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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 = 0.05, 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))

________________________________________________________________________________________

fricas [A]  time = 1.19, size = 40, normalized size = 0.89 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x \arctan \left (a x\right ) + 1\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

________________________________________________________________________________________

maple [C]  time = 0.46, size = 98, normalized size = 2.18 \[ \frac {\left (i+\arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2} a}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )-i\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2} a} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 41, normalized size = 0.91 \[ \frac {x \arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c} c} + \frac {1}{\sqrt {a^{2} c x^{2} + c} a c} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

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)

________________________________________________________________________________________