3.144 \(\int \frac{1}{\sqrt{a+b x^2} \tan ^{-1}(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}})^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}{e \sqrt{a+b x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}\right )} \]

[Out]

-(Sqrt[-((a*e^2)/b) - e^2*x^2]/(e*Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]))

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Rubi [A]  time = 0.101419, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {5157, 5155} \[ -\frac{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}{e \sqrt{a+b x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^2),x]

[Out]

-(Sqrt[-((a*e^2)/b) - e^2*x^2]/(e*Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]))

Rule 5157

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]^(m_.)/Sqrt[(d_.) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a
+ b*x^2]/Sqrt[d + e*x^2], Int[ArcTan[(c*x)/Sqrt[a + b*x^2]]^m/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c, d, e
, m}, x] && EqQ[b + c^2, 0] && EqQ[b*d - a*e, 0]

Rule 5155

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]^(m_.)/Sqrt[(a_.) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcTan
[(c*x)/Sqrt[a + b*x^2]]^(m + 1)/(c*(m + 1)), x] /; FreeQ[{a, b, c, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0914722, size = 60, normalized size = 0.91 \[ \frac{e \sqrt{a+b x^2}}{b \sqrt{-\frac{e^2 \left (a+b x^2\right )}{b}} \tan ^{-1}\left (\frac{e x}{\sqrt{-\frac{e^2 \left (a+b x^2\right )}{b}}}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^2),x]

[Out]

(e*Sqrt[a + b*x^2])/(b*Sqrt[-((e^2*(a + b*x^2))/b)]*ArcTan[(e*x)/Sqrt[-((e^2*(a + b*x^2))/b)]])

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Maple [F]  time = 0.728, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arctan \left ({ex{\frac{1}{\sqrt{-{\frac{a{e}^{2}}{b}}-{e}^{2}{x}^{2}}}}} \right ) \right ) ^{-2}{\frac{1}{\sqrt{b{x}^{2}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^2/(b*x^2+a)^(1/2),x)

[Out]

int(1/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^2/(b*x^2+a)^(1/2),x)

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Maxima [A]  time = 1.51578, size = 62, normalized size = 0.94 \begin{align*} -\frac{\sqrt{-b x^{2} - a}}{\sqrt{b x^{2} + a} \sqrt{b} \arctan \left (\sqrt{b} x, \sqrt{-b x^{2} - a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*x^2 - a)/(sqrt(b*x^2 + a)*sqrt(b)*arctan2(sqrt(b)*x, sqrt(-b*x^2 - a)))

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Fricas [A]  time = 1.8452, size = 163, normalized size = 2.47 \begin{align*} \frac{\sqrt{b x^{2} + a} \sqrt{-\frac{b e^{2} x^{2} + a e^{2}}{b}}}{{\left (b e x^{2} + a e\right )} \arctan \left (\frac{b x \sqrt{-\frac{b e^{2} x^{2} + a e^{2}}{b}}}{b e x^{2} + a e}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2 + a)*sqrt(-(b*e^2*x^2 + a*e^2)/b)/((b*e*x^2 + a*e)*arctan(b*x*sqrt(-(b*e^2*x^2 + a*e^2)/b)/(b*e*x^2
+ a*e)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/atan(e*x/(-a*e**2/b-e**2*x**2)**(1/2))**2/(b*x**2+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*x**2)*atan(e*x/sqrt(-a*e**2/b - e**2*x**2))**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2} + a} \arctan \left (\frac{e x}{\sqrt{-e^{2} x^{2} - \frac{a e^{2}}{b}}}\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x^2 + a)*arctan(e*x/sqrt(-e^2*x^2 - a*e^2/b))^2), x)