Optimal. Leaf size=103 \[ \frac{(b c-a d)^2 \log (a \cos (e+f x)+b \sin (e+f x))}{b f \left (a^2+b^2\right )}+\frac{a x (b c-a d)^2}{b^2 \left (a^2+b^2\right )}+\frac{d x (2 b c-a d)}{b^2}-\frac{d^2 \log (\cos (e+f x))}{b f} \]
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Rubi [A] time = 0.123844, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3541, 3475, 3484, 3530} \[ \frac{(b c-a d)^2 \log (a \cos (e+f x)+b \sin (e+f x))}{b f \left (a^2+b^2\right )}+\frac{a x (b c-a d)^2}{b^2 \left (a^2+b^2\right )}+\frac{d x (2 b c-a d)}{b^2}-\frac{d^2 \log (\cos (e+f x))}{b f} \]
Antiderivative was successfully verified.
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Rule 3541
Rule 3475
Rule 3484
Rule 3530
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^2}{a+b \tan (e+f x)} \, dx &=\frac{d (2 b c-a d) x}{b^2}+\frac{d^2 \int \tan (e+f x) \, dx}{b}+\frac{(b c-a d)^2 \int \frac{1}{a+b \tan (e+f x)} \, dx}{b^2}\\ &=\frac{a (b c-a d)^2 x}{b^2 \left (a^2+b^2\right )}+\frac{d (2 b c-a d) x}{b^2}-\frac{d^2 \log (\cos (e+f x))}{b f}+\frac{(b c-a d)^2 \int \frac{b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac{a (b c-a d)^2 x}{b^2 \left (a^2+b^2\right )}+\frac{d (2 b c-a d) x}{b^2}-\frac{d^2 \log (\cos (e+f x))}{b f}+\frac{(b c-a d)^2 \log (a \cos (e+f x)+b \sin (e+f x))}{b \left (a^2+b^2\right ) f}\\ \end{align*}
Mathematica [C] time = 0.153347, size = 108, normalized size = 1.05 \[ \frac{\frac{2 (b c-a d)^2 \log (a+b \tan (e+f x))}{b \left (a^2+b^2\right )}-\frac{(c-i d)^2 \log (\tan (e+f x)+i)}{b+i a}+\frac{(c+i d)^2 \log (-\tan (e+f x)+i)}{-b+i a}}{2 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 249, normalized size = 2.4 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) acd}{f \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) b{c}^{2}}{2\,f \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) b{d}^{2}}{2\,f \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) a{c}^{2}}{f \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) a{d}^{2}}{f \left ({a}^{2}+{b}^{2} \right ) }}+2\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) bcd}{f \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{\ln \left ( a+b\tan \left ( fx+e \right ) \right ){a}^{2}{d}^{2}}{f \left ({a}^{2}+{b}^{2} \right ) b}}-2\,{\frac{\ln \left ( a+b\tan \left ( fx+e \right ) \right ) acd}{f \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{b\ln \left ( a+b\tan \left ( fx+e \right ) \right ){c}^{2}}{f \left ({a}^{2}+{b}^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68731, size = 166, normalized size = 1.61 \begin{align*} \frac{\frac{2 \,{\left (a c^{2} + 2 \, b c d - a d^{2}\right )}{\left (f x + e\right )}}{a^{2} + b^{2}} + \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{2} b + b^{3}} - \frac{{\left (b c^{2} - 2 \, a c d - b d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55674, size = 294, normalized size = 2.85 \begin{align*} -\frac{{\left (a^{2} + b^{2}\right )} d^{2} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \,{\left (a b c^{2} + 2 \, b^{2} c d - a b d^{2}\right )} f x -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \,{\left (a^{2} b + b^{3}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.91538, size = 1025, normalized size = 9.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45284, size = 171, normalized size = 1.66 \begin{align*} \frac{\frac{2 \,{\left (a c^{2} + 2 \, b c d - a d^{2}\right )}{\left (f x + e\right )}}{a^{2} + b^{2}} - \frac{{\left (b c^{2} - 2 \, a c d - b d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{2} b + b^{3}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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