3.484 \(\int \frac{1}{(c+a^2 c x^2)^2 \tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=33 \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a c^2}+\frac{\log \left (\tan ^{-1}(a x)\right )}{2 a c^2} \]

[Out]

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

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Rubi [A]  time = 0.0664349, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4904, 3312, 3302} \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a c^2}+\frac{\log \left (\tan ^{-1}(a x)\right )}{2 a c^2} \]

Antiderivative was successfully verified.

[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 4904

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 3312

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 3302

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]

Rubi steps

\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a c^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a c^2}\\ &=\frac{\log \left (\tan ^{-1}(a x)\right )}{2 a c^2}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a c^2}\\ &=\frac{\text{Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a c^2}+\frac{\log \left (\tan ^{-1}(a x)\right )}{2 a c^2}\\ \end{align*}

Mathematica [A]  time = 0.0269469, size = 23, normalized size = 0.7 \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )+\log \left (\tan ^{-1}(a x)\right )}{2 a c^2} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.058, size = 30, normalized size = 0.9 \begin{align*}{\frac{{\it Ci} \left ( 2\,\arctan \left ( ax \right ) \right ) }{2\,a{c}^{2}}}+{\frac{\ln \left ( \arctan \left ( ax \right ) \right ) }{2\,a{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [C]  time = 1.64666, size = 194, normalized size = 5.88 \begin{align*} \frac{2 \, \log \left (\arctan \left (a x\right )\right ) + \logintegral \left (-\frac{a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + \logintegral \left (-\frac{a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{4 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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