3.134 \(\int \frac{\tan ^{-1}(\frac{c x}{\sqrt{a-c^2 x^2}})^m}{\sqrt{d-\frac{c^2 d x^2}{a}}} \, dx\)

Optimal. Leaf size=63 \[ \frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )^{m+1}}{c (m+1) \sqrt{d-\frac{c^2 d x^2}{a}}} \]

[Out]

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

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Rubi [A]  time = 0.106523, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {5157, 5155} \[ \frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )^{m+1}}{c (m+1) \sqrt{d-\frac{c^2 d x^2}{a}}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^m/Sqrt[d - (c^2*d*x^2)/a],x]

[Out]

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

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

Mathematica [A]  time = 0.0604835, size = 63, normalized size = 1. \[ \frac{\sqrt{a-c^2 x^2} \tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )^{m+1}}{c (m+1) \sqrt{d-\frac{c^2 d x^2}{a}}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^m/Sqrt[d - (c^2*d*x^2)/a],x]

[Out]

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

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Maple [A]  time = 0.756, size = 73, normalized size = 1.2 \begin{align*} -{\frac{{c}^{2}{x}^{2}-a}{ \left ( 1+m \right ) c} \left ( \arctan \left ({cx{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+a}}}} \right ) \right ) ^{1+m}{\frac{1}{\sqrt{-{\frac{d \left ({c}^{2}{x}^{2}-a \right ) }{a}}}}}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(c*x/(-c^2*x^2+a)^(1/2))^m/(d-c^2*d*x^2/a)^(1/2),x)

[Out]

-arctan(c*x/(-c^2*x^2+a)^(1/2))^(1+m)/(1+m)*(c^2*x^2-a)/(-d*(c^2*x^2-a)/a)^(1/2)/(-c^2*x^2+a)^(1/2)/c

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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(c*x/(-c^2*x^2+a)^(1/2))^m/(d-c^2*d*x^2/a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 2.43247, size = 252, normalized size = 4. \begin{align*} -\frac{\sqrt{-c^{2} x^{2} + a} a \left (-\arctan \left (\frac{\sqrt{-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )\right )^{m} \sqrt{-\frac{c^{2} d x^{2} - a d}{a}} \arctan \left (\frac{\sqrt{-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )}{a c d m + a c d -{\left (c^{3} d m + c^{3} d\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-c^2*x^2 + a)*a*(-arctan(sqrt(-c^2*x^2 + a)*c*x/(c^2*x^2 - a)))^m*sqrt(-(c^2*d*x^2 - a*d)/a)*arctan(sqrt
(-c^2*x^2 + a)*c*x/(c^2*x^2 - a))/(a*c*d*m + a*c*d - (c^3*d*m + c^3*d)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arctan(c*x/sqrt(-c^2*x^2 + a))^m/sqrt(-c^2*d*x^2/a + d), x)