3.145 \(\int \frac{1}{\sqrt{a+b x^2} \tan ^{-1}(\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}})^3} \, dx\)

Optimal. Leaf size=68 \[ -\frac{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}{2 e \sqrt{a+b x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.100236, antiderivative size = 68, 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} \[ -\frac{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}{2 e \sqrt{a+b x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^3),x]

[Out]

-Sqrt[-((a*e^2)/b) - e^2*x^2]/(2*e*Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^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 )^3} \, 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 )^3} \, dx}{\sqrt{a+b x^2}}\\ &=-\frac{\sqrt{-\frac{a e^2}{b}-e^2 x^2}}{2 e \sqrt{a+b x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{-\frac{a e^2}{b}-e^2 x^2}}\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.088125, size = 62, normalized size = 0.91 \[ -\frac{\sqrt{-\frac{e^2 \left (a+b x^2\right )}{b}}}{2 e \sqrt{a+b x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{-\frac{e^2 \left (a+b x^2\right )}{b}}}\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^3),x]

[Out]

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

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Maple [F]  time = 0.711, 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 ) ^{-3}{\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))^3/(b*x^2+a)^(1/2),x)

[Out]

int(1/arctan(e*x/(-a*e^2/b-e^2*x^2)^(1/2))^3/(b*x^2+a)^(1/2),x)

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Maxima [A]  time = 1.71051, size = 80, normalized size = 1.18 \begin{align*} -\frac{\sqrt{b x^{2} + a} \sqrt{-b x^{2} - a} \sqrt{b}}{2 \,{\left (b^{2} x^{2} + a b\right )} \arctan \left (\sqrt{b} x, \sqrt{-b x^{2} - a}\right )^{2}} \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))^3/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(b*x^2 + a)*sqrt(-b*x^2 - a)*sqrt(b)/((b^2*x^2 + a*b)*arctan2(sqrt(b)*x, sqrt(-b*x^2 - a))^2)

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Fricas [A]  time = 1.91877, size = 173, normalized size = 2.54 \begin{align*} -\frac{\sqrt{b x^{2} + a} \sqrt{-\frac{b e^{2} x^{2} + a e^{2}}{b}}}{2 \,{\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 )^{2}} \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))^3/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

-1/2*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))^2)

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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}^{3}{\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))**3/(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))**3), 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 )^{3}}\,{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))^3/(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))^3), x)