3.139 \(\int \frac{1}{\sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}(\frac{c x}{\sqrt{a-c^2 x^2}})^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.10101, antiderivative size = 59, 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} \[ -\frac{\sqrt{a-c^2 x^2}}{2 c \sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^3),x]

[Out]

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

Mathematica [A]  time = 0.0261237, size = 59, normalized size = 1. \[ -\frac{\sqrt{a-c^2 x^2}}{2 c \sqrt{d-\frac{c^2 d x^2}{a}} \tan ^{-1}\left (\frac{c x}{\sqrt{a-c^2 x^2}}\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^3),x]

[Out]

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

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Maple [A]  time = 0.675, size = 72, normalized size = 1.2 \begin{align*}{\frac{a}{2\,d \left ({c}^{2}{x}^{2}-a \right ) c}\sqrt{-{\frac{d \left ({c}^{2}{x}^{2}-a \right ) }{a}}}\sqrt{-{c}^{2}{x}^{2}+a} \left ( \arctan \left ({cx{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+a}}}} \right ) \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arctan(c*x/(-c^2*x^2+a)^(1/2))^3/(d-c^2*d*x^2/a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.69293, size = 39, normalized size = 0.66 \begin{align*} -\frac{\sqrt{a}}{2 \, c \sqrt{d} \arctan \left (c x, \sqrt{-c^{2} x^{2} + a}\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctan(c*x/(-c^2*x^2+a)^(1/2))^3/(d-c^2*d*x^2/a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.88121, size = 165, normalized size = 2.8 \begin{align*} \frac{\sqrt{-c^{2} x^{2} + a} a \sqrt{-\frac{c^{2} d x^{2} - a d}{a}}}{2 \,{\left (c^{3} d x^{2} - a c d\right )} \arctan \left (\frac{\sqrt{-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctan(c*x/(-c^2*x^2+a)^(1/2))^3/(d-c^2*d*x^2/a)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(-c^2*x^2 + a)*a*sqrt(-(c^2*d*x^2 - a*d)/a)/((c^3*d*x^2 - a*c*d)*arctan(sqrt(-c^2*x^2 + a)*c*x/(c^2*x^
2 - a))^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- d \left (-1 + \frac{c^{2} x^{2}}{a}\right )} \operatorname{atan}^{3}{\left (\frac{c x}{\sqrt{a - c^{2} x^{2}}} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/atan(c*x/(-c**2*x**2+a)**(1/2))**3/(d-c**2*d*x**2/a)**(1/2),x)

[Out]

Integral(1/(sqrt(-d*(-1 + c**2*x**2/a))*atan(c*x/sqrt(a - c**2*x**2))**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\frac{c^{2} d x^{2}}{a} + d} \arctan \left (\frac{c x}{\sqrt{-c^{2} x^{2} + a}}\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arctan(c*x/(-c^2*x^2+a)^(1/2))^3/(d-c^2*d*x^2/a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-c^2*d*x^2/a + d)*arctan(c*x/sqrt(-c^2*x^2 + a))^3), x)