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]

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

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Rubi [A]  time = 0.11, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.06, size = 63, normalized size = 1.00 \[ \frac {\sqrt {a-c^2 x^2} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^{m+1}}{c (m+1) \sqrt {d-\frac {c^2 d x^2}{a}}} \]

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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fricas [B]  time = 0.79, size = 126, normalized size = 2.00 \[ -\frac {\sqrt {-c^{2} x^{2} + a} a \left (-\arctan \left (\frac {\sqrt {-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )\right )^{m} \sqrt {-\frac {c^{2} d x^{2} - a d}{a}} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )}{a c d m + a c d - {\left (c^{3} d m + c^{3} d\right )} x^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2} + a}}\right )^{m}}{\sqrt {-\frac {c^{2} d x^{2}}{a} + d}}\,{d x} \]

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)

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maple [A]  time = 0.52, size = 73, normalized size = 1.16 \[ -\frac {\arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2}+a}}\right )^{1+m} \left (c^{2} x^{2}-a \right )}{\left (1+m \right ) \sqrt {-\frac {d \left (c^{2} x^{2}-a \right )}{a}}\, \sqrt {-c^{2} x^{2}+a}\, c} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.73, size = 57, normalized size = 0.90 \[ \frac {{\mathrm {atan}\left (\frac {c\,x}{\sqrt {a-c^2\,x^2}}\right )}^{m+1}\,\sqrt {a-c^2\,x^2}}{c\,\left (m+1\right )\,\sqrt {d-\frac {c^2\,d\,x^2}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}^{m}{\left (\frac {c x}{\sqrt {a - c^{2} x^{2}}} \right )}}{\sqrt {- d \left (-1 + \frac {c^{2} x^{2}}{a}\right )}}\, dx \]

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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