Optimal. Leaf size=169 \[ -\frac{b^{5/2} (7 a-5 b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{2 a^{7/2} f (a-b)^2}+\frac{\left (2 a^2+2 a b-5 b^2\right ) \cot (e+f x)}{2 a^3 f (a-b)}-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 f (a-b)}-\frac{b \cot ^3(e+f x)}{2 a f (a-b) \left (a+b \tan ^2(e+f x)\right )}+\frac{x}{(a-b)^2} \]
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Rubi [A] time = 0.287193, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3670, 472, 583, 522, 203, 205} \[ -\frac{b^{5/2} (7 a-5 b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{2 a^{7/2} f (a-b)^2}+\frac{\left (2 a^2+2 a b-5 b^2\right ) \cot (e+f x)}{2 a^3 f (a-b)}-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 f (a-b)}-\frac{b \cot ^3(e+f x)}{2 a f (a-b) \left (a+b \tan ^2(e+f x)\right )}+\frac{x}{(a-b)^2} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 472
Rule 583
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\cot ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{b \cot ^3(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{2 a-5 b-5 b x^2}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 a (a-b) f}\\ &=-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 (a-b) f}-\frac{b \cot ^3(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{3 \left (2 a^2+2 a b-5 b^2\right )+3 (2 a-5 b) b x^2}{x^2 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 a^2 (a-b) f}\\ &=\frac{\left (2 a^2+2 a b-5 b^2\right ) \cot (e+f x)}{2 a^3 (a-b) f}-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 (a-b) f}-\frac{b \cot ^3(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (2 a^3+2 a^2 b+2 a b^2-5 b^3\right )+3 b \left (2 a^2+2 a b-5 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 a^3 (a-b) f}\\ &=\frac{\left (2 a^2+2 a b-5 b^2\right ) \cot (e+f x)}{2 a^3 (a-b) f}-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 (a-b) f}-\frac{b \cot ^3(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{(a-b)^2 f}-\frac{\left ((7 a-5 b) b^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{2 a^3 (a-b)^2 f}\\ &=\frac{x}{(a-b)^2}-\frac{(7 a-5 b) b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{2 a^{7/2} (a-b)^2 f}+\frac{\left (2 a^2+2 a b-5 b^2\right ) \cot (e+f x)}{2 a^3 (a-b) f}-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 (a-b) f}-\frac{b \cot ^3(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 3.0002, size = 137, normalized size = 0.81 \[ \frac{\frac{3 b^{5/2} (5 b-7 a) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{a^{7/2} (a-b)^2}+\frac{3 \left (2 (e+f x)-\frac{b^3 (a-b) \sin (2 (e+f x))}{a^3 ((a-b) \cos (2 (e+f x))+a+b)}\right )}{(a-b)^2}-\frac{2 \cot (e+f x) \left (a \csc ^2(e+f x)-4 a-6 b\right )}{a^3}}{6 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 218, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,f{a}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}+{\frac{1}{f{a}^{2}\tan \left ( fx+e \right ) }}+2\,{\frac{b}{f{a}^{3}\tan \left ( fx+e \right ) }}-{\frac{{b}^{3}\tan \left ( fx+e \right ) }{2\,f{a}^{2} \left ( a-b \right ) ^{2} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }}+{\frac{{b}^{4}\tan \left ( fx+e \right ) }{2\,f{a}^{3} \left ( a-b \right ) ^{2} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }}-{\frac{7\,{b}^{3}}{2\,f{a}^{2} \left ( a-b \right ) ^{2}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{4}}{2\,f{a}^{3} \left ( a-b \right ) ^{2}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{f \left ( a-b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80331, size = 1335, normalized size = 7.9 \begin{align*} \left [\frac{24 \, a^{3} b f x \tan \left (f x + e\right )^{5} + 24 \, a^{4} f x \tan \left (f x + e\right )^{3} + 12 \,{\left (2 \, a^{3} b - 7 \, a b^{3} + 5 \, b^{4}\right )} \tan \left (f x + e\right )^{4} - 8 \, a^{4} + 16 \, a^{3} b - 8 \, a^{2} b^{2} + 8 \,{\left (3 \, a^{4} - a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \tan \left (f x + e\right )^{2} - 3 \,{\left ({\left (7 \, a b^{3} - 5 \, b^{4}\right )} \tan \left (f x + e\right )^{5} +{\left (7 \, a^{2} b^{2} - 5 \, a b^{3}\right )} \tan \left (f x + e\right )^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b^{2} \tan \left (f x + e\right )^{4} - 6 \, a b \tan \left (f x + e\right )^{2} + a^{2} + 4 \,{\left (a b \tan \left (f x + e\right )^{3} - a^{2} \tan \left (f x + e\right )\right )} \sqrt{-\frac{b}{a}}}{b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}}\right )}{24 \,{\left ({\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} f \tan \left (f x + e\right )^{5} +{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \tan \left (f x + e\right )^{3}\right )}}, \frac{12 \, a^{3} b f x \tan \left (f x + e\right )^{5} + 12 \, a^{4} f x \tan \left (f x + e\right )^{3} + 6 \,{\left (2 \, a^{3} b - 7 \, a b^{3} + 5 \, b^{4}\right )} \tan \left (f x + e\right )^{4} - 4 \, a^{4} + 8 \, a^{3} b - 4 \, a^{2} b^{2} + 4 \,{\left (3 \, a^{4} - a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3}\right )} \tan \left (f x + e\right )^{2} - 3 \,{\left ({\left (7 \, a b^{3} - 5 \, b^{4}\right )} \tan \left (f x + e\right )^{5} +{\left (7 \, a^{2} b^{2} - 5 \, a b^{3}\right )} \tan \left (f x + e\right )^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{{\left (b \tan \left (f x + e\right )^{2} - a\right )} \sqrt{\frac{b}{a}}}{2 \, b \tan \left (f x + e\right )}\right )}{12 \,{\left ({\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} f \tan \left (f x + e\right )^{5} +{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \tan \left (f x + e\right )^{3}\right )}}\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 = 1.43984, size = 242, normalized size = 1.43 \begin{align*} -\frac{\frac{3 \, b^{3} \tan \left (f x + e\right )}{{\left (a^{4} - a^{3} b\right )}{\left (b \tan \left (f x + e\right )^{2} + a\right )}} + \frac{3 \,{\left (7 \, a b^{3} - 5 \, b^{4}\right )}{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b}}\right )\right )}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt{a b}} - \frac{6 \,{\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}} - \frac{2 \,{\left (3 \, a \tan \left (f x + e\right )^{2} + 6 \, b \tan \left (f x + e\right )^{2} - a\right )}}{a^{3} \tan \left (f x + e\right )^{3}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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