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

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

[Out]

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

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Rubi [A]  time = 0.0904266, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4905, 4904, 3302} \[ \frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{a c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4905

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[(d^(q + 1/2)*Sqrt[1
+ c^2*x^2])/Sqrt[d + e*x^2], Int[(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] && ILtQ[2*(q + 1), 0] &&  !(IntegerQ[q] || GtQ[d, 0])

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

Mathematica [A]  time = 0.166972, size = 39, normalized size = 1. \[ \frac{\sqrt{a^2 c x^2+c} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{a c^2 \sqrt{a^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.287, size = 136, normalized size = 3.5 \begin{align*}{\frac{-{\frac{i}{2}}{\it csgn} \left ( \arctan \left ( ax \right ) \right ){\it csgn} \left ( i\arctan \left ( ax \right ) \right ) \pi }{{c}^{2}a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{{\frac{i}{2}}{\it csgn} \left ( i\arctan \left ( ax \right ) \right ) \pi }{{c}^{2}a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{{\it Ci} \left ( \arctan \left ( ax \right ) \right ) }{{c}^{2}a}\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(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x),x)

[Out]

-1/2*I*csgn(arctan(a*x))*csgn(I*arctan(a*x))*Pi/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)/c^2/a+1/2*I*csgn(I
*arctan(a*x))*Pi/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)/c^2/a+Ci(arctan(a*x))/(a^2*x^2+1)^(1/2)*(c*(a*x-I
)*(a*x+I))^(1/2)/c^2/a

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

[Out]

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

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

[Out]

Integral(1/((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{1}{{\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(1/(a^2*c*x^2+c)^(3/2)/arctan(a*x),x, algorithm="giac")

[Out]

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