\(\int \frac {x}{(c+a^2 c x^2)^{3/2} \arctan (a x)^2} \, dx\) [581]

Optimal result

Integrand size = 22, antiderivative size = 69 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=-\frac {x}{a c \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {c+a^2 c x^2}} \]

[Out]

-x/a/c/arctan(a*x)/(a^2*c*x^2+c)^(1/2)+Ci(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 = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5062, 5025, 5024, 3383} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

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

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 5024

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

Rule 5025

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^(q + 1/2)*(Sqrt[1
+ c^2*x^2]/Sqrt[d + e*x^2]), Int[(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] && ILtQ[2*(q + 1), 0] &&  !(IntegerQ[q] || GtQ[d, 0])

Rule 5062

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[
(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Dist[f*(m/(b*c*(p + 1))), Int[
(f*x)^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e
, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=\frac {-a x+\sqrt {1+a^2 x^2} \arctan (a x) \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 6.16 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {\left (\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}+\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+2 \sqrt {a^{2} x^{2}+1}\, a x +\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right ) a^{2} c^{2}}\)	\(126\)

[In]

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

[Out]

-1/2*(arctan(a*x)*Ei(1,-I*arctan(a*x))*a^2*x^2+arctan(a*x)*Ei(1,I*arctan(a*x))*a^2*x^2+2*(a^2*x^2+1)^(1/2)*a*x
+Ei(1,-I*arctan(a*x))*arctan(a*x)+Ei(1,I*arctan(a*x))*arctan(a*x))/(a^2*x^2+1)^(3/2)*(c*(a*x-I)*(I+a*x))^(1/2)
/arctan(a*x)/a^2/c^2

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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