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

Optimal. Leaf size=45 \[ \frac{1}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.0248004, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {4894} \[ \frac{1}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0451827, size = 38, normalized size = 0.84 \[ \frac{\sqrt{a^2 c x^2+c} \left (a x \tan ^{-1}(a x)+1\right )}{c^2 \left (a^3 x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.231, size = 98, normalized size = 2.2 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) +i \right ) \left ( ax-i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( ax+i \right ) \left ( \arctan \left ( ax \right ) -i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02519, size = 55, normalized size = 1.22 \begin{align*} \frac{x \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} c} + \frac{1}{\sqrt{a^{2} c x^{2} + c} a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.31497, size = 88, normalized size = 1.96 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x \arctan \left (a x\right ) + 1\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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) + 1)/(a^3*c^2*x^2 + a*c^2)

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

[Out]

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

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Giac [A]  time = 1.38002, size = 55, normalized size = 1.22 \begin{align*} \frac{x \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} c} + \frac{1}{\sqrt{a^{2} c x^{2} + c} a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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