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

Optimal. Leaf size=72 \[ \frac{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}} \tan ^{-1}\left (\frac{e x}{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}\right )^{m+1}}{e (m+1) \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.107659, antiderivative size = 72, 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} \[ \frac{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}} \tan ^{-1}\left (\frac{e x}{\sqrt{e^2 \left (-x^2\right )-\frac{a e^2}{b}}}\right )^{m+1}}{e (m+1) \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.221558, size = 66, normalized size = 0.92 \[ \frac{\sqrt{-\frac{e^2 \left (a+b x^2\right )}{b}} \tan ^{-1}\left (\frac{e x}{\sqrt{-\frac{e^2 \left (a+b x^2\right )}{b}}}\right )^{m+1}}{e (m+1) \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.9, 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 ) ^{m}{\frac{1}{\sqrt{b{x}^{2}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.13816, size = 270, normalized size = 3.75 \begin{align*} -\frac{\sqrt{b x^{2} + a} \left (-\arctan \left (\frac{b x \sqrt{-\frac{b e^{2} x^{2} + a e^{2}}{b}}}{b e x^{2} + a e}\right )\right )^{m} \sqrt{-\frac{b e^{2} x^{2} + a e^{2}}{b}} \arctan \left (\frac{b x \sqrt{-\frac{b e^{2} x^{2} + a e^{2}}{b}}}{b e x^{2} + a e}\right )}{a e m +{\left (b e m + b e\right )} x^{2} + a e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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