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

Optimal. Leaf size=55 \[ \frac {\sqrt {a-c^2 x^2} \log \left (\tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )\right )}{c \sqrt {d-\frac {c^2 d x^2}{a}}} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 55, 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, 5153} \[ \frac {\sqrt {a-c^2 x^2} \log \left (\tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )\right )}{c \sqrt {d-\frac {c^2 d x^2}{a}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5153

Int[1/(ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*Sqrt[(a_.) + (b_.)*(x_)^2]), x_Symbol] :> Simp[(1*Log[A
rcTan[(c*x)/Sqrt[a + b*x^2]]])/c, x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]

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

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Mathematica [A]  time = 0.06, size = 55, normalized size = 1.00 \[ \frac {\sqrt {a-c^2 x^2} \log \left (\tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )\right )}{c \sqrt {d-\frac {c^2 d x^2}{a}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 83, normalized size = 1.51 \[ -\frac {\sqrt {-c^{2} x^{2} + a} a \sqrt {-\frac {c^{2} d x^{2} - a d}{a}} \log \left (2 \, \arctan \left (\frac {\sqrt {-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )\right )}{c^{3} d x^{2} - a c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 1.78, size = 71, normalized size = 1.29 \[ -\frac {\sqrt {-\frac {d \left (c^{2} x^{2}-a \right )}{a}}\, \sqrt {-c^{2} x^{2}+a}\, \ln \left (\arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2}+a}}\right )\right ) a}{d \left (c^{2} x^{2}-a \right ) c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.60, size = 49, normalized size = 0.89 \[ \frac {\ln \left (\mathrm {atan}\left (\frac {c\,x}{\sqrt {a-c^2\,x^2}}\right )\right )\,\sqrt {a-c^2\,x^2}}{c\,\sqrt {d-\frac {c^2\,d\,x^2}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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