Optimal. Leaf size=107 \[ \frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right )}{b^2 f} \]
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Rubi [A] time = 0.08, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {4004} \[ \frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right )}{b^2 f} \]
Antiderivative was successfully verified.
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Rule 4004
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx &=\frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) E\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}}}{b^2 f}\\ \end {align*}
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Mathematica [A] time = 7.94, size = 211, normalized size = 1.97 \[ \frac {A (a+b) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left (\sqrt {\frac {\cos (e+f x)}{\cos (e+f x)+1}} \sqrt {\sec (e+f x)+1} E\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )-\sin (e+f x) \sqrt {\frac {1}{\cos (e+f x)+1}} \sqrt {\sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}}\right )}{b f \left (\frac {1}{\cos (e+f x)+1}\right )^{3/2} \sqrt {a+b \sec (e+f x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {A \sec \left (f x + e\right )^{2} - A \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}}, x\right ) \]
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 \[ \int -\frac {{\left (A \sec \left (f x + e\right ) - A\right )} \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.92, size = 457, normalized size = 4.27 \[ -\frac {2 A \sqrt {\frac {b +a \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \left (1+\cos \left (f x +e \right )\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (\cos \left (f x +e \right ) \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sin \left (f x +e \right ) a +\cos \left (f x +e \right ) \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sin \left (f x +e \right ) b +\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sin \left (f x +e \right ) a +\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sin \left (f x +e \right ) b -a \left (\cos ^{2}\left (f x +e \right )\right )+a \cos \left (f x +e \right )-b \cos \left (f x +e \right )+b \right )}{f \sin \left (f x +e \right )^{5} \left (b +a \cos \left (f x +e \right )\right ) b} \]
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 \[ -\int \frac {{\left (A \sec \left (f x + e\right ) - A\right )} \sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a}}\,{d x} \]
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 \[ \int \frac {A-\frac {A}{\cos \left (e+f\,x\right )}}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]
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 \[ - A \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\right )\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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