\(\int \frac {\arctan (a x)}{(c+a^2 c x^2)^{3/2}} \, dx\) [235]

Optimal result

Integrand size = 19, antiderivative size = 45 \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {1}{a c \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)}{c \sqrt {c+a^2 c x^2}} \]

[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] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {5014} \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}} \]

[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 5014

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*} \text {integral}& = \frac {1}{a c \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)}{c \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84 \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} (1+a x \arctan (a x))}{c^2 \left (a+a^3 x^2\right )} \]

[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))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.18

method	result	size
default	\(\frac {\left (\arctan \left (a x \right )+i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2} a}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )-i\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2} a}\)	\(98\)

[In]

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

[Out]

1/2*(arctan(a*x)+I)*(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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89 \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (a x \arctan \left (a x\right ) + 1\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \]

[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)

Sympy [F]

\[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arctan \left (a x\right )}{\sqrt {a^{2} c x^{2} + c} c} + \frac {1}{\sqrt {a^{2} c x^{2} + c} a c} \]

[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)

Giac [F]

\[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[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)