Optimal. Leaf size=42 \[ -\frac{(a d+b c) \log (\cos (e+f x))}{f}+x (a c-b d)+\frac{b d \tan (e+f x)}{f} \]
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Rubi [A] time = 0.0253722, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3525, 3475} \[ -\frac{(a d+b c) \log (\cos (e+f x))}{f}+x (a c-b d)+\frac{b d \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int (a+b \tan (e+f x)) (c+d \tan (e+f x)) \, dx &=(a c-b d) x+\frac{b d \tan (e+f x)}{f}+(b c+a d) \int \tan (e+f x) \, dx\\ &=(a c-b d) x-\frac{(b c+a d) \log (\cos (e+f x))}{f}+\frac{b d \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0326185, size = 59, normalized size = 1.4 \[ a c x-\frac{a d \log (\cos (e+f x))}{f}-\frac{b c \log (\cos (e+f x))}{f}-\frac{b d \tan ^{-1}(\tan (e+f x))}{f}+\frac{b d \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 77, normalized size = 1.8 \begin{align*}{\frac{\tan \left ( fx+e \right ) bd}{f}}+{\frac{a\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) d}{2\,f}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) bc}{2\,f}}+{\frac{a\arctan \left ( \tan \left ( fx+e \right ) \right ) c}{f}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) bd}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57196, size = 68, normalized size = 1.62 \begin{align*} \frac{2 \, b d \tan \left (f x + e\right ) + 2 \,{\left (a c - b d\right )}{\left (f x + e\right )} +{\left (b c + a d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50945, size = 122, normalized size = 2.9 \begin{align*} \frac{2 \,{\left (a c - b d\right )} f x + 2 \, b d \tan \left (f x + e\right ) -{\left (b c + a d\right )} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.222615, size = 73, normalized size = 1.74 \begin{align*} \begin{cases} a c x + \frac{a d \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac{b c \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - b d x + \frac{b d \tan{\left (e + f x \right )}}{f} & \text{for}\: f \neq 0 \\x \left (a + b \tan{\left (e \right )}\right ) \left (c + d \tan{\left (e \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.37717, size = 479, normalized size = 11.4 \begin{align*} \frac{2 \, a c f x \tan \left (f x\right ) \tan \left (e\right ) - 2 \, b d f x \tan \left (f x\right ) \tan \left (e\right ) - b c \log \left (\frac{4 \,{\left (\tan \left (e\right )^{2} + 1\right )}}{\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1}\right ) \tan \left (f x\right ) \tan \left (e\right ) - a d \log \left (\frac{4 \,{\left (\tan \left (e\right )^{2} + 1\right )}}{\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1}\right ) \tan \left (f x\right ) \tan \left (e\right ) - 2 \, a c f x + 2 \, b d f x + b c \log \left (\frac{4 \,{\left (\tan \left (e\right )^{2} + 1\right )}}{\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1}\right ) + a d \log \left (\frac{4 \,{\left (\tan \left (e\right )^{2} + 1\right )}}{\tan \left (f x\right )^{4} \tan \left (e\right )^{2} - 2 \, \tan \left (f x\right )^{3} \tan \left (e\right ) + \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + \tan \left (f x\right )^{2} - 2 \, \tan \left (f x\right ) \tan \left (e\right ) + 1}\right ) - 2 \, b d \tan \left (f x\right ) - 2 \, b d \tan \left (e\right )}{2 \,{\left (f \tan \left (f x\right ) \tan \left (e\right ) - f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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