3.510 \(\int \frac{x}{(c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{a^2 c \sqrt{a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.168539, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4971, 4970, 3299} \[ \frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{a^2 c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4971

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])

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

Mathematica [A]  time = 0.11586, size = 37, normalized size = 0.95 \[ \frac{\left (a^2 x^2+1\right )^{3/2} \text{Si}\left (\tan ^{-1}(a x)\right )}{a^2 \left (c \left (a^2 x^2+1\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.296, size = 82, normalized size = 2.1 \begin{align*} -{\frac{{\it csgn} \left ( \arctan \left ( ax \right ) \right ) \pi }{2\,{c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{{\it Si} \left ( \arctan \left ( ax \right ) \right ) }{{c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*csgn(arctan(a*x))*Pi/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)/c^2/a^2+Si(arctan(a*x))/(a^2*x^2+1)^(1/2
)*(c*(a*x-I)*(a*x+I))^(1/2)/c^2/a^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 )}^{\frac{3}{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)^(3/2)/arctan(a*x),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} x}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{\frac{3}{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)^(3/2)/arctan(a*x),x, algorithm="giac")

[Out]

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