (a^2 c x^2+c\right )^{3/2}}{30 a^2}+\frac{3 c \sqrt{a^2 c x^2+c}}{20 a^2}+\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac{x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{10 a}-\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c*Sqrt[c + a^2*c*x^2])/(20*a^2) + (c + a^2*c*x^2)^(3/2)/(30*a^2) - (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/
(20*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(10*a) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(5*a^2*c) + (((3
*I)/10)*c^2*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]) -
(((3*I)/20)*c^2*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
]) + (((3*I)/20)*c^2*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 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx &=\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac{2 \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx}{5 a}\\ &=\frac{\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac{(3 c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx}{10 a}\\ &=\frac{3 c \sqrt{c+a^2 c x^2}}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a}-\frac{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{20 a}\\ &=\frac{3 c \sqrt{c+a^2 c x^2}}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a}-\frac{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac{\left (3 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{20 a \sqrt{c+a^2 c x^2}}\\ &=\frac{3 c \sqrt{c+a^2 c x^2}}{20 a^2}+\frac{\left (c+a^2 c x^2\right )^{3/2}}{30 a^2}-\frac{3 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a}-\frac{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{10 a}+\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}+\frac{3 i c^2 \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 )}{10 a^2 \sqrt{c+a^2 c x^2}}-\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}+\frac{3 i c^2 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{20 a^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 4.04465, size = 601, normalized size = 1.8 \[ \frac{c \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c} \left (80 \left (-\frac{4 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\frac{4 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}-\frac{3 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 x^2+1}}+\frac{3 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 x^2+1}}+4 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+2 \cos \left (2 \tan ^{-1}(a x)\right )-\tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+\tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+2\right )-\left (a^2 x^2+1\right ) \left (-\frac{176 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}+\frac{176 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}-\frac{110 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 x^2+1}}+\frac{110 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )}{\sqrt{a^2 x^2+1}}-32 \tan ^{-1}(a x)^2+4 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )-22 \tan ^{-1}(a x) \sin \left (4 \tan ^{-1}(a x)\right )+160 \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )+72 \cos \left (2 \tan ^{-1}(a x)\right )+22 \cos \left (4 \tan ^{-1}(a x)\right )-55 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )-11 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (5 \tan ^{-1}(a x)\right )+55 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+11 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (5 \tan ^{-1}(a x)\right )+50\right )\right )}{960 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*(80*(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*Arc
Tan[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*ArcTa
n[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]]) - (1 + a^2*x^2)*(50 - 32*ArcTan[a*x]^2 + 72*Cos[
2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(
I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTa
n[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1
+ a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x
]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*
PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4
*ArcTan[a*x]])))/(960*a^2)

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Maple [A]  time = 0.309, size = 237, normalized size = 0.7 \begin{align*}{\frac{c \left ( 12\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}-6\,\arctan \left ( ax \right ){x}^{3}{a}^{3}+24\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+2\,{a}^{2}{x}^{2}-15\,\arctan \left ( ax \right ) xa+12\, \left ( \arctan \left ( ax \right ) \right ) ^{2}+11 \right ) }{60\,{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{3\,c}{20\,{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*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2,x)

[Out]

1/60*c/a^2*(c*(a*x-I)*(a*x+I))^(1/2)*(12*arctan(a*x)^2*x^4*a^4-6*arctan(a*x)*x^3*a^3+24*arctan(a*x)^2*x^2*a^2+
2*a^2*x^2-15*arctan(a*x)*x*a+12*arctan(a*x)^2+11)+3/20*c*(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*dilog(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{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x \arctan \left (a x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError