Optimal. Leaf size=65 \[ \frac {(c+2 d) \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {(c-d) \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2} \]
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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 = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {4000, 3794} \[ \frac {(c+2 d) \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {(c-d) \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3794
Rule 4000
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx &=\frac {(c-d) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {(c+2 d) \int \frac {\sec (e+f x)}{a+a \sec (e+f x)} \, dx}{3 a}\\ &=\frac {(c-d) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {(c+2 d) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 76, normalized size = 1.17 \[ \frac {\sec \left (\frac {e}{2}\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \left ((2 c+d) \sin \left (e+\frac {3 f x}{2}\right )+3 (c+d) \sin \left (\frac {f x}{2}\right )-3 c \sin \left (e+\frac {f x}{2}\right )\right )}{3 a^2 f (\cos (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 58, normalized size = 0.89 \[ \frac {{\left ({\left (2 \, c + d\right )} \cos \left (f x + e\right ) + c + 2 \, d\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 64, normalized size = 0.98 \[ -\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{6 \, a^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 60, normalized size = 0.92 \[ \frac {-\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) c}{3}+\frac {\left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right ) d}{3}+\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c +\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) d}{2 f \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 93, normalized size = 1.43 \[ \frac {\frac {d {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} + \frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.71, size = 45, normalized size = 0.69 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (c+d\right )}{2\,a^2\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (c-d\right )}{6\,a^2\,f} \]
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 \[ \frac {\int \frac {c \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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