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

Optimal. Leaf size=56 \[ -\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )}{\sqrt{c}} \]

[Out]

-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c]

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Rubi [A]  time = 0.0926184, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4944, 266, 63, 208} \[ -\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c]

Rule 4944

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0921398, size = 62, normalized size = 1.11 \[ \frac{a \left (\log (x)-\log \left (\sqrt{c} \sqrt{a^2 c x^2+c}+c\right )\right )}{\sqrt{c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.369, size = 139, normalized size = 2.5 \begin{align*} -{\frac{\arctan \left ( ax \right ) }{cx}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{a}{c}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{a}{c}\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-1 \right ) \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(arctan(a*x)/x^2/(a^2*c*x^2+c)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.39954, size = 163, normalized size = 2.91 \begin{align*} \frac{a \sqrt{c} x \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{a^{2} c x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )}{2 \, c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*sqrt(c)*x*log(-(a^2*c*x^2 - 2*sqrt(a^2*c*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*sqrt(a^2*c*x^2 + c)*arctan(a*
x))/(c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23342, size = 136, normalized size = 2.43 \begin{align*}{\left (\frac{a \arctan \left (x{\left | a \right |}\right )}{\sqrt{c}{\left | a \right |}} - \frac{2 \, \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}}\right )}{\left | a \right |} + \frac{2 \, \sqrt{c}{\left | a \right |} \arctan \left (a x\right )}{{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} + c}\right )}^{2} - c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*arctan(x*abs(a))/(sqrt(c)*abs(a)) - 2*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c))*abs
(a) + 2*sqrt(c)*abs(a)*arctan(a*x)/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 + c))^2 - c)