Optimal. Leaf size=184 \[ -\frac{x \left (2 b c d-a \left (c^2-d^2\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^2}+\frac{b^3 \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)^2}+\frac{d^2}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac{d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2 (b c-a d)^2} \]
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Rubi [A] time = 0.507971, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3569, 3651, 3530} \[ -\frac{x \left (2 b c d-a \left (c^2-d^2\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^2}+\frac{b^3 \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)^2}+\frac{d^2}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac{d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 3569
Rule 3651
Rule 3530
Rubi steps
\begin{align*} \int \frac{1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx &=\frac{d^2}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{\int \frac{-a c d+b \left (c^2+d^2\right )-d (b c-a d) \tan (e+f x)+b d^2 \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{(b c-a d) \left (c^2+d^2\right )}\\ &=-\frac{\left (2 b c d-a \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^2}+\frac{d^2}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{b^3 \int \frac{b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right ) (b c-a d)^2}+\frac{\left (d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )\right ) \int \frac{d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(b c-a d)^2 \left (c^2+d^2\right )^2}\\ &=-\frac{\left (2 b c d-a \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^2}+\frac{b^3 \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right ) (b c-a d)^2 f}+\frac{d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^2 \left (c^2+d^2\right )^2 f}+\frac{d^2}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 2.15867, size = 306, normalized size = 1.66 \[ \frac{\frac{2 b d^2 \left (a^2+b^2\right ) \left (b \left (3 c^2+d^2\right )-2 a c d\right ) \log (c+d \tan (e+f x))-2 b^4 \left (c^2+d^2\right )^2 \log (a+b \tan (e+f x))+(b c-a d)^2 \left (a \sqrt{-b^2} \left (c^2-d^2\right )+2 b c d \left (a-\sqrt{-b^2}\right )+b^2 \left (c^2-d^2\right )\right ) \log \left (\sqrt{-b^2}-b \tan (e+f x)\right )+(b c-a d)^2 \left (a \sqrt{-b^2} \left (d^2-c^2\right )+2 b c d \left (a+\sqrt{-b^2}\right )+b^2 \left (c^2-d^2\right )\right ) \log \left (\sqrt{-b^2}+b \tan (e+f x)\right )}{2 b \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)}-\frac{d^2}{c+d \tan (e+f x)}}{f \left (c^2+d^2\right ) (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 412, normalized size = 2.2 \begin{align*} -{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) acd}{f \left ({a}^{2}+{b}^{2} \right ) \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) b{c}^{2}}{2\,f \left ({a}^{2}+{b}^{2} \right ) \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) b{d}^{2}}{2\,f \left ({a}^{2}+{b}^{2} \right ) \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) a{c}^{2}}{f \left ({a}^{2}+{b}^{2} \right ) \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) a{d}^{2}}{f \left ({a}^{2}+{b}^{2} \right ) \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-2\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) bcd}{f \left ({a}^{2}+{b}^{2} \right ) \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-{\frac{{d}^{2}}{f \left ( ad-bc \right ) \left ({c}^{2}+{d}^{2} \right ) \left ( c+d\tan \left ( fx+e \right ) \right ) }}+2\,{\frac{{d}^{3}\ln \left ( c+d\tan \left ( fx+e \right ) \right ) ac}{f \left ( ad-bc \right ) ^{2} \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-3\,{\frac{{d}^{2}\ln \left ( c+d\tan \left ( fx+e \right ) \right ) b{c}^{2}}{f \left ( ad-bc \right ) ^{2} \left ({c}^{2}+{d}^{2} \right ) ^{2}}}-{\frac{{d}^{4}\ln \left ( c+d\tan \left ( fx+e \right ) \right ) b}{f \left ( ad-bc \right ) ^{2} \left ({c}^{2}+{d}^{2} \right ) ^{2}}}+{\frac{{b}^{3}\ln \left ( a+b\tan \left ( fx+e \right ) \right ) }{f \left ({a}^{2}+{b}^{2} \right ) \left ( ad-bc \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.86785, size = 518, normalized size = 2.82 \begin{align*} \frac{\frac{2 \, b^{3} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{2} b^{2} + b^{4}\right )} c^{2} - 2 \,{\left (a^{3} b + a b^{3}\right )} c d +{\left (a^{4} + a^{2} b^{2}\right )} d^{2}} + \frac{2 \,{\left (a c^{2} - 2 \, b c d - a d^{2}\right )}{\left (f x + e\right )}}{{\left (a^{2} + b^{2}\right )} c^{4} + 2 \,{\left (a^{2} + b^{2}\right )} c^{2} d^{2} +{\left (a^{2} + b^{2}\right )} d^{4}} + \frac{2 \, d^{2}}{b c^{4} - a c^{3} d + b c^{2} d^{2} - a c d^{3} +{\left (b c^{3} d - a c^{2} d^{2} + b c d^{3} - a d^{4}\right )} \tan \left (f x + e\right )} - \frac{2 \,{\left (3 \, b c^{2} d^{2} - 2 \, a c d^{3} + b d^{4}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{2} c^{6} - 2 \, a b c^{5} d - 4 \, a b c^{3} d^{3} - 2 \, a b c d^{5} + a^{2} d^{6} +{\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{2} +{\left (2 \, a^{2} + b^{2}\right )} c^{2} d^{4}} - \frac{{\left (b c^{2} + 2 \, a c d - b d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{2} + b^{2}\right )} c^{4} + 2 \,{\left (a^{2} + b^{2}\right )} c^{2} d^{2} +{\left (a^{2} + b^{2}\right )} d^{4}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.99267, size = 1486, normalized size = 8.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38032, size = 730, normalized size = 3.97 \begin{align*} \frac{\frac{2 \, b^{4} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{2} b^{3} c^{2} + b^{5} c^{2} - 2 \, a^{3} b^{2} c d - 2 \, a b^{4} c d + a^{4} b d^{2} + a^{2} b^{3} d^{2}} + \frac{2 \,{\left (a c^{2} - 2 \, b c d - a d^{2}\right )}{\left (f x + e\right )}}{a^{2} c^{4} + b^{2} c^{4} + 2 \, a^{2} c^{2} d^{2} + 2 \, b^{2} c^{2} d^{2} + a^{2} d^{4} + b^{2} d^{4}} - \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} c^{4} + b^{2} c^{4} + 2 \, a^{2} c^{2} d^{2} + 2 \, b^{2} c^{2} d^{2} + a^{2} d^{4} + b^{2} d^{4}} - \frac{2 \,{\left (3 \, b c^{2} d^{3} - 2 \, a c d^{4} + b d^{5}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3} + 2 \, b^{2} c^{4} d^{3} - 4 \, a b c^{3} d^{4} + 2 \, a^{2} c^{2} d^{5} + b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}} + \frac{2 \,{\left (3 \, b c^{2} d^{3} \tan \left (f x + e\right ) - 2 \, a c d^{4} \tan \left (f x + e\right ) + b d^{5} \tan \left (f x + e\right ) + 4 \, b c^{3} d^{2} - 3 \, a c^{2} d^{3} + 2 \, b c d^{4} - a d^{5}\right )}}{{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + 2 \, b^{2} c^{4} d^{2} - 4 \, a b c^{3} d^{3} + 2 \, a^{2} c^{2} d^{4} + b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )}{\left (d \tan \left (f x + e\right ) + c\right )}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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