Optimal. Leaf size=65 \[ -\frac {a c \tan ^3(e+f x) \cos ^2(e+f x)^{\frac {p+3}{2}} (g \sec (e+f x))^p \, _2F_1\left (\frac {3}{2},\frac {p+3}{2};\frac {5}{2};\sin ^2(e+f x)\right )}{3 f} \]
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Rubi [A] time = 0.09, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3962, 2617} \[ -\frac {a c \tan ^3(e+f x) \cos ^2(e+f x)^{\frac {p+3}{2}} (g \sec (e+f x))^p \, _2F_1\left (\frac {3}{2},\frac {p+3}{2};\frac {5}{2};\sin ^2(e+f x)\right )}{3 f} \]
Antiderivative was successfully verified.
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Rule 2617
Rule 3962
Rubi steps
\begin {align*} \int (g \sec (e+f x))^p (a+a \sec (e+f x)) (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int (g \sec (e+f x))^p \tan ^2(e+f x) \, dx\right )\\ &=-\frac {a c \cos ^2(e+f x)^{\frac {3+p}{2}} \, _2F_1\left (\frac {3}{2},\frac {3+p}{2};\frac {5}{2};\sin ^2(e+f x)\right ) (g \sec (e+f x))^p \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 72, normalized size = 1.11 \[ -\frac {a c \tan (e+f x) (g \sec (e+f x))^p \left (\frac {\, _2F_1\left (\frac {1}{2},\frac {p}{2};\frac {p+2}{2};\sec ^2(e+f x)\right )}{\sqrt {-\tan ^2(e+f x)}}+p\right )}{f p (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c \sec \left (f x + e\right )^{2} - a c\right )} \left (g \sec \left (f x + e\right )\right )^{p}, x\right ) \]
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 \[ \int -{\left (a \sec \left (f x + e\right ) + a\right )} {\left (c \sec \left (f x + e\right ) - c\right )} \left (g \sec \left (f x + e\right )\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.62, size = 0, normalized size = 0.00 \[ \int \left (g \sec \left (f x +e \right )\right )^{p} \left (a +a \sec \left (f x +e \right )\right ) \left (c -c \sec \left (f x +e \right )\right )\, dx \]
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 \[ -\int {\left (a \sec \left (f x + e\right ) + a\right )} {\left (c \sec \left (f x + e\right ) - c\right )} \left (g \sec \left (f x + e\right )\right )^{p}\,{d x} \]
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 \[ \int \left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )\,\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )\,{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^p \,d x \]
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 \[ - a c \left (\int \left (- \left (g \sec {\left (e + f x \right )}\right )^{p}\right )\, dx + \int \left (g \sec {\left (e + f x \right )}\right )^{p} \sec ^{2}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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