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

Optimal. Leaf size=49 \[ \frac{x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.0553146, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4930, 191} \[ \frac{x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]/(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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0495469, size = 42, normalized size = 0.86 \[ \frac{\sqrt{a^2 c x^2+c} \left (a x-\tan ^{-1}(a x)\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.266, size = 100, normalized size = 2. \begin{align*} -{\frac{ \left ( \arctan \left ( ax \right ) +i \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 ( \arctan \left ( ax \right ) -i \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)/(a^2*c*x^2+c)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.74233, size = 38, normalized size = 0.78 \begin{align*} \frac{a x - \arctan \left (a x\right )}{\sqrt{a^{2} x^{2} + 1} a^{2} c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x - arctan(a*x))/(sqrt(a^2*x^2 + 1)*a^2*c^(3/2))

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Fricas [A]  time = 2.2285, size = 88, normalized size = 1.8 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x - \arctan \left (a x\right )\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)/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

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

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.3689, size = 61, normalized size = 1.24 \begin{align*} \frac{x}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )}{\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)/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

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