Optimal. Leaf size=100 \[ -\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^m}{f \left (4 m^2+8 m+3\right ) \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) \sqrt {c-c \sec (e+f x)} (a \sec (e+f x)+a)^m}{f (2 m+3)} \]
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Rubi [A] time = 0.22, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3955, 3953} \[ -\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a)^m}{f \left (4 m^2+8 m+3\right ) \sqrt {c-c \sec (e+f x)}}-\frac {2 c \tan (e+f x) \sqrt {c-c \sec (e+f x)} (a \sec (e+f x)+a)^m}{f (2 m+3)} \]
Antiderivative was successfully verified.
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Rule 3953
Rule 3955
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{3/2} \, dx &=-\frac {2 c (a+a \sec (e+f x))^m \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f (3+2 m)}+\frac {(4 c) \int \sec (e+f x) (a+a \sec (e+f x))^m \sqrt {c-c \sec (e+f x)} \, dx}{3+2 m}\\ &=-\frac {8 c^2 (a+a \sec (e+f x))^m \tan (e+f x)}{f \left (3+8 m+4 m^2\right ) \sqrt {c-c \sec (e+f x)}}-\frac {2 c (a+a \sec (e+f x))^m \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f (3+2 m)}\\ \end {align*}
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Mathematica [F] time = 38.44, size = 0, normalized size = 0.00 \[ \int \sec (e+f x) (a+a \sec (e+f x))^m (c-c \sec (e+f x))^{3/2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 112, normalized size = 1.12 \[ \frac {2 \, {\left ({\left (2 \, c m + 5 \, c\right )} \cos \left (f x + e\right )^{2} - 2 \, c m + 4 \, c \cos \left (f x + e\right ) - c\right )} \left (\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}\right )^{m} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{{\left (4 \, f m^{2} + 8 \, f m + 3 \, f\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right )} \]
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 \[ \int {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}} {\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.63, size = 0, normalized size = 0.00 \[ \int \sec \left (f x +e \right ) \left (a +a \sec \left (f x +e \right )\right )^{m} \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 171, normalized size = 1.71 \[ -\frac {2 \, {\left (\sqrt {2} 2^{m + 2} \left (-a\right )^{m} c^{\frac {3}{2}} - \frac {\sqrt {2} {\left (2^{m + 2} m + 3 \cdot 2^{m + 1}\right )} \left (-a\right )^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} e^{\left (-m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )\right )}}{{\left (4 \, m^{2} + 8 \, m + 3\right )} f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 154, normalized size = 1.54 \[ -\frac {2\,c\,{\left (\frac {a\,\left (\cos \left (e+f\,x\right )+1\right )}{\cos \left (e+f\,x\right )}\right )}^m\,\sqrt {\frac {c\,\left (\cos \left (e+f\,x\right )-1\right )}{\cos \left (e+f\,x\right )}}\,\left (5\,\sin \left (e+f\,x\right )-2\,\sin \left (2\,e+2\,f\,x\right )+5\,\sin \left (3\,e+3\,f\,x\right )+2\,m\,\sin \left (e+f\,x\right )-4\,m\,\sin \left (2\,e+2\,f\,x\right )+2\,m\,\sin \left (3\,e+3\,f\,x\right )\right )}{f\,\left (4\,m^2+8\,m+3\right )\,\left (3\,\cos \left (e+f\,x\right )-2\,\cos \left (2\,e+2\,f\,x\right )+\cos \left (3\,e+3\,f\,x\right )-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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