Optimal. Leaf size=116 \[ \frac {g \cot (e+f x) \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}{a c f}-\frac {g^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{\sqrt {2} \sqrt {a} c f} \]
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Rubi [A] time = 0.30, antiderivative size = 150, normalized size of antiderivative = 1.29, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3964, 94, 93, 205} \[ \frac {g^{3/2} \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right )}{\sqrt {2} \sqrt {c} f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {g \tan (e+f x) \sqrt {g \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 205
Rule 3964
Rubi steps
\begin {align*} \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx &=-\frac {(a c g \tan (e+f x)) \operatorname {Subst}\left (\int \frac {\sqrt {g x}}{(a+a x) (c-c x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {g \sqrt {g \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {\left (a g^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {g x} (a+a x) \sqrt {c-c x}} \, dx,x,\sec (e+f x)\right )}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {g \sqrt {g \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {\left (a g^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{a g+2 a c x^2} \, dx,x,\frac {\sqrt {g \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}}\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {g \sqrt {g \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac {g^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {g \sec (e+f x)}}{\sqrt {g} \sqrt {c-c \sec (e+f x)}}\right ) \tan (e+f x)}{\sqrt {2} \sqrt {c} f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 2.98, size = 236, normalized size = 2.03 \[ -\frac {a \sin ^3\left (\frac {1}{2} (e+f x)\right ) \cos \left (\frac {1}{2} (e+f x)\right ) (g \sec (e+f x))^{5/2} \left (-4 \sec (e+f x)+\frac {\sqrt {\tan ^2(e+f x)} \left (\log \left (-3 \sec ^2(e+f x)-2 \sec (e+f x)-2 \sqrt {2} \sqrt {\tan ^2(e+f x)} \sqrt {\sec (e+f x)+1} \sqrt {\sec (e+f x)}+1\right )-\log \left (-3 \sec ^2(e+f x)-2 \sec (e+f x)+2 \sqrt {2} \sqrt {\tan ^2(e+f x)} \sqrt {\sec (e+f x)+1} \sqrt {\sec (e+f x)}+1\right )\right )}{\sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )}}-4\right )}{c f g (\sec (e+f x)-1)^2 (a (\sec (e+f x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 330, normalized size = 2.84 \[ \left [\frac {\sqrt {2} a g \sqrt {\frac {g}{a}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + g \cos \left (f x + e\right )^{2} - 2 \, g \cos \left (f x + e\right ) - 3 \, g}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 4 \, g \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{4 \, a c f \sin \left (f x + e\right )}, \frac {\sqrt {2} a g \sqrt {-\frac {g}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {-\frac {g}{a}} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{g \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, g \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{2 \, a c f \sin \left (f x + e\right )}\right ] \]
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 \[ \int -\frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (c \sec \left (f x + e\right ) - c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.70, size = 152, normalized size = 1.31 \[ \frac {\left (\cos \left (f x +e \right ) \sqrt {2}\, \arcsinh \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-\sqrt {2}\, \arcsinh \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-2 \sin \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\right ) \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (-1+\cos \left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}}{2 c f \left (\frac {1}{1+\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \sin \left (f x +e \right )^{4} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.09, size = 536, normalized size = 4.62 \[ \frac {{\left (4 \, g \cos \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 4 \, g \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) \sin \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - {\left (g \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + g \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, g \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + g\right )} \log \left (\cos \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + {\left (g \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + g \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, g \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + g\right )} \log \left (\cos \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 4 \, g \sin \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {g}}{2 \, {\left (\sqrt {2} c \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sqrt {2} c \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sqrt {2} c \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + \sqrt {2} c\right )} \sqrt {a} f} \]
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 \[ \int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )} \,d x \]
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 \[ - \frac {\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} - \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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