Optimal. Leaf size=19 \[ -\frac {\sec ^2(e+f x) \tan (e+f x)}{f} \]
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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 = {4128}
\begin {gather*} -\frac {\tan (e+f x) \sec ^2(e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 4128
Rubi steps
\begin {align*} \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx &=-\frac {\sec ^2(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 \begin {gather*} -\frac {\sec ^2(e+f x) \tan (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 34, normalized size = 1.79
method | result | size |
derivativedivides | \(\frac {2 \tan \left (f x +e \right )+3 \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(34\) |
default | \(\frac {2 \tan \left (f x +e \right )+3 \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(34\) |
risch | \(\frac {4 i \left ({\mathrm e}^{4 i \left (f x +e \right )}-{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}\) | \(41\) |
norman | \(\frac {\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {4 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 22, normalized size = 1.16 \begin {gather*} -\frac {\tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.67, size = 21, normalized size = 1.11 \begin {gather*} -\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{3}} \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*} - \int \left (- 2 \sec ^{2}{\left (e + f x \right )}\right )\, dx - \int 3 \sec ^{4}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 20, normalized size = 1.05 \begin {gather*} -\frac {\tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.40, size = 21, normalized size = 1.11 \begin {gather*} -\frac {\mathrm {tan}\left (e+f\,x\right )\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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