\(\int \frac {1}{(c+a^2 c x^2)^{3/2} \arctan (a x)^2} \, dx\) [582]

Optimal result

Integrand size = 21, antiderivative size = 69 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=-\frac {1}{a c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a c \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5022, 5091, 5090, 3380} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}} \]

[In]

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

[Out]

-(1/(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(a*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 5022

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

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx=-\frac {1+\sqrt {1+a^2 x^2} \arctan (a x) \text {Si}(\arctan (a x))}{a c \sqrt {c+a^2 c x^2} \arctan (a x)} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 5.79 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.86

method	result	size
default	\(-\frac {i \left (\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}-\arctan \left (a x \right ) \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )-2 i \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right ) a \,c^{2}}\)	\(128\)

[In]

int(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

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

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2,x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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