Optimal. Leaf size=83 \[ \frac {2 g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \Pi \left (\frac {2 a}{a+b};\frac {1}{2} (e+f x)|\frac {2 c}{c+d}\right )}{f (a+b) \sqrt {c+d \sec (e+f x)}} \]
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Rubi [A] time = 0.43, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3975, 2807, 2805} \[ \frac {2 g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \Pi \left (\frac {2 a}{a+b};\frac {1}{2} (e+f x)|\frac {2 c}{c+d}\right )}{f (a+b) \sqrt {c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2805
Rule 2807
Rule 3975
Rubi steps
\begin {align*} \int \frac {(g \sec (e+f x))^{3/2}}{(a+b \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx &=\frac {\left (g \sqrt {d+c \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{(b+a \cos (e+f x)) \sqrt {d+c \cos (e+f x)}} \, dx}{\sqrt {c+d \sec (e+f x)}}\\ &=\frac {\left (g \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{(b+a \cos (e+f x)) \sqrt {\frac {d}{c+d}+\frac {c \cos (e+f x)}{c+d}}} \, dx}{\sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 g \sqrt {\frac {d+c \cos (e+f x)}{c+d}} \Pi \left (\frac {2 a}{a+b};\frac {1}{2} (e+f x)|\frac {2 c}{c+d}\right ) \sqrt {g \sec (e+f x)}}{(a+b) f \sqrt {c+d \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 83, normalized size = 1.00 \[ \frac {2 g \sqrt {g \sec (e+f x)} \sqrt {\frac {c \cos (e+f x)+d}{c+d}} \Pi \left (\frac {2 a}{a+b};\frac {1}{2} (e+f x)|\frac {2 c}{c+d}\right )}{f (a+b) \sqrt {c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \sec \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.15, size = 236, normalized size = 2.84 \[ -\frac {2 i \left (a \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {c -d}{c +d}}\right )+b \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {c -d}{c +d}}\right )-2 a \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -\frac {a -b}{a +b}, i \sqrt {\frac {c -d}{c +d}}\right )\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (c +d \right )}}\, \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}}{f \left (d +c \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \left (a -b \right ) \left (a +b \right )} \]
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 \[ \int \frac {\left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (b \sec \left (f x + e\right ) + a\right )} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
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 \[ \int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (a + b \sec {\left (e + f x \right )}\right ) \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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