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 = 0.07, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 264
Rule 4132
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*}
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Mathematica [A] time = 0.11, 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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fricas [A] time = 0.67, size = 47, normalized size = 1.42 \[ -\frac {\sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{{\left (a + b\right )} f \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{2}}{\sqrt {b \sec \left (f x + e\right )^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.70, size = 48, normalized size = 1.45 \[ -\frac {\sqrt {\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}}\, \cos \left (f x +e \right )}{f \sin \left (f x +e \right ) \left (a +b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 33, normalized size = 1.00 \[ -\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a + b}}{{\left (a + b\right )} f \tan \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.65, size = 74, normalized size = 2.24 \[ -\frac {\left (2\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )\right )\,\sqrt {\frac {a+2\,b+a\,\cos \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{2\,f\,{\sin \left (2\,e+2\,f\,x\right )}^2\,\left (a+b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sec ^{2}{\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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