Optimal. Leaf size=89 \[ -\frac{\left (2 a c d+b \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}-x \left (2 b c d-a \left (c^2-d^2\right )\right )+\frac{d (a d+b c) \tan (e+f x)}{f}+\frac{b (c+d \tan (e+f x))^2}{2 f} \]
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Rubi [A] time = 0.0844995, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3528, 3525, 3475} \[ -\frac{\left (2 a c d+b \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}-x \left (2 b c d-a \left (c^2-d^2\right )\right )+\frac{d (a d+b c) \tan (e+f x)}{f}+\frac{b (c+d \tan (e+f x))^2}{2 f} \]
Antiderivative was successfully verified.
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Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^2 \, dx &=\frac{b (c+d \tan (e+f x))^2}{2 f}+\int (c+d \tan (e+f x)) (a c-b d+(b c+a d) \tan (e+f x)) \, dx\\ &=-\left (2 b c d-a \left (c^2-d^2\right )\right ) x+\frac{d (b c+a d) \tan (e+f x)}{f}+\frac{b (c+d \tan (e+f x))^2}{2 f}+\left (2 a c d+b \left (c^2-d^2\right )\right ) \int \tan (e+f x) \, dx\\ &=-\left (2 b c d-a \left (c^2-d^2\right )\right ) x-\frac{\left (2 a c d+b \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}+\frac{d (b c+a d) \tan (e+f x)}{f}+\frac{b (c+d \tan (e+f x))^2}{2 f}\\ \end{align*}
Mathematica [C] time = 0.443712, size = 96, normalized size = 1.08 \[ \frac{2 d (a d+2 b c) \tan (e+f x)+(b+i a) (c-i d)^2 \log (\tan (e+f x)+i)+(b-i a) (c+i d)^2 \log (-\tan (e+f x)+i)+b d^2 \tan ^2(e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 151, normalized size = 1.7 \begin{align*}{\frac{b{d}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{2\,f}}+{\frac{a\tan \left ( fx+e \right ){d}^{2}}{f}}+2\,{\frac{bcd\tan \left ( fx+e \right ) }{f}}+{\frac{a\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) cd}{f}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) b{c}^{2}}{2\,f}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) b{d}^{2}}{2\,f}}+{\frac{a\arctan \left ( \tan \left ( fx+e \right ) \right ){c}^{2}}{f}}-{\frac{a\arctan \left ( \tan \left ( fx+e \right ) \right ){d}^{2}}{f}}-2\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) bcd}{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78973, size = 123, normalized size = 1.38 \begin{align*} \frac{b d^{2} \tan \left (f x + e\right )^{2} + 2 \,{\left (a c^{2} - 2 \, b c d - a d^{2}\right )}{\left (f x + e\right )} +{\left (b c^{2} + 2 \, a c d - b d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 2 \,{\left (2 \, b c d + a d^{2}\right )} \tan \left (f x + e\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46949, size = 209, normalized size = 2.35 \begin{align*} \frac{b d^{2} \tan \left (f x + e\right )^{2} + 2 \,{\left (a c^{2} - 2 \, b c d - a d^{2}\right )} f x -{\left (b c^{2} + 2 \, a c d - b d^{2}\right )} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left (2 \, b c d + a d^{2}\right )} \tan \left (f x + e\right )}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.3717, size = 143, normalized size = 1.61 \begin{align*} \begin{cases} a c^{2} x + \frac{a c d \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - a d^{2} x + \frac{a d^{2} \tan{\left (e + f x \right )}}{f} + \frac{b c^{2} \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 2 b c d x + \frac{2 b c d \tan{\left (e + f x \right )}}{f} - \frac{b d^{2} \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac{b d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} & \text{for}\: f \neq 0 \\x \left (a + b \tan{\left (e \right )}\right ) \left (c + d \tan{\left (e \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.91577, size = 1307, normalized size = 14.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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