Optimal. Leaf size=78 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f \sqrt {c-d}} \]
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Rubi [A] time = 0.16, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {3983, 203} \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f \sqrt {c-d}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3983
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{2+(a c-a d) x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}\\ &=\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c-d} f}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 107, normalized size = 1.37 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \sqrt {c \cos (e+f x)+d} \tan ^{-1}\left (\frac {\sqrt {c-d} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c \cos (e+f x)+d}}\right )}{f \sqrt {c-d} \sqrt {a (\sec (e+f x)+1)} \sqrt {c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 246, normalized size = 3.15 \[ \left [\frac {\sqrt {2} \sqrt {-\frac {1}{a c - a d}} \log \left (-\frac {2 \, \sqrt {2} {\left (c - d\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a c - a d}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (3 \, c - d\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (c + d\right )} \cos \left (f x + e\right ) + c - 3 \, d}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right )}{2 \, f}, -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a c - a d} \sin \left (f x + e\right )}\right )}{\sqrt {a c - a d} f}\right ] \]
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 \[ \int \frac {\sec \left (f x + e\right )}{\sqrt {a \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.95, size = 170, normalized size = 2.18 \[ \frac {2 \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \ln \left (-\frac {\sqrt {c -d}\, \cos \left (f x +e \right )-\sin \left (f x +e \right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}-\sqrt {c -d}}{\sin \left (f x +e \right )}\right )}{f \sin \left (f x +e \right )^{2} \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, a \sqrt {c -d}} \]
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 \[ \int \frac {\sec \left (f x + e\right )}{\sqrt {a \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
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 \[ \int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
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 \[ \int \frac {\sec {\left (e + f x \right )}}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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