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

Optimal. Leaf size=298 \[ \frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 c x^2+c}}{8 a}-\frac{\left (a^2 c x^2+c\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]

[Out]

(-3*c*Sqrt[c + a^2*c*x^2])/(8*a) - (c + a^2*c*x^2)^(3/2)/(12*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/8 +
(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/
Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])
/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/S
qrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.138201, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4878, 4890, 4886} \[ \frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{8 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 c x^2+c}}{8 a}-\frac{\left (a^2 c x^2+c\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*c*Sqrt[c + a^2*c*x^2])/(8*a) - (c + a^2*c*x^2)^(3/2)/(12*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/8 +
(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/
Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])
/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/S
qrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])

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 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx &=-\frac{\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{4} (3 c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac{3 c \sqrt{c+a^2 c x^2}}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac{1}{8} \left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{3 c \sqrt{c+a^2 c x^2}}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\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}{8 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 c \sqrt{c+a^2 c x^2}}{8 a}-\frac{\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\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 )}{4 a \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 )}{8 a \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 )}{8 a \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 2.58904, size = 351, normalized size = 1.18 \[ \frac{c \sqrt{a^2 c x^2+c} \left (72 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-72 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+2 \left (a^2 x^2+1\right )^{3/2}+96 \sqrt{a^2 x^2+1} \left (a x \tan ^{-1}(a x)-1\right )+6 \left (a^2 x^2+1\right )^2 \cos \left (3 \tan ^{-1}(a x)\right )-3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \left (-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+2 \sin \left (3 \tan ^{-1}(a x)\right )+4 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+\left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )\right )+96 \tan ^{-1}(a x) \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )\right )}{192 a \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[c + a^2*c*x^2]*(2*(1 + a^2*x^2)^(3/2) + 96*Sqrt[1 + a^2*x^2]*(-1 + a*x*ArcTan[a*x]) + 6*(1 + a^2*x^2)^
2*Cos[3*ArcTan[a*x]] + 96*ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + (72*I)*P
olyLog[2, (-I)*E^(I*ArcTan[a*x])] - (72*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - 3*(1 + a^2*x^2)^2*ArcTan[a*x]*((-
14*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])
] - Log[1 + I*E^(I*ArcTan[a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a
*x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]])))/(192*a*Sqrt[1 + a^2*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.292, size = 201, normalized size = 0.7 \begin{align*}{\frac{c \left ( 6\,\arctan \left ( ax \right ){x}^{3}{a}^{3}-2\,{a}^{2}{x}^{2}+15\,\arctan \left ( ax \right ) xa-11 \right ) }{24\,a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{3\,c}{8\,a}\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((a^2*c*x^2+c)^(3/2)*arctan(a*x),x)

[Out]

1/24*c/a*(c*(a*x-I)*(a*x+I))^(1/2)*(6*arctan(a*x)*x^3*a^3-2*a^2*x^2+15*arctan(a*x)*x*a-11)-3/8*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/(a^2*x^2+1)^(1/2)

________________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError