Optimal. Leaf size=17 \[ -\frac {a c \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3962, 2607, 30} \[ -\frac {a c \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 3962
Rubi steps
\begin {align*} \int \sec ^2(e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx\right )\\ &=-\frac {(a c) \operatorname {Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a c \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ -\frac {a c \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 35, normalized size = 2.06 \[ \frac {{\left (a c \cos \left (f x + e\right )^{2} - a c\right )} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.39, size = 16, normalized size = 0.94 \[ -\frac {a c \tan \left (f x + e\right )^{3}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.94, size = 36, normalized size = 2.12 \[ \frac {c a \tan \left (f x +e \right )+c a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 36, normalized size = 2.12 \[ -\frac {{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a c - 3 \, a c \tan \left (f x + e\right )}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.70, size = 15, normalized size = 0.88 \[ -\frac {a\,c\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.30, size = 51, normalized size = 3.00 \[ \begin {cases} \frac {- a c \left (\frac {\tan ^{3}{\left (e + f x \right )}}{3} + \tan {\left (e + f x \right )}\right ) + a c \tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sec {\relax (e )} + a\right ) \left (- c \sec {\relax (e )} + c\right ) \sec ^{2}{\relax (e )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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