Optimal. Leaf size=47 \[ a^2 c^2 x-\frac {a^2 c^2 \tan (e+f x)}{f}+\frac {a^2 c^2 \tan ^3(e+f x)}{3 f} \]
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Rubi [A]
time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3989, 3554, 8}
\begin {gather*} \frac {a^2 c^2 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^2 \tan (e+f x)}{f}+a^2 c^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3989
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \tan ^4(e+f x) \, dx\\ &=\frac {a^2 c^2 \tan ^3(e+f x)}{3 f}-\left (a^2 c^2\right ) \int \tan ^2(e+f x) \, dx\\ &=-\frac {a^2 c^2 \tan (e+f x)}{f}+\frac {a^2 c^2 \tan ^3(e+f x)}{3 f}+\left (a^2 c^2\right ) \int 1 \, dx\\ &=a^2 c^2 x-\frac {a^2 c^2 \tan (e+f x)}{f}+\frac {a^2 c^2 \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 45, normalized size = 0.96 \begin {gather*} a^2 c^2 \left (\frac {\text {ArcTan}(\tan (e+f x))}{f}-\frac {\tan (e+f x)}{f}+\frac {\tan ^3(e+f x)}{3 f}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 58, normalized size = 1.23
method | result | size |
derivativedivides | \(\frac {-a^{2} c^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-2 a^{2} c^{2} \tan \left (f x +e \right )+a^{2} c^{2} \left (f x +e \right )}{f}\) | \(58\) |
default | \(\frac {-a^{2} c^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-2 a^{2} c^{2} \tan \left (f x +e \right )+a^{2} c^{2} \left (f x +e \right )}{f}\) | \(58\) |
risch | \(a^{2} c^{2} x -\frac {4 i a^{2} c^{2} \left (3 \,{\mathrm e}^{4 i \left (f x +e \right )}+3 \,{\mathrm e}^{2 i \left (f x +e \right )}+2\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}\) | \(59\) |
norman | \(\frac {a^{2} c^{2} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-a^{2} c^{2} x +\frac {2 a^{2} c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {20 a^{2} c^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {2 a^{2} c^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+3 a^{2} c^{2} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-3 a^{2} c^{2} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\) | \(150\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 61, normalized size = 1.30 \begin {gather*} \frac {{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{2} + 3 \, {\left (f x + e\right )} a^{2} c^{2} - 6 \, a^{2} c^{2} \tan \left (f x + e\right )}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.50, size = 69, normalized size = 1.47 \begin {gather*} \frac {3 \, a^{2} c^{2} f x \cos \left (f x + e\right )^{3} - {\left (4 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} - a^{2} c^{2}\right )} \sin \left (f x + e\right )}{3 \, 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*} a^{2} c^{2} \left (\int 1\, dx + \int \left (- 2 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 48, normalized size = 1.02 \begin {gather*} \frac {a^{2} c^{2} \tan \left (f x + e\right )^{3} + 3 \, {\left (f x + e\right )} a^{2} c^{2} - 3 \, a^{2} c^{2} \tan \left (f x + e\right )}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.71, size = 84, normalized size = 1.79 \begin {gather*} a^2\,c^2\,x+\frac {2\,a^2\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-\frac {20\,a^2\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+2\,a^2\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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