Optimal. Leaf size=126 \[ -\frac{(b c-a d)^2}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac{2 (a c+b d) (b c-a d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2}-\frac{x (b (c-d)-a (c+d)) (a (c-d)+b (c+d))}{\left (c^2+d^2\right )^2} \]
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Rubi [A] time = 0.222707, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3542, 3531, 3530} \[ -\frac{(b c-a d)^2}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}-\frac{2 (a c+b d) (b c-a d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2}-\frac{x (b (c-d)-a (c+d)) (a (c-d)+b (c+d))}{\left (c^2+d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3542
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^2} \, dx &=-\frac{(b c-a d)^2}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{\int \frac{a^2 c-b^2 c+2 a b d+\left (2 a b c-a^2 d+b^2 d\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{c^2+d^2}\\ &=-\frac{(b (c-d)-a (c+d)) (a (c-d)+b (c+d)) x}{\left (c^2+d^2\right )^2}-\frac{(b c-a d)^2}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{(2 (b c-a d) (a c+b d)) \int \frac{d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{\left (c^2+d^2\right )^2}\\ &=-\frac{(b (c-d)-a (c+d)) (a (c-d)+b (c+d)) x}{\left (c^2+d^2\right )^2}-\frac{2 (b c-a d) (a c+b d) \log (c \cos (e+f x)+d \sin (e+f x))}{\left (c^2+d^2\right )^2 f}-\frac{(b c-a d)^2}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 1.83363, size = 320, normalized size = 2.54 \[ \frac{(a+b \tan (e+f x))^2 (c \cos (e+f x)+d \sin (e+f x)) \left (2 i (e+f x) \left (a^2 c d+a b \left (d^2-c^2\right )-b^2 c d\right ) (c \cos (e+f x)+d \sin (e+f x))+\left (a^2 c d+a b \left (d^2-c^2\right )-b^2 c d\right ) (c \cos (e+f x)+d \sin (e+f x)) \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )+2 i \left (a^2 (-c) d+a b \left (c^2-d^2\right )+b^2 c d\right ) \tan ^{-1}(\tan (e+f x)) (c \cos (e+f x)+d \sin (e+f x))+\frac{\left (c^2+d^2\right ) (b c-a d)^2 \sin (e+f x)}{c}+(e+f x) (a (c+d)+b (d-c)) (a (c-d)+b (c+d)) (c \cos (e+f x)+d \sin (e+f x))\right )}{f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))^2 (a \cos (e+f x)+b \sin (e+f x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 465, normalized size = 3.7 \begin{align*} -{\frac{{a}^{2}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) cd}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ab{c}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ab{d}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){b}^{2}cd}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+{\frac{{a}^{2}\arctan \left ( \tan \left ( fx+e \right ) \right ){c}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-{\frac{{a}^{2}\arctan \left ( \tan \left ( fx+e \right ) \right ){d}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+4\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) abcd}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ){b}^{2}{c}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ){b}^{2}{d}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-{\frac{{a}^{2}d}{f \left ({c}^{2}+{d}^{2} \right ) \left ( c+d\tan \left ( fx+e \right ) \right ) }}+2\,{\frac{abc}{f \left ({c}^{2}+{d}^{2} \right ) \left ( c+d\tan \left ( fx+e \right ) \right ) }}-{\frac{{b}^{2}{c}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) d \left ( c+d\tan \left ( fx+e \right ) \right ) }}+2\,{\frac{{a}^{2}\ln \left ( c+d\tan \left ( fx+e \right ) \right ) cd}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-2\,{\frac{\ln \left ( c+d\tan \left ( fx+e \right ) \right ) ab{c}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+2\,{\frac{\ln \left ( c+d\tan \left ( fx+e \right ) \right ) ab{d}^{2}}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-2\,{\frac{\ln \left ( c+d\tan \left ( fx+e \right ) \right ){b}^{2}cd}{f \left ({c}^{2}+{d}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78939, size = 309, normalized size = 2.45 \begin{align*} \frac{\frac{{\left (4 \, a b c d +{\left (a^{2} - b^{2}\right )} c^{2} -{\left (a^{2} - b^{2}\right )} d^{2}\right )}{\left (f x + e\right )}}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} - \frac{2 \,{\left (a b c^{2} - a b d^{2} -{\left (a^{2} - b^{2}\right )} c d\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac{{\left (a b c^{2} - a b d^{2} -{\left (a^{2} - b^{2}\right )} c d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{3} d + c d^{3} +{\left (c^{2} d^{2} + d^{4}\right )} \tan \left (f x + e\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55563, size = 612, normalized size = 4.86 \begin{align*} -\frac{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3} -{\left (4 \, a b c^{2} d +{\left (a^{2} - b^{2}\right )} c^{3} -{\left (a^{2} - b^{2}\right )} c d^{2}\right )} f x +{\left (a b c^{3} - a b c d^{2} -{\left (a^{2} - b^{2}\right )} c^{2} d +{\left (a b c^{2} d - a b d^{3} -{\left (a^{2} - b^{2}\right )} c d^{2}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac{d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) -{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (4 \, a b c d^{2} +{\left (a^{2} - b^{2}\right )} c^{2} d -{\left (a^{2} - b^{2}\right )} d^{3}\right )} f x\right )} \tan \left (f x + e\right )}{{\left (c^{4} d + 2 \, c^{2} d^{3} + d^{5}\right )} f \tan \left (f x + e\right ) +{\left (c^{5} + 2 \, c^{3} d^{2} + c d^{4}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.59364, size = 447, normalized size = 3.55 \begin{align*} \frac{\frac{{\left (a^{2} c^{2} - b^{2} c^{2} + 4 \, a b c d - a^{2} d^{2} + b^{2} d^{2}\right )}{\left (f x + e\right )}}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac{{\left (a b c^{2} - a^{2} c d + b^{2} c d - a b d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} - \frac{2 \,{\left (a b c^{2} d - a^{2} c d^{2} + b^{2} c d^{2} - a b d^{3}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{4} d + 2 \, c^{2} d^{3} + d^{5}} + \frac{2 \, a b c^{2} d^{2} \tan \left (f x + e\right ) - 2 \, a^{2} c d^{3} \tan \left (f x + e\right ) + 2 \, b^{2} c d^{3} \tan \left (f x + e\right ) - 2 \, a b d^{4} \tan \left (f x + e\right ) - b^{2} c^{4} + 4 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + b^{2} c^{2} d^{2} - a^{2} d^{4}}{{\left (c^{4} d + 2 \, c^{2} d^{3} + d^{5}\right )}{\left (d \tan \left (f x + e\right ) + c\right )}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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