Optimal. Leaf size=32 \[ \frac {a A c \tan (e+f x)}{f}+\frac {a B c \tan ^2(e+f x)}{2 f} \]
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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {3669}
\begin {gather*} \frac {a A c \tan (e+f x)}{f}+\frac {a B c \tan ^2(e+f x)}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3669
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx &=\frac {(a c) \text {Subst}(\int (A+B x) \, dx,x,\tan (e+f x))}{f}\\ &=\frac {a A c \tan (e+f x)}{f}+\frac {a B c \tan ^2(e+f x)}{2 f}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 32, normalized size = 1.00 \begin {gather*} \frac {a B c \sec ^2(e+f x)}{2 f}+\frac {a A c \tan (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 27, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {a c \left (\frac {B \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f}\) | \(27\) |
default | \(\frac {a c \left (\frac {B \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f}\) | \(27\) |
norman | \(\frac {a A c \tan \left (f x +e \right )}{f}+\frac {a B c \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}\) | \(31\) |
risch | \(\frac {2 a c \left (i A \,{\mathrm e}^{2 i \left (f x +e \right )}+B \,{\mathrm e}^{2 i \left (f x +e \right )}+i A \right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 31, normalized size = 0.97 \begin {gather*} \frac {B a c \tan \left (f x + e\right )^{2} + 2 \, A a c \tan \left (f x + e\right )}{2 \, 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 = 2.82, size = 57, normalized size = 1.78 \begin {gather*} -\frac {2 \, {\left ({\left (-i \, A - B\right )} a c e^{\left (2 i \, f x + 2 i \, e\right )} - i \, A a c\right )}}{f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \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 = 0.16, size = 82, normalized size = 2.56 \begin {gather*} \frac {2 i A a c + \left (2 i A a c e^{2 i e} + 2 B a c e^{2 i e}\right ) e^{2 i f x}}{f e^{4 i e} e^{4 i f x} + 2 f e^{2 i e} e^{2 i f x} + 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. 113 vs.
\(2 (32) = 64\).
time = 0.54, size = 113, normalized size = 3.53 \begin {gather*} \frac {B a c \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 2 \, A a c \tan \left (f x\right )^{2} \tan \left (e\right ) - 2 \, A a c \tan \left (f x\right ) \tan \left (e\right )^{2} + B a c \tan \left (f x\right )^{2} + B a c \tan \left (e\right )^{2} + 2 \, A a c \tan \left (f x\right ) + 2 \, A a c \tan \left (e\right ) + B a c}{2 \, {\left (f \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 2 \, 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 = 8.42, size = 25, normalized size = 0.78 \begin {gather*} \frac {a\,c\,\mathrm {tan}\left (e+f\,x\right )\,\left (2\,A+B\,\mathrm {tan}\left (e+f\,x\right )\right )}{2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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