Optimal. Leaf size=62 \[ \frac {a^2 A c^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 A c^2 \tan (e+f x)}{f}+\frac {a^2 B c^2 \sec ^4(e+f x)}{4 f} \]
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Rubi [A] time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3588, 73, 641} \[ \frac {a^2 A c^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 A c^2 \tan (e+f x)}{f}+\frac {a^2 B c^2 \sec ^4(e+f x)}{4 f} \]
Antiderivative was successfully verified.
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Rule 73
Rule 641
Rule 3588
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx &=\frac {(a c) \operatorname {Subst}(\int (a+i a x) (A+B x) (c-i c x) \, dx,x,\tan (e+f x))}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int (A+B x) \left (a c+a c x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^2 B c^2 \sec ^4(e+f x)}{4 f}+\frac {(a A c) \operatorname {Subst}\left (\int \left (a c+a c x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^2 B c^2 \sec ^4(e+f x)}{4 f}+\frac {a^2 A c^2 \tan (e+f x)}{f}+\frac {a^2 A c^2 \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 53, normalized size = 0.85 \[ \frac {a^2 A c^2 \left (\frac {1}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{f}+\frac {a^2 B c^2 \sec ^4(e+f x)}{4 f} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.68, size = 104, normalized size = 1.68 \[ \frac {{\left (12 i \, A + 12 \, B\right )} a^{2} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 16 i \, A a^{2} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, A a^{2} c^{2}}{3 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.87, size = 411, normalized size = 6.63 \[ \frac {3 \, B a^{2} c^{2} \tan \left (f x\right )^{4} \tan \relax (e)^{4} - 12 \, A a^{2} c^{2} \tan \left (f x\right )^{4} \tan \relax (e)^{3} - 12 \, A a^{2} c^{2} \tan \left (f x\right )^{3} \tan \relax (e)^{4} + 6 \, B a^{2} c^{2} \tan \left (f x\right )^{4} \tan \relax (e)^{2} + 6 \, B a^{2} c^{2} \tan \left (f x\right )^{2} \tan \relax (e)^{4} - 4 \, A a^{2} c^{2} \tan \left (f x\right )^{4} \tan \relax (e) + 24 \, A a^{2} c^{2} \tan \left (f x\right )^{3} \tan \relax (e)^{2} + 24 \, A a^{2} c^{2} \tan \left (f x\right )^{2} \tan \relax (e)^{3} - 4 \, A a^{2} c^{2} \tan \left (f x\right ) \tan \relax (e)^{4} + 3 \, B a^{2} c^{2} \tan \left (f x\right )^{4} + 12 \, B a^{2} c^{2} \tan \left (f x\right )^{2} \tan \relax (e)^{2} + 3 \, B a^{2} c^{2} \tan \relax (e)^{4} + 4 \, A a^{2} c^{2} \tan \left (f x\right )^{3} - 24 \, A a^{2} c^{2} \tan \left (f x\right )^{2} \tan \relax (e) - 24 \, A a^{2} c^{2} \tan \left (f x\right ) \tan \relax (e)^{2} + 4 \, A a^{2} c^{2} \tan \relax (e)^{3} + 6 \, B a^{2} c^{2} \tan \left (f x\right )^{2} + 6 \, B a^{2} c^{2} \tan \relax (e)^{2} + 12 \, A a^{2} c^{2} \tan \left (f x\right ) + 12 \, A a^{2} c^{2} \tan \relax (e) + 3 \, B a^{2} c^{2}}{12 \, {\left (f \tan \left (f x\right )^{4} \tan \relax (e)^{4} - 4 \, f \tan \left (f x\right )^{3} \tan \relax (e)^{3} + 6 \, f \tan \left (f x\right )^{2} \tan \relax (e)^{2} - 4 \, f \tan \left (f x\right ) \tan \relax (e) + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 53, normalized size = 0.85 \[ \frac {a^{2} c^{2} \left (\frac {B \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {A \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 72, normalized size = 1.16 \[ \frac {3 \, B a^{2} c^{2} \tan \left (f x + e\right )^{4} + 4 \, A a^{2} c^{2} \tan \left (f x + e\right )^{3} + 6 \, B a^{2} c^{2} \tan \left (f x + e\right )^{2} + 12 \, A a^{2} c^{2} \tan \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.49, size = 82, normalized size = 1.32 \[ \frac {a^2\,c^2\,\sin \left (e+f\,x\right )\,\left (12\,A\,{\cos \left (e+f\,x\right )}^3+6\,B\,{\cos \left (e+f\,x\right )}^2\,\sin \left (e+f\,x\right )+4\,A\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^2+3\,B\,{\sin \left (e+f\,x\right )}^3\right )}{12\,f\,{\cos \left (e+f\,x\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.60, size = 167, normalized size = 2.69 \[ \frac {16 A a^{2} c^{2} e^{2 i e} e^{2 i f x} + 4 A a^{2} c^{2} + \left (12 A a^{2} c^{2} e^{4 i e} - 12 i B a^{2} c^{2} e^{4 i e}\right ) e^{4 i f x}}{- 3 i f e^{8 i e} e^{8 i f x} - 12 i f e^{6 i e} e^{6 i f x} - 18 i f e^{4 i e} e^{4 i f x} - 12 i f e^{2 i e} e^{2 i f x} - 3 i f} \]
Verification of antiderivative is not currently implemented for this CAS.
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