Optimal. Leaf size=61 \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {d} f} \]
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Rubi [A] time = 0.15, antiderivative size = 61, 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 = {3980, 206} \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {d} f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3980
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 a) \operatorname {Subst}\left (\int \frac {1}{1-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 {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {d} f}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 102, normalized size = 1.67 \[ \frac {\sqrt {2} \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (\sec (e+f x)+1)} \sqrt {c \cos (e+f x)+d} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c \cos (e+f x)+d}}\right )}{\sqrt {d} f \sqrt {c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 307, normalized size = 5.03 \[ \left [\frac {\sqrt {\frac {a}{d}} \log \left (-\frac {8 \, a c d \cos \left (f x + e\right ) + {\left (a c^{2} - 6 \, a c d + a d^{2}\right )} \cos \left (f x + e\right )^{3} + 4 \, {\left (2 \, d^{2} \cos \left (f x + e\right ) + {\left (c d - d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {a}{d}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 8 \, a d^{2} + {\left (a c^{2} + 2 \, a c d - 7 \, a d^{2}\right )} \cos \left (f x + e\right )^{2}}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right )}{2 \, f}, \frac {\sqrt {-\frac {a}{d}} \arctan \left (-\frac {2 \, d \sqrt {-\frac {a}{d}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{{\left (a c - a d\right )} \cos \left (f x + e\right )^{2} + 2 \, a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}\right )}{f}\right ] \]
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 \[ \int \frac {\sqrt {a \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{\sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.19, size = 301, normalized size = 4.93 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )\right ) \left (\ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {-d}\, \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )-c \sin \left (f x +e \right )-d \sin \left (f x +e \right )-c \cos \left (f x +e \right )+d \cos \left (f x +e \right )+c -d \right )}{-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}\right )-\ln \left (\frac {2 \sqrt {2}\, \sqrt {-d}\, \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )-2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )+2 c \cos \left (f x +e \right )-2 d \cos \left (f x +e \right )-2 c +2 d}{1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}\right )\right ) \sqrt {2}}{f \sin \left (f x +e \right )^{2} \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, \sqrt {-d}} \]
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 \[ \int \frac {\sqrt {a \sec \left (f x + e\right ) + a} \sec \left (f x + e\right )}{\sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}}{\cos \left (e+f\,x\right )\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
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 \[ \int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sec {\left (e + f x \right )}}{\sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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