Optimal. Leaf size=82 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{2 a^{5/2} f}-\frac{3 \cot (e+f x)}{2 a^2 f}+\frac{\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )} \]
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Rubi [A] time = 0.0741918, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3663, 290, 325, 205} \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{2 a^{5/2} f}-\frac{3 \cot (e+f x)}{2 a^2 f}+\frac{\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )} \]
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 )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 a f}\\ &=-\frac{3 \cot (e+f x)}{2 a^2 f}+\frac{\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{2 a^2 f}\\ &=-\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{2 a^{5/2} f}-\frac{3 \cot (e+f x)}{2 a^2 f}+\frac{\cot (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 0.569976, size = 83, normalized size = 1.01 \[ \frac{\sqrt{a} \left (-\frac{b \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}-2 \cot (e+f x)\right )-3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right )}{2 a^{5/2} f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 75, normalized size = 0.9 \begin{align*} -{\frac{1}{f{a}^{2}\tan \left ( fx+e \right ) }}-{\frac{b\tan \left ( fx+e \right ) }{2\,f{a}^{2} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }}-{\frac{3\,b}{2\,f{a}^{2}}\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 = 1.97383, size = 888, normalized size = 10.83 \begin{align*} \left [-\frac{4 \,{\left (2 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + b\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 ) + 12 \, b \cos \left (f x + e\right )}{8 \,{\left (a^{2} b f +{\left (a^{3} - a^{2} b\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, -\frac{2 \,{\left (2 \, a - 3 \, b\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + b\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 ) + 6 \, b \cos \left (f x + e\right )}{4 \,{\left (a^{2} b f +{\left (a^{3} - a^{2} b\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 = 1.41373, size = 126, normalized size = 1.54 \begin{align*} -\frac{\frac{3 \,{\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^{2}} + \frac{3 \, b \tan \left (f x + e\right )^{2} + 2 \, a}{{\left (b \tan \left (f x + e\right )^{3} + a \tan \left (f x + e\right )\right )} a^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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