Optimal. Leaf size=112 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 a^{7/2} f}+\frac{5 \cot (e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )}-\frac{15 \cot (e+f x)}{8 a^3 f}+\frac{\cot (e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^2} \]
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Rubi [A] time = 0.0877426, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3663, 290, 325, 205} \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 a^{7/2} f}+\frac{5 \cot (e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )}-\frac{15 \cot (e+f x)}{8 a^3 f}+\frac{\cot (e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{\csc ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\cot (e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^2}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 a f}\\ &=\frac{\cot (e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^2}+\frac{5 \cot (e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )}+\frac{15 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 a^2 f}\\ &=-\frac{15 \cot (e+f x)}{8 a^3 f}+\frac{\cot (e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^2}+\frac{5 \cot (e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )}-\frac{(15 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^3 f}\\ &=-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{8 a^{7/2} f}-\frac{15 \cot (e+f x)}{8 a^3 f}+\frac{\cot (e+f x)}{4 a f \left (a+b \tan ^2(e+f x)\right )^2}+\frac{5 \cot (e+f x)}{8 a^2 f \left (a+b \tan ^2(e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 0.826774, size = 144, normalized size = 1.29 \[ \frac{\frac{4 a^{3/2} b^2 \sin (2 (e+f x))}{(a-b) ((a-b) \cos (2 (e+f x))+a+b)^2}-15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )-\frac{\sqrt{a} b (9 a-7 b) \sin (2 (e+f x))}{(a-b) ((a-b) \cos (2 (e+f x))+a+b)}-8 \sqrt{a} \cot (e+f x)}{8 a^{7/2} f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 108, normalized size = 1. \begin{align*} -{\frac{1}{f{a}^{3}\tan \left ( fx+e \right ) }}-{\frac{7\,{b}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{8\,f{a}^{3} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{9\,b\tan \left ( fx+e \right ) }{8\,f{a}^{2} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{15\,b}{8\,f{a}^{3}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 [B] time = 2.34278, size = 1300, normalized size = 11.61 \begin{align*} \left [-\frac{4 \,{\left (8 \, a^{2} - 25 \, a b + 15 \, b^{2}\right )} \cos \left (f x + e\right )^{5} + 20 \,{\left (5 \, a b - 6 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \,{\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \,{\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{3} - a b \cos \left (f x + e\right )\right )} \sqrt{-\frac{b}{a}} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) + 60 \, b^{2} \cos \left (f x + e\right )}{32 \,{\left (a^{3} b^{2} f +{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \,{\left (a^{4} b - a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, -\frac{2 \,{\left (8 \, a^{2} - 25 \, a b + 15 \, b^{2}\right )} \cos \left (f x + e\right )^{5} + 10 \,{\left (5 \, a b - 6 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \,{\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt{\frac{b}{a}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 30 \, b^{2} \cos \left (f x + e\right )}{16 \,{\left (a^{3} b^{2} f +{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{4} + 2 \,{\left (a^{4} b - a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\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 = 1.62092, size = 147, normalized size = 1.31 \begin{align*} -\frac{\frac{15 \,{\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 )} b}{\sqrt{a b} a^{3}} + \frac{7 \, b^{2} \tan \left (f x + e\right )^{3} + 9 \, a b \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2} a^{3}} + \frac{8}{a^{3} \tan \left (f x + e\right )}}{8 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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