Optimal. Leaf size=19 \[ -\frac {\tan (e+f x) \sec ^4(e+f x)}{f} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4043} \[ -\frac {\tan (e+f x) \sec ^4(e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 4043
Rubi steps
\begin {align*} \int \sec ^4(e+f x) \left (4-5 \sec ^2(e+f x)\right ) \, dx &=-\frac {\sec ^4(e+f x) \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 19, normalized size = 1.00 \[ -\frac {\tan (e+f x) \sec ^4(e+f x)}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 19, normalized size = 1.00 \[ -\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 33, normalized size = 1.74 \[ -\frac {\tan \left (f x + e\right )^{5} + 2 \, \tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.17, size = 56, normalized size = 2.95 \[ \frac {-4 \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )+5 \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 30, normalized size = 1.58 \[ -\frac {\tan \left (f x + e\right )^{5} + 2 \, \tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.39, size = 19, normalized size = 1.00 \[ -\frac {\sin \left (e+f\,x\right )}{f\,{\cos \left (e+f\,x\right )}^5} \]
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 \[ - \int \left (- 4 \sec ^{4}{\left (e + f x \right )}\right )\, dx - \int 5 \sec ^{6}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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