Optimal. Leaf size=209 \[ \frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {a+b \sec (e+f x)} E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {a-b}{a+b}\right )}{c f (a-b) \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}-\frac {2 \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{c f (a-b)} \]
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Rubi [A] time = 0.29, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3972, 3832, 3968} \[ \frac {\sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {a+b \sec (e+f x)} E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {a-b}{a+b}\right )}{c f (a-b) \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}}-\frac {2 \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{c f (a-b)} \]
Antiderivative was successfully verified.
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Rule 3832
Rule 3968
Rule 3972
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 {b \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{(a-b) c}-\frac {c \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx}{-a c+b c}\\ &=-\frac {2 \sqrt {a+b} \cot (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) c f}+\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)}}{(a-b) c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}}\\ \end {align*}
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Mathematica [A] time = 14.70, size = 375, normalized size = 1.79 \[ \frac {\cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2(e+f x) (a \cos (e+f x)+b) \left (\frac {2 \sin (e+f x)}{b-a}-\frac {2 \tan \left (\frac {1}{2} (e+f x)\right )}{b-a}\right )}{f (c \sec (e+f x)+c) \sqrt {a+b \sec (e+f x)}}-\frac {2 \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right ) \sec ^{\frac {3}{2}}(e+f x) \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)} \left (\sqrt {2} \sqrt {\frac {a-b}{a+b}} \sqrt {\frac {\cos (e+f x)}{\cos (e+f x)+1}} \tan \left (\frac {1}{2} (e+f x)\right ) (a \cos (e+f x)+b)+(a-b) \sqrt {\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) (a \cos (e+f x)+b)}{a+b}} E\left (\sin ^{-1}\left (\sqrt {\frac {a-b}{a+b}} \tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a+b}{a-b}\right )\right )}{f \left (\frac {a-b}{a+b}\right )^{3/2} (a+b) (c \sec (e+f x)+c) \sqrt {\cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )} \sqrt {a+b \sec (e+f x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{b c \sec \left (f x + e\right )^{2} + {\left (a + b\right )} c \sec \left (f x + e\right ) + a c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.01, size = 225, normalized size = 1.08 \[ -\frac {\sqrt {\frac {b +a \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \left (1+\cos \left (f x +e \right )\right )^{2} \left (-1+\cos \left (f x +e \right )\right ) \left (2 \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) b -a \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right )-b \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{c f \left (b +a \cos \left (f x +e \right )\right ) \sin \left (f x +e \right )^{2} \left (a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )} + \sqrt {a + b \sec {\left (e + f x \right )}}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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