Optimal. Leaf size=59 \[ \frac {a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac {2 a^3 c^3 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^3 \tan (e+f x)}{f} \]
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Rubi [A] time = 0.07, antiderivative size = 59, 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 = {3522, 3767} \[ \frac {a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac {2 a^3 c^3 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^3 \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3767
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^3 \, dx &=\left (a^3 c^3\right ) \int \sec ^6(e+f x) \, dx\\ &=-\frac {\left (a^3 c^3\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=\frac {a^3 c^3 \tan (e+f x)}{f}+\frac {2 a^3 c^3 \tan ^3(e+f x)}{3 f}+\frac {a^3 c^3 \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 41, normalized size = 0.69 \[ \frac {a^3 c^3 \left (\frac {1}{5} \tan ^5(e+f x)+\frac {2}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.41, size = 108, normalized size = 1.83 \[ \frac {160 i \, a^{3} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 80 i \, a^{3} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 16 i \, a^{3} c^{3}}{15 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.07, size = 371, normalized size = 6.29 \[ -\frac {15 \, a^{3} c^{3} \tan \left (f x\right )^{5} \tan \relax (e)^{4} + 15 \, a^{3} c^{3} \tan \left (f x\right )^{4} \tan \relax (e)^{5} + 10 \, a^{3} c^{3} \tan \left (f x\right )^{5} \tan \relax (e)^{2} - 30 \, a^{3} c^{3} \tan \left (f x\right )^{4} \tan \relax (e)^{3} - 30 \, a^{3} c^{3} \tan \left (f x\right )^{3} \tan \relax (e)^{4} + 10 \, a^{3} c^{3} \tan \left (f x\right )^{2} \tan \relax (e)^{5} + 3 \, a^{3} c^{3} \tan \left (f x\right )^{5} - 5 \, a^{3} c^{3} \tan \left (f x\right )^{4} \tan \relax (e) + 60 \, a^{3} c^{3} \tan \left (f x\right )^{3} \tan \relax (e)^{2} + 60 \, a^{3} c^{3} \tan \left (f x\right )^{2} \tan \relax (e)^{3} - 5 \, a^{3} c^{3} \tan \left (f x\right ) \tan \relax (e)^{4} + 3 \, a^{3} c^{3} \tan \relax (e)^{5} + 10 \, a^{3} c^{3} \tan \left (f x\right )^{3} - 30 \, a^{3} c^{3} \tan \left (f x\right )^{2} \tan \relax (e) - 30 \, a^{3} c^{3} \tan \left (f x\right ) \tan \relax (e)^{2} + 10 \, a^{3} c^{3} \tan \relax (e)^{3} + 15 \, a^{3} c^{3} \tan \left (f x\right ) + 15 \, a^{3} c^{3} \tan \relax (e)}{15 \, {\left (f \tan \left (f x\right )^{5} \tan \relax (e)^{5} - 5 \, f \tan \left (f x\right )^{4} \tan \relax (e)^{4} + 10 \, f \tan \left (f x\right )^{3} \tan \relax (e)^{3} - 10 \, f \tan \left (f x\right )^{2} \tan \relax (e)^{2} + 5 \, f \tan \left (f x\right ) \tan \relax (e) - f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 38, normalized size = 0.64 \[ \frac {a^{3} c^{3} \left (\frac {\left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {2 \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\tan \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 52, normalized size = 0.88 \[ \frac {3 \, a^{3} c^{3} \tan \left (f x + e\right )^{5} + 10 \, a^{3} c^{3} \tan \left (f x + e\right )^{3} + 15 \, a^{3} c^{3} \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.73, size = 39, normalized size = 0.66 \[ \frac {a^3\,c^3\,\mathrm {tan}\left (e+f\,x\right )\,\left (3\,{\mathrm {tan}\left (e+f\,x\right )}^4+10\,{\mathrm {tan}\left (e+f\,x\right )}^2+15\right )}{15\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.52, size = 156, normalized size = 2.64 \[ \frac {160 i a^{3} c^{3} e^{4 i e} e^{4 i f x} + 80 i a^{3} c^{3} e^{2 i e} e^{2 i f x} + 16 i a^{3} c^{3}}{15 f e^{10 i e} e^{10 i f x} + 75 f e^{8 i e} e^{8 i f x} + 150 f e^{6 i e} e^{6 i f x} + 150 f e^{4 i e} e^{4 i f x} + 75 f e^{2 i e} e^{2 i f x} + 15 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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