Optimal. Leaf size=102 \[ \frac {2 \tan (e+f x) \sqrt {\frac {c+d \sec (e+f x)}{c+d}} \Pi \left (\frac {2 a}{a+b};\sin ^{-1}\left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right )|\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {-\tan ^2(e+f x)} \sqrt {c+d \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2829, 3973} \[ \frac {2 \tan (e+f x) \sqrt {\frac {c+d \sec (e+f x)}{c+d}} \Pi \left (\frac {2 a}{a+b};\sin ^{-1}\left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right )|\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {-\tan ^2(e+f x)} \sqrt {c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2829
Rule 3973
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx &=\int \frac {\sec (e+f x)}{(b+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx\\ &=\frac {2 \Pi \left (\frac {2 a}{a+b};\sin ^{-1}\left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sec (e+f x)}{c+d}} \tan (e+f x)}{(a+b) f \sqrt {c+d \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.40, size = 187, normalized size = 1.83 \[ -\frac {2 \sqrt {\sec (e+f x)} \sqrt {\sec (e+f x)+1} \sqrt {\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left ((a+b) F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )-2 a \Pi \left (\frac {b-a}{a+b};\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )\right )}{f (a-b) (a+b) \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 2.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \sec \left (f x + e\right ) + c}}{b c \cos \left (f x + e\right ) + a c + {\left (b d \cos \left (f x + e\right ) + a d\right )} \sec \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.58, size = 239, normalized size = 2.34 \[ \frac {2 \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (c +d \right )}}\, \left (1+\cos \left (f x +e \right )\right )^{2} \left (2 a \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, -\frac {a -b}{a +b}, \sqrt {\frac {c -d}{c +d}}\right )-a \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right )-b \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right )\right ) \left (-1+\cos \left (f x +e \right )\right )}{f \left (d +c \cos \left (f x +e \right )\right ) \sin \left (f x +e \right )^{2} \left (a -b \right ) \left (a +b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cos \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}\,\left (a+b\,\cos \left (e+f\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \cos {\left (e + f x \right )}\right ) \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________