Optimal. Leaf size=38 \[ \frac {\tan (e+f x) (c-c \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 38, 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 = {3950} \[ \frac {\tan (e+f x) (c-c \sec (e+f x))^2}{5 f (a \sec (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3950
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^3} \, dx &=\frac {(c-c \sec (e+f x))^2 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 25, normalized size = 0.66 \[ \frac {c^2 \tan ^5\left (\frac {1}{2} (e+f x)\right )}{5 a^3 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 82, normalized size = 2.16 \[ \frac {{\left (c^{2} \cos \left (f x + e\right )^{2} - 2 \, c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \sin \left (f x + e\right )}{5 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.80, size = 23, normalized size = 0.61 \[ \frac {c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}}{5 \, a^{3} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 23, normalized size = 0.61 \[ \frac {c^{2} \left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{5 f \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 185, normalized size = 4.87 \[ \frac {\frac {c^{2} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac {c^{2} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {6 \, c^{2} {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 22, normalized size = 0.58 \[ \frac {c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,a^3\,f} \]
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 \[ \frac {c^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {2 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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