Optimal. Leaf size=192 \[ \frac {2 \sqrt {a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d} (b c-a d)} \]
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Rubi [A] time = 0.17, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {3984} \[ \frac {2 \sqrt {a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 3984
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx &=\frac {2 \sqrt {a+b} \cot (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{\sqrt {c+d} (b c-a d) f}\\ \end {align*}
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Mathematica [A] time = 3.60, size = 233, normalized size = 1.21 \[ \frac {4 \sin ^2\left (\frac {1}{2} (e+f x)\right ) \csc (e+f x) \sqrt {a+b \sec (e+f x)} \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(a+b) \csc ^2\left (\frac {1}{2} (e+f x)\right ) (c \cos (e+f x)+d)}{a d-b c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{a d-b c}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right )}{f (a+b) \sqrt {c+d \sec (e+f x)} \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} (e+f x)\right ) (a \cos (e+f x)+b)}{b c-a d}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c} \sec \left (f x + e\right )}{b d \sec \left (f x + e\right )^{2} + a c + {\left (b c + a d\right )} \sec \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.09, size = 219, normalized size = 1.14 \[ \frac {2 \left (\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (c +d \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}}{f \left (-1+\cos \left (f x +e \right )\right ) \left (d +c \cos \left (f x +e \right )\right ) \left (b +a \cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}} \]
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 \[ \int \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
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 \[ \int \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}} \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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