Optimal. Leaf size=36 \[ \frac {(c-c \sec (e+f x)) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {4035}
\begin {gather*} \frac {\tan (e+f x) (c-c \sec (e+f x))}{3 f (a \sec (e+f x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 4035
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx &=\frac {(c-c \sec (e+f x)) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 23, normalized size = 0.64 \begin {gather*} -\frac {c \tan ^3\left (\frac {1}{2} (e+f x)\right )}{3 a^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 21, normalized size = 0.58
method | result | size |
derivativedivides | \(-\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f \,a^{2}}\) | \(21\) |
default | \(-\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f \,a^{2}}\) | \(21\) |
risch | \(\frac {2 i c \left (3 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}\) | \(37\) |
norman | \(\frac {\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}-\frac {c \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (38) = 76\).
time = 0.27, size = 102, normalized size = 2.83 \begin {gather*} -\frac {\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}} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.41, size = 57, normalized size = 1.58 \begin {gather*} \frac {{\left (c \cos \left (f x + e\right ) - c\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 )}} \end {gather*}
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 \begin {gather*} - \frac {c \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 20, normalized size = 0.56 \begin {gather*} -\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}{3 \, a^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.57, size = 20, normalized size = 0.56 \begin {gather*} -\frac {c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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