Optimal. Leaf size=57 \[ -\frac {\sqrt {a-c^2 x^2}}{c \sqrt {d-\frac {c^2 d x^2}{a}} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 57, 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} \[ -\frac {\sqrt {a-c^2 x^2}}{c \sqrt {d-\frac {c^2 d x^2}{a}} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5155
Rule 5157
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 )^2} \, 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 )^2} \, dx}{\sqrt {d-\frac {c^2 d x^2}{a}}}\\ &=-\frac {\sqrt {a-c^2 x^2}}{c \sqrt {d-\frac {c^2 d x^2}{a}} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 57, normalized size = 1.00 \[ -\frac {\sqrt {a-c^2 x^2}}{c \sqrt {d-\frac {c^2 d x^2}{a}} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 82, normalized size = 1.44 \[ -\frac {\sqrt {-c^{2} x^{2} + a} a \sqrt {-\frac {c^{2} d x^{2} - a d}{a}}}{{\left (c^{3} d x^{2} - a c d\right )} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\frac {c^{2} d x^{2}}{a} + d} \arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2} + a}}\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.42, size = 71, normalized size = 1.25 \[ \frac {\sqrt {-\frac {d \left (c^{2} x^{2}-a \right )}{a}}\, \sqrt {-c^{2} x^{2}+a}\, a}{d \left (c^{2} x^{2}-a \right ) c \arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2}+a}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 29, normalized size = 0.51 \[ -\frac {\sqrt {a}}{c \sqrt {d} \arctan \left (c x, \sqrt {-c^{2} x^{2} + a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.60, size = 51, normalized size = 0.89 \[ -\frac {\sqrt {a-c^2\,x^2}}{c\,\mathrm {atan}\left (\frac {c\,x}{\sqrt {a-c^2\,x^2}}\right )\,\sqrt {d-\frac {c^2\,d\,x^2}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- d \left (-1 + \frac {c^{2} x^{2}}{a}\right )} \operatorname {atan}^{2}{\left (\frac {c x}{\sqrt {a - c^{2} x^{2}}} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________