\(\int \frac {x}{(c+a^2 c x^2)^{3/2} \arctan (a x)} \, dx\) [510]

Optimal result

Integrand size = 22, antiderivative size = 39 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a^2 c \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5091, 5090, 3380} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

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

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 5090

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c^(m
+ 1), Subst[Int[(a + b*x)^p*(Sin[x]^m/Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d,
e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 5091

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^(q + 1
/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]), Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] &&  !(IntegerQ[q] || GtQ[d, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^2 c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a^2 c \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a^2 c \sqrt {c \left (1+a^2 x^2\right )}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.62 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.10

method	result	size
default	\(-\frac {\operatorname {csgn}\left (\arctan \left (a x \right )\right ) \pi \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a^{2} c^{2}}+\frac {\operatorname {Si}\left (\arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{2} c^{2}}\)	\(82\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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