\(\int \frac {1}{(c+a^2 c x^2)^2 \arctan (a x)} \, dx\) [484]

Optimal result

Integrand size = 19, antiderivative size = 33 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx=\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{2 a c^2}+\frac {\log (\arctan (a x))}{2 a c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5024, 3393, 3383} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx=\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{2 a c^2}+\frac {\log (\arctan (a x))}{2 a c^2} \]

[In]

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

[Out]

CosIntegral[2*ArcTan[a*x]]/(2*a*c^2) + Log[ArcTan[a*x]]/(2*a*c^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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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])

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx=\frac {\operatorname {CosIntegral}(2 \arctan (a x))+\log (\arctan (a x))}{2 a c^2} \]

[In]

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

[Out]

(CosIntegral[2*ArcTan[a*x]] + Log[ArcTan[a*x]])/(2*a*c^2)

Maple [A] (verified)

Time = 3.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(\frac {\ln \left (\arctan \left (a x \right )\right )+\operatorname {Ci}\left (2 \arctan \left (a x \right )\right )}{2 a \,c^{2}}\)	\(22\)
default	\(\frac {\ln \left (\arctan \left (a x \right )\right )+\operatorname {Ci}\left (2 \arctan \left (a x \right )\right )}{2 a \,c^{2}}\)	\(22\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx=\frac {2 \, \log \left (\arctan \left (a x\right )\right ) + \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{4 \, a c^{2}} \]

[In]

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

[Out]

1/4*(2*log(arctan(a*x)) + log_integral(-(a^2*x^2 + 2*I*a*x - 1)/(a^2*x^2 + 1)) + log_integral(-(a^2*x^2 - 2*I*
a*x - 1)/(a^2*x^2 + 1)))/(a*c^2)

Sympy [F]

\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {atan}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}{\left (a x \right )} + \operatorname {atan}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

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

[Out]

Integral(1/(a**4*x**4*atan(a*x) + 2*a**2*x**2*atan(a*x) + atan(a*x)), x)/c**2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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