Optimal. Leaf size=59 \[ \frac {\sqrt {a-c^2 x^2} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^2}{2 c \sqrt {d-\frac {c^2 d x^2}{a}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {5157, 5155} \[ \frac {\sqrt {a-c^2 x^2} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^2}{2 c \sqrt {d-\frac {c^2 d x^2}{a}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5155
Rule 5157
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )}{\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 )}{\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 )^2}{2 c \sqrt {d-\frac {c^2 d x^2}{a}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 59, normalized size = 1.00 \[ \frac {\sqrt {a-c^2 x^2} \tan ^{-1}\left (\frac {c x}{\sqrt {a-c^2 x^2}}\right )^2}{2 c \sqrt {d-\frac {c^2 d x^2}{a}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a \sqrt {-\frac {c^{2} d x^{2} - a d}{a}} \arctan \left (\frac {\sqrt {-c^{2} x^{2} + a} c x}{c^{2} x^{2} - a}\right )}{c^{2} d x^{2} - a d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2} + a}}\right )}{\sqrt {-\frac {c^{2} d x^{2}}{a} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.43, size = 72, normalized size = 1.22 \[ -\frac {\sqrt {-\frac {d \left (c^{2} x^{2}-a \right )}{a}}\, \sqrt {-c^{2} x^{2}+a}\, \arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2}+a}}\right )^{2} a}{2 d \left (c^{2} x^{2}-a \right ) c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (\frac {c x}{\sqrt {-c^{2} x^{2} + a}}\right )}{\sqrt {-\frac {c^{2} d x^{2}}{a} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atan}\left (\frac {c\,x}{\sqrt {a-c^2\,x^2}}\right )}{\sqrt {d-\frac {c^2\,d\,x^2}{a}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}{\left (\frac {c x}{\sqrt {a - c^{2} x^{2}}} \right )}}{\sqrt {- d \left (-1 + \frac {c^{2} x^{2}}{a}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________