Optimal. Leaf size=38 \[ \frac {a^2 c^2 \tan (e+f x)}{f}+\frac {a^2 c^2 \tan ^3(e+f x)}{3 f} \]
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Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {3603, 3852}
\begin {gather*} \frac {a^2 c^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 c^2 \tan (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3603
Rule 3852
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \sec ^4(e+f x) \, dx\\ &=-\frac {\left (a^2 c^2\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=\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 = 0.06, size = 29, normalized size = 0.76 \begin {gather*} \frac {a^2 c^2 \left (\tan (e+f x)+\frac {1}{3} \tan ^3(e+f x)\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 28, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {a^{2} c^{2} \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}+\tan \left (f x +e \right )\right )}{f}\) | \(28\) |
default | \(\frac {a^{2} c^{2} \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}+\tan \left (f x +e \right )\right )}{f}\) | \(28\) |
norman | \(\frac {a^{2} c^{2} \tan \left (f x +e \right )}{f}+\frac {a^{2} c^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}\) | \(37\) |
risch | \(\frac {4 i a^{2} c^{2} \left (3 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 37, normalized size = 0.97 \begin {gather*} \frac {a^{2} c^{2} \tan \left (f x + e\right )^{3} + 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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Fricas [C] Result contains complex when optimal does not.
time = 1.17, size = 71, normalized size = 1.87 \begin {gather*} -\frac {4 \, {\left (-3 i \, a^{2} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{2} c^{2}\right )}}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.17, size = 94, normalized size = 2.47 \begin {gather*} \frac {12 i a^{2} c^{2} e^{2 i e} e^{2 i f x} + 4 i a^{2} c^{2}}{3 f e^{6 i e} e^{6 i f x} + 9 f e^{4 i e} e^{4 i f x} + 9 f e^{2 i e} e^{2 i f x} + 3 f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 166 vs.
\(2 (38) = 76\).
time = 0.56, size = 166, normalized size = 4.37 \begin {gather*} -\frac {3 \, a^{2} c^{2} \tan \left (f x\right )^{3} \tan \left (e\right )^{2} + 3 \, a^{2} c^{2} \tan \left (f x\right )^{2} \tan \left (e\right )^{3} + a^{2} c^{2} \tan \left (f x\right )^{3} - 3 \, a^{2} c^{2} \tan \left (f x\right )^{2} \tan \left (e\right ) - 3 \, a^{2} c^{2} \tan \left (f x\right ) \tan \left (e\right )^{2} + a^{2} c^{2} \tan \left (e\right )^{3} + 3 \, a^{2} c^{2} \tan \left (f x\right ) + 3 \, a^{2} c^{2} \tan \left (e\right )}{3 \, {\left (f \tan \left (f x\right )^{3} \tan \left (e\right )^{3} - 3 \, f \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + 3 \, f \tan \left (f x\right ) \tan \left (e\right ) - f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.64, size = 27, normalized size = 0.71 \begin {gather*} \frac {a^2\,c^2\,\mathrm {tan}\left (e+f\,x\right )\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2+3\right )}{3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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