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

Optimal. Leaf size=17 \[ \frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0709479, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4970, 4406, 12, 3299} \[ \frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4970

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 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3299

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]

Rubi steps

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

Mathematica [A]  time = 0.0380637, size = 17, normalized size = 1. \[ \frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.055, size = 16, normalized size = 0.9 \begin{align*}{\frac{{\it Si} \left ( 2\,\arctan \left ( ax \right ) \right ) }{2\,{a}^{2}{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\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(x/(a^2*c*x^2+c)^2/arctan(a*x),x, algorithm="maxima")

[Out]

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

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Fricas [C]  time = 1.75612, size = 174, normalized size = 10.24 \begin{align*} \frac{i \, \logintegral \left (-\frac{a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - i \, \logintegral \left (-\frac{a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{4 \, a^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{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(x/(a**2*c*x**2+c)**2/atan(a*x),x)

[Out]

Integral(x/(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{x}{{\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(x/(a^2*c*x^2+c)^2/arctan(a*x),x, algorithm="giac")

[Out]

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