Optimal. Leaf size=62 \[ \frac {\tan (e+f x)}{2 f \sqrt {a \cos ^2(e+f x)}}-\frac {\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 f \sqrt {a \cos ^2(e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3176, 3207, 2611, 3770} \[ \frac {\tan (e+f x)}{2 f \sqrt {a \cos ^2(e+f x)}}-\frac {\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 f \sqrt {a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2611
Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tan ^2(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx &=\int \frac {\tan ^2(e+f x)}{\sqrt {a \cos ^2(e+f x)}} \, dx\\ &=\frac {\cos (e+f x) \int \sec (e+f x) \tan ^2(e+f x) \, dx}{\sqrt {a \cos ^2(e+f x)}}\\ &=\frac {\tan (e+f x)}{2 f \sqrt {a \cos ^2(e+f x)}}-\frac {\cos (e+f x) \int \sec (e+f x) \, dx}{2 \sqrt {a \cos ^2(e+f x)}}\\ &=-\frac {\tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{2 f \sqrt {a \cos ^2(e+f x)}}+\frac {\tan (e+f x)}{2 f \sqrt {a \cos ^2(e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 43, normalized size = 0.69 \[ \frac {\tan (e+f x)-\cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{2 f \sqrt {a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 67, normalized size = 1.08 \[ -\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (\cos \left (f x + e\right )^{2} \log \left (-\frac {\sin \left (f x + e\right ) + 1}{\sin \left (f x + e\right ) - 1}\right ) - 2 \, \sin \left (f x + e\right )\right )}}{4 \, a f \cos \left (f x + e\right )^{3}} \]
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 = 1.50, size = 65, normalized size = 1.05 \[ \frac {\frac {\sin \left (f x +e \right )}{2}+\frac {\left (\ln \left (\sin \left (f x +e \right )-1\right )-\ln \left (1+\sin \left (f x +e \right )\right )\right ) \left (\cos ^{2}\left (f x +e \right )\right )}{4}}{\cos \left (f x +e \right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.50, size = 527, normalized size = 8.50 \[ \frac {4 \, {\left (\sin \left (3 \, f x + 3 \, e\right ) - \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) + 1\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) + 1\right ) - 4 \, {\left (\cos \left (3 \, f x + 3 \, e\right ) - \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + 4 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \sin \left (3 \, f x + 3 \, e\right ) - 8 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 8 \, \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 8 \, \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - 4 \, \sin \left (f x + e\right )}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \sqrt {a} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}{\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________