Optimal. Leaf size=148 \[ \frac{b \left (3 a^2-3 a b+b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^3 f (a-b)^3}-\frac{b (2 a-b)}{2 a^2 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}+\frac{\log (\tan (e+f x))}{a^3 f}-\frac{b}{4 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac{\log (\cos (e+f x))}{f (a-b)^3} \]
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Rubi [A] time = 0.165032, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3670, 446, 72} \[ \frac{b \left (3 a^2-3 a b+b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^3 f (a-b)^3}-\frac{b (2 a-b)}{2 a^2 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}+\frac{\log (\tan (e+f x))}{a^3 f}-\frac{b}{4 a f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac{\log (\cos (e+f x))}{f (a-b)^3} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (1+x) (a+b x)^3} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x}-\frac{1}{(a-b)^3 (1+x)}+\frac{b^2}{a (a-b) (a+b x)^3}+\frac{(2 a-b) b^2}{a^2 (a-b)^2 (a+b x)^2}+\frac{b^2 \left (3 a^2-3 a b+b^2\right )}{a^3 (a-b)^3 (a+b x)}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{\log (\cos (e+f x))}{(a-b)^3 f}+\frac{\log (\tan (e+f x))}{a^3 f}+\frac{b \left (3 a^2-3 a b+b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^3 (a-b)^3 f}-\frac{b}{4 a (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{(2 a-b) b}{2 a^2 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 1.6598, size = 126, normalized size = 0.85 \[ \frac{\frac{\frac{b \left (2 \left (3 a^2-3 a b+b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )-\frac{a (a-b) \left (2 b (2 a-b) \tan ^2(e+f x)+a (5 a-3 b)\right )}{\left (a+b \tan ^2(e+f x)\right )^2}\right )}{(a-b)^3}+4 \log (\tan (e+f x))}{a^3}+\frac{4 \log (\cos (e+f x))}{(a-b)^3}}{4 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.098, size = 289, normalized size = 2. \begin{align*}{\frac{\ln \left ( \cos \left ( fx+e \right ) +1 \right ) }{2\,f{a}^{3}}}+{\frac{3\,{b}^{2}}{2\,fa \left ( a-b \right ) ^{3} \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) }}-{\frac{{b}^{3}}{2\,f{a}^{2} \left ( a-b \right ) ^{3} \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) }}+{\frac{3\,b\ln \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) }{2\,fa \left ( a-b \right ) ^{3}}}-{\frac{3\,{b}^{2}\ln \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) }{2\,f{a}^{2} \left ( a-b \right ) ^{3}}}+{\frac{{b}^{3}\ln \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) }{2\,f{a}^{3} \left ( a-b \right ) ^{3}}}-{\frac{{b}^{3}}{4\,fa \left ( a-b \right ) ^{3} \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) ^{2}}}+{\frac{\ln \left ( \cos \left ( fx+e \right ) -1 \right ) }{2\,f{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11028, size = 338, normalized size = 2.28 \begin{align*} \frac{\frac{2 \,{\left (3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}} + \frac{6 \, a^{2} b^{2} - 3 \, a b^{3} - 2 \,{\left (3 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \sin \left (f x + e\right )^{2}}{a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3} +{\left (a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right )} \sin \left (f x + e\right )^{4} - 2 \,{\left (a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sin \left (f x + e\right )^{2}} + \frac{2 \, \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{3}}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89519, size = 895, normalized size = 6.05 \begin{align*} \frac{6 \, a^{3} b^{2} - 3 \, a^{2} b^{3} +{\left (5 \, a^{2} b^{3} - 2 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} + 2 \,{\left (3 \, a^{3} b^{2} + a^{2} b^{3} - a b^{4}\right )} \tan \left (f x + e\right )^{2} + 2 \,{\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3} +{\left (a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right )} \tan \left (f x + e\right )^{4} + 2 \,{\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} \tan \left (f x + e\right )^{2}\right )} \log \left (\frac{\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left (3 \, a^{4} b - 3 \, a^{3} b^{2} + a^{2} b^{3} +{\left (3 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} \tan \left (f x + e\right )^{4} + 2 \,{\left (3 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right )} \tan \left (f x + e\right )^{2}\right )} \log \left (\frac{b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{2} + 1}\right )}{4 \,{\left ({\left (a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right )} f \tan \left (f x + e\right )^{4} + 2 \,{\left (a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right )} f \tan \left (f x + e\right )^{2} +{\left (a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right )} f\right )}} \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 [A] time = 1.63073, size = 360, normalized size = 2.43 \begin{align*} \frac{\frac{2 \,{\left (3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left ({\left | -a \sin \left (f x + e\right )^{2} + b \sin \left (f x + e\right )^{2} + a \right |}\right )}{a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}} - \frac{9 \, a^{3} b \sin \left (f x + e\right )^{4} - 18 \, a^{2} b^{2} \sin \left (f x + e\right )^{4} + 12 \, a b^{3} \sin \left (f x + e\right )^{4} - 3 \, b^{4} \sin \left (f x + e\right )^{4} - 18 \, a^{3} b \sin \left (f x + e\right )^{2} + 24 \, a^{2} b^{2} \sin \left (f x + e\right )^{2} - 8 \, a b^{3} \sin \left (f x + e\right )^{2} + 9 \, a^{3} b - 6 \, a^{2} b^{2}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )}{\left (a \sin \left (f x + e\right )^{2} - b \sin \left (f x + e\right )^{2} - a\right )}^{2}} + \frac{2 \, \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{3}}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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