Optimal. Leaf size=95 \[ \frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {a+b \sec (e+f x)} E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {a-b}{a+b}\right )}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {3968} \[ \frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {a+b \sec (e+f x)} E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {a-b}{a+b}\right )}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3968
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx &=\frac {E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}}{c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 5.75, size = 264, normalized size = 2.78 \[ \frac {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 \sqrt {\frac {\cos (e+f x)}{\cos (e+f x)+1}} \sqrt {\sec (e+f x)+1} \sec ^4\left (\frac {1}{2} (e+f x)\right ) E\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )}{\left (\frac {1}{\cos (e+f x)+1}\right )^{3/2} \sqrt {\frac {a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}}}+\frac {\left (\sin \left (\frac {3}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\sec (e+f x)+1} \sec ^5\left (\frac {1}{2} (e+f x)\right )}{\left (\frac {1}{\cos (e+f x)+1}\right )^{3/2}}-8 \sqrt {\sec (e+f x)} \left (\sin (e+f x)-\tan \left (\frac {1}{2} (e+f x)\right )\right )\right )}{4 c f (\sec (e+f x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{c \sec \left (f x + e\right ) + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{c \sec \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.97, size = 153, normalized size = 1.61 \[ -\frac {\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \left (1+\cos \left (f x +e \right )\right )^{2} \left (-a -b \right )}{c f \left (b +a \cos \left (f x +e \right )\right ) \sin \left (f x +e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{c \sec \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}}{\cos \left (e+f\,x\right )\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {a + b \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________