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

Optimal. Leaf size=78 \[ \frac{2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.110577, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4930, 4894} \[ \frac{2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(a^2*c*Sqrt[c + a^2*c*x^2]) + (2*x*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^2/(a^2*c*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 4894

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b/(c*d*Sqrt[d + e*x^2]),
x] + Simp[(x*(a + b*ArcTan[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rubi steps

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

Mathematica [A]  time = 0.0748785, size = 50, normalized size = 0.64 \[ \frac{\sqrt{a^2 c x^2+c} \left (-\tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+2\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.265, size = 116, normalized size = 1.5 \begin{align*} -{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2+2\,i\arctan \left ( ax \right ) \right ) \left ( 1+iax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -1+iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \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)^(3/2),x)

[Out]

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

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Maxima [A]  time = 2.53735, size = 99, normalized size = 1.27 \begin{align*} \sqrt{c}{\left (\frac{2 \, x \arctan \left (a x\right )}{\sqrt{a^{2} x^{2} + 1} a c^{2}} - \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1} a^{2} c^{2}} + \frac{2}{\sqrt{a^{2} x^{2} + 1} a^{2} c^{2}}\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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.24516, size = 115, normalized size = 1.47 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (2 \, a x \arctan \left (a x\right ) - \arctan \left (a x\right )^{2} + 2\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \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)^(3/2),x, algorithm="fricas")

[Out]

sqrt(a^2*c*x^2 + c)*(2*a*x*arctan(a*x) - arctan(a*x)^2 + 2)/(a^4*c^2*x^2 + a^2*c^2)

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

[Out]

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

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Giac [A]  time = 1.22588, size = 97, normalized size = 1.24 \begin{align*} \frac{2 \, x \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} a^{2} c} + \frac{2}{\sqrt{a^{2} c x^{2} + c} a^{2} c} \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)^(3/2),x, algorithm="giac")

[Out]

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