Optimal. Leaf size=33 \[ -\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f (a+b)} \]
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Rubi [A] time = 0.0710493, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {4132, 264} \[ -\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{f (a+b)} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 264
Rubi steps
\begin{align*} \int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot (e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{(a+b) f}\\ \end{align*}
Mathematica [A] time = 0.112694, size = 55, normalized size = 1.67 \[ -\frac{\csc (e+f x) \sec (e+f x) (a \cos (2 (e+f x))+a+2 b)}{2 f (a+b) \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.322, size = 48, normalized size = 1.5 \begin{align*} -{\frac{\cos \left ( fx+e \right ) }{f \left ( a+b \right ) \sin \left ( fx+e \right ) }\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( \cos \left ( fx+e \right ) \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 = 0.529402, size = 113, normalized size = 3.42 \begin{align*} -\frac{\sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{{\left (a + b\right )} f \sin \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (e + f x \right )}}{\sqrt{a + b \sec ^{2}{\left (e + f x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (f x + e\right )^{2}}{\sqrt{b \sec \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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