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

Optimal. Leaf size=59 \[ \frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^2 c}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{a^2 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0576372, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4930, 217, 206} \[ \frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^2 c}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{a^2 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4930

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0679329, size = 60, normalized size = 1.02 \[ \frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\sqrt{c} \log \left (\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right )}{a^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.431, size = 144, normalized size = 2.4 \begin{align*}{\frac{\arctan \left ( ax \right ) }{{a}^{2}c}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{1}{{a}^{2}c}\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+i \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{1}{{a}^{2}c}\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-i \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(x*arctan(a*x)/(a^2*c*x^2+c)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.77494, size = 82, normalized size = 1.39 \begin{align*} \frac{2 \, \sqrt{a^{2} x^{2} + 1} \arctan \left (a x\right ) - \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right )}{2 \, a^{2} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.42572, size = 158, normalized size = 2.68 \begin{align*} \frac{2 \, \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right ) + \sqrt{c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt{a^{2} c x^{2} + c} a \sqrt{c} x - c\right )}{2 \, a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.15659, size = 81, normalized size = 1.37 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )}{a^{2} c} + \frac{\log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} + c} \right |}\right )}{a \sqrt{c}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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