Optimal. Leaf size=319 \[ \frac {2 \sqrt {c+d} \cot (e+f x) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (\sec (e+f x)+1)}{c-d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right )}{a f (c-d)}-\frac {2 \sqrt {c+d} \cot (e+f x) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (\sec (e+f x)+1)}{c-d}} \Pi \left (\frac {c+d}{c};\sin ^{-1}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right )}{a c f}-\frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {c+d \sec (e+f x)} E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {c-d}{c+d}\right )}{a f (c-d) \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3929, 3921, 3784, 3832, 3968} \[ \frac {2 \sqrt {c+d} \cot (e+f x) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (\sec (e+f x)+1)}{c-d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right )}{a f (c-d)}-\frac {2 \sqrt {c+d} \cot (e+f x) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (\sec (e+f x)+1)}{c-d}} \Pi \left (\frac {c+d}{c};\sin ^{-1}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right )}{a c f}-\frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {c+d \sec (e+f x)} E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {c-d}{c+d}\right )}{a f (c-d) \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3784
Rule 3832
Rule 3921
Rule 3929
Rule 3968
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx &=-\frac {\int \frac {-a c+a d-a d \sec (e+f x)}{\sqrt {c+d \sec (e+f x)}} \, dx}{a^2 (c-d)}+\frac {a \int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx}{-a c+a d}\\ &=-\frac {E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}}+\frac {\int \frac {1}{\sqrt {c+d \sec (e+f x)}} \, dx}{a}+\frac {d \int \frac {\sec (e+f x)}{\sqrt {c+d \sec (e+f x)}} \, dx}{a (c-d)}\\ &=\frac {2 \sqrt {c+d} \cot (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a (c-d) f}-\frac {2 \sqrt {c+d} \cot (e+f x) \Pi \left (\frac {c+d}{c};\sin ^{-1}\left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right )|\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a c f}-\frac {E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 12.71, size = 187, normalized size = 0.59 \[ \frac {2 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{\cos (e+f x)+1}} \sec (e+f x) \sqrt {\frac {c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left (2 (c-2 d) F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )+(c+d) E\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )+4 (d-c) \Pi \left (-1;\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )\right )}{a f (d-c) \sqrt {c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \sec \left (f x + e\right ) + c}}{a d \sec \left (f x + e\right )^{2} + a c + {\left (a c + a d\right )} \sec \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \sec \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 2.10, size = 327, normalized size = 1.03 \[ -\frac {\sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (c +d \right )}}\, \left (1+\cos \left (f x +e \right )\right )^{2} \left (2 \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right ) c -4 \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right ) d +c \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right )+d \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right )-4 c \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, -1, \sqrt {\frac {c -d}{c +d}}\right )+4 \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, -1, \sqrt {\frac {c -d}{c +d}}\right ) d \right ) \left (-1+\cos \left (f x +e \right )\right )}{a f \left (d +c \cos \left (f x +e \right )\right ) \sin \left (f x +e \right )^{2} \left (c -d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \sec \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{\sqrt {c + d \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )} + \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________