Optimal. Leaf size=19 \[ -\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3176, 3205, 16, 32} \[ -\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 32
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \sqrt {a-a \sin ^2(e+f x)} \tan (e+f x) \, dx &=\int \sqrt {a \cos ^2(e+f x)} \tan (e+f x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a x}}{x} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\sqrt {a \cos ^2(e+f x)}}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 19, normalized size = 1.00 \[ -\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 17, normalized size = 0.89 \[ -\frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.17, size = 21, normalized size = 1.11 \[ -\frac {\sqrt {a -a \left (\sin ^{2}\left (f x +e \right )\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 20, normalized size = 1.05 \[ -\frac {\sqrt {-a \sin \left (f x + e\right )^{2} + a}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 15.25, size = 20, normalized size = 1.05 \[ -\frac {\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \tan {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________