Optimal. Leaf size=30 \[ -\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f} \]
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Rubi [A]
time = 0.05, antiderivative size = 30, 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 = {3744, 270}
\begin {gather*} -\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 3744
Rubi steps
\begin {align*} \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 49, normalized size = 1.63 \begin {gather*} -\frac {\cot (e+f x) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}}{\sqrt {2} a f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 57, normalized size = 1.90
method | result | size |
default | \(-\frac {\sqrt {\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}}\, \cos \left (f x +e \right )}{f \sin \left (f x +e \right ) a}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 32, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{a f \tan \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.86, size = 53, normalized size = 1.77 \begin {gather*} -\frac {\sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a f \sin \left (f x + e\right )} \end {gather*}
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 \begin {gather*} \int \frac {\csc ^{2}{\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.86, size = 36, normalized size = 1.20 \begin {gather*} -\frac {\mathrm {cot}\left (e+f\,x\right )\,\sqrt {a+\frac {b\,{\sin \left (e+f\,x\right )}^2}{{\cos \left (e+f\,x\right )}^2}}}{a\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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