Optimal. Leaf size=140 \[ \frac {\sqrt {2} \sqrt {c-d} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f} \]
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Rubi [A] time = 0.46, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3981, 3983, 203, 3980, 206} \[ \frac {\sqrt {2} \sqrt {c-d} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 3980
Rule 3981
Rule 3983
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}} \, dx &=\frac {d \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx}{a}-(-c+d) \int \frac {\sec (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx\\ &=-\frac {(2 (c-d)) \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 {(2 d) \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 {\sqrt {2} \sqrt {c-d} \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} f}+\frac {2 \sqrt {d} \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 {a} f}\\ \end {align*}
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Mathematica [A] time = 17.02, size = 187, normalized size = 1.34 \[ \frac {\sqrt {c} \sin (e+f x) \sqrt {c+d \sec (e+f x)} \left (2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {c \cos (e+f x)+d}}{\sqrt {d} \sqrt {c-c \cos (e+f x)}}\right )-\sqrt {2} \sqrt {c-d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c \cos (e+f x)+d}}{\sqrt {c-d} \sqrt {c-c \cos (e+f x)}}\right )\right )}{f \sqrt {a (\sec (e+f x)+1)} \sqrt {c-c \cos (e+f x)} \sqrt {c \cos (e+f x)+d}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 1048, normalized size = 7.49 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {d \sec \left (f x + e\right ) + c} \sec \left (f x + e\right )}{\sqrt {a \sec \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.99, size = 503, normalized size = 3.59 \[ \frac {\sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )\right ) \left (d \ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {-d}\, \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )-c \sin \left (f x +e \right )-d \sin \left (f x +e \right )-c \cos \left (f x +e \right )+d \cos \left (f x +e \right )+c -d \right )}{-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}\right ) \sqrt {c -d}-d \ln \left (\frac {2 \sqrt {2}\, \sqrt {-d}\, \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )-2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )+2 c \cos \left (f x +e \right )-2 d \cos \left (f x +e \right )-2 c +2 d}{1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}\right ) \sqrt {c -d}+\ln \left (\frac {\sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, \sqrt {c -d}\, \sin \left (f x +e \right )-c \cos \left (f x +e \right )+d \cos \left (f x +e \right )+c -d}{\sin \left (f x +e \right ) \sqrt {c -d}}\right ) \sqrt {2}\, \sqrt {-d}\, c -\ln \left (\frac {\sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, \sqrt {c -d}\, \sin \left (f x +e \right )-c \cos \left (f x +e \right )+d \cos \left (f x +e \right )+c -d}{\sin \left (f x +e \right ) \sqrt {c -d}}\right ) \sqrt {2}\, \sqrt {-d}\, d \right ) \sqrt {2}}{f \sin \left (f x +e \right )^{2} \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, a \sqrt {c -d}\, \sqrt {-d}} \]
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 \[ \int \frac {\sqrt {d \sec \left (f x + e\right ) + c} \sec \left (f x + e\right )}{\sqrt {a \sec \left (f x + e\right ) + a}}\,{d x} \]
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 \[ \int \frac {\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}} \,d x \]
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 \[ \int \frac {\sqrt {c + d \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )}}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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