Optimal. Leaf size=210 \[ -\frac{b^3 (4 a-3 b)}{2 a^4 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac{b^3}{4 a^3 f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac{b^3 \left (10 a^2-15 a b+6 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^5 f (a-b)^3}+\frac{\left (a^2+3 a b+6 b^2\right ) \log (\tan (e+f x))}{a^5 f}+\frac{(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac{\cot ^4(e+f x)}{4 a^3 f}+\frac{\log (\cos (e+f x))}{f (a-b)^3} \]
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Rubi [A] time = 0.237839, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 446, 88} \[ -\frac{b^3 (4 a-3 b)}{2 a^4 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac{b^3}{4 a^3 f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac{b^3 \left (10 a^2-15 a b+6 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^5 f (a-b)^3}+\frac{\left (a^2+3 a b+6 b^2\right ) \log (\tan (e+f x))}{a^5 f}+\frac{(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac{\cot ^4(e+f x)}{4 a^3 f}+\frac{\log (\cos (e+f x))}{f (a-b)^3} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^5 \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^3 (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^3}+\frac{-a-3 b}{a^4 x^2}+\frac{a^2+3 a b+6 b^2}{a^5 x}-\frac{1}{(a-b)^3 (1+x)}+\frac{b^4}{a^3 (a-b) (a+b x)^3}+\frac{(4 a-3 b) b^4}{a^4 (a-b)^2 (a+b x)^2}+\frac{b^4 \left (10 a^2-15 a b+6 b^2\right )}{a^5 (a-b)^3 (a+b x)}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac{\cot ^4(e+f x)}{4 a^3 f}+\frac{\log (\cos (e+f x))}{(a-b)^3 f}+\frac{\left (a^2+3 a b+6 b^2\right ) \log (\tan (e+f x))}{a^5 f}+\frac{b^3 \left (10 a^2-15 a b+6 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^5 (a-b)^3 f}-\frac{b^3}{4 a^3 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac{(4 a-3 b) b^3}{2 a^4 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 2.4705, size = 178, normalized size = 0.85 \[ \frac{\frac{\frac{b^3 \left (2 \left (10 a^2-15 a b+6 b^2\right ) \log \left (a+b \tan ^2(e+f x)\right )-\frac{a (a-b) \left (2 b (4 a-3 b) \tan ^2(e+f x)+a (9 a-7 b)\right )}{\left (a+b \tan ^2(e+f x)\right )^2}\right )}{(a-b)^3}+4 \left (a^2+3 a b+6 b^2\right ) \log (\tan (e+f x))}{2 a^5}+\frac{(a+3 b) \cot ^2(e+f x)}{a^4}-\frac{\cot ^4(e+f x)}{2 a^3}+\frac{2 \log (\cos (e+f x))}{(a-b)^3}}{2 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.109, size = 477, normalized size = 2.3 \begin{align*} -{\frac{1}{16\,f{a}^{3} \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}+{\frac{7}{16\,f{a}^{3} \left ( \cos \left ( fx+e \right ) +1 \right ) }}+{\frac{3\,b}{4\,f{a}^{4} \left ( \cos \left ( fx+e \right ) +1 \right ) }}+{\frac{\ln \left ( \cos \left ( fx+e \right ) +1 \right ) }{2\,f{a}^{3}}}+{\frac{3\,\ln \left ( \cos \left ( fx+e \right ) +1 \right ) b}{2\,f{a}^{4}}}+3\,{\frac{\ln \left ( \cos \left ( fx+e \right ) +1 \right ){b}^{2}}{f{a}^{5}}}+5\,{\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 ) }{f{a}^{3} \left ( a-b \right ) ^{3}}}-{\frac{15\,{b}^{4}\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}^{4} \left ( a-b \right ) ^{3}}}+3\,{\frac{{b}^{5}\ln \left ( a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b \right ) }{f{a}^{5} \left ( a-b \right ) ^{3}}}+{\frac{5\,{b}^{4}}{2\,f{a}^{3} \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}^{5}}{2\,f{a}^{4} \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}^{5}}{4\,f{a}^{3} \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{1}{16\,f{a}^{3} \left ( \cos \left ( fx+e \right ) -1 \right ) ^{2}}}-{\frac{7}{16\,f{a}^{3} \left ( \cos \left ( fx+e \right ) -1 \right ) }}-{\frac{3\,b}{4\,f{a}^{4} \left ( \cos \left ( fx+e \right ) -1 \right ) }}+{\frac{\ln \left ( \cos \left ( fx+e \right ) -1 \right ) }{2\,f{a}^{3}}}+{\frac{3\,\ln \left ( \cos \left ( fx+e \right ) -1 \right ) b}{2\,f{a}^{4}}}+3\,{\frac{\ln \left ( \cos \left ( fx+e \right ) -1 \right ){b}^{2}}{f{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18268, size = 562, normalized size = 2.68 \begin{align*} \frac{\frac{2 \,{\left (10 \, a^{2} b^{3} - 15 \, a b^{4} + 6 \, b^{5}\right )} \log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}} + \frac{2 \,{\left (2 \, a^{6} - 7 \, a^{5} b + 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 25 \, a^{2} b^{4} + 21 \, a b^{5} - 6 \, b^{6}\right )} \sin \left (f x + e\right )^{6} - a^{6} + 3 \, a^{5} b - 3 \, a^{4} b^{2} + a^{3} b^{3} -{\left (9 \, a^{6} - 25 \, a^{5} b + 10 \, a^{4} b^{2} + 30 \, a^{3} b^{3} - 45 \, a^{2} b^{4} + 18 \, a b^{5}\right )} \sin \left (f x + e\right )^{4} + 2 \,{\left (3 \, a^{6} - 7 \, a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - 2 \, a^{2} b^{4}\right )} \sin \left (f x + e\right )^{2}}{{\left (a^{9} - 5 \, a^{8} b + 10 \, a^{7} b^{2} - 10 \, a^{6} b^{3} + 5 \, a^{5} b^{4} - a^{4} b^{5}\right )} \sin \left (f x + e\right )^{8} - 2 \,{\left (a^{9} - 4 \, a^{8} b + 6 \, a^{7} b^{2} - 4 \, a^{6} b^{3} + a^{5} b^{4}\right )} \sin \left (f x + e\right )^{6} +{\left (a^{9} - 3 \, a^{8} b + 3 \, a^{7} b^{2} - a^{6} b^{3}\right )} \sin \left (f x + e\right )^{4}} + \frac{2 \,{\left (a^{2} + 3 \, a b + 6 \, b^{2}\right )} \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{5}}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.32967, size = 1324, normalized size = 6.3 \begin{align*} \frac{3 \,{\left (a^{5} b^{2} - a^{4} b^{3} - 3 \, a^{3} b^{4} + 8 \, a^{2} b^{5} - 4 \, a b^{6}\right )} \tan \left (f x + e\right )^{8} - a^{7} + 3 \, a^{6} b - 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \,{\left (3 \, a^{6} b - 2 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 14 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - 6 \, a b^{6}\right )} \tan \left (f x + e\right )^{6} +{\left (3 \, a^{7} + a^{6} b - 10 \, a^{5} b^{2} - 6 \, a^{4} b^{3} + 33 \, a^{3} b^{4} - 18 \, a^{2} b^{5}\right )} \tan \left (f x + e\right )^{4} + 2 \,{\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right )} \tan \left (f x + e\right )^{2} + 2 \,{\left ({\left (a^{5} b^{2} - 10 \, a^{2} b^{5} + 15 \, a b^{6} - 6 \, b^{7}\right )} \tan \left (f x + e\right )^{8} + 2 \,{\left (a^{6} b - 10 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6}\right )} \tan \left (f x + e\right )^{6} +{\left (a^{7} - 10 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5}\right )} \tan \left (f x + e\right )^{4}\right )} \log \left (\frac{\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left ({\left (10 \, a^{2} b^{5} - 15 \, a b^{6} + 6 \, b^{7}\right )} \tan \left (f x + e\right )^{8} + 2 \,{\left (10 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 6 \, a b^{6}\right )} \tan \left (f x + e\right )^{6} +{\left (10 \, a^{4} b^{3} - 15 \, a^{3} b^{4} + 6 \, a^{2} b^{5}\right )} \tan \left (f x + e\right )^{4}\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^{8} b^{2} - 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - a^{5} b^{5}\right )} f \tan \left (f x + e\right )^{8} + 2 \,{\left (a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} - a^{6} b^{4}\right )} f \tan \left (f x + e\right )^{6} +{\left (a^{10} - 3 \, a^{9} b + 3 \, a^{8} b^{2} - a^{7} b^{3}\right )} f \tan \left (f x + e\right )^{4}\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 [B] time = 1.77064, size = 2016, normalized size = 9.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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