Optimal. Leaf size=30 \[ -\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{a f} \]
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Rubi [A] time = 0.0420239, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4134, 264} \[ -\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{a f} \]
Antiderivative was successfully verified.
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Rule 4134
Rule 264
Rubi steps
\begin{align*} \int \frac{\sin (e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)}}{a f}\\ \end{align*}
Mathematica [A] time = 0.114465, size = 48, normalized size = 1.6 \[ -\frac{\sec (e+f x) (a \cos (2 e+2 f x)+a+2 b)}{2 a f \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 31, normalized size = 1. \begin{align*} -{\frac{1}{fa\sec \left ( fx+e \right ) }\sqrt{a+b \left ( \sec \left ( fx+e \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995049, size = 38, normalized size = 1.27 \begin{align*} -\frac{\sqrt{a + \frac{b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.511495, size = 88, normalized size = 2.93 \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 )}{a f} \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{\sin{\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 [B] time = 1.66257, size = 81, normalized size = 2.7 \begin{align*} \frac{\sqrt{b} \mathrm{sgn}\left (f\right ) \mathrm{sgn}\left (\cos \left (f x + e\right )\right )}{a{\left | f \right |}} - \frac{\sqrt{a \cos \left (f x + e\right )^{2} + b}}{a{\left | f \right |} \mathrm{sgn}\left (f\right ) \mathrm{sgn}\left (\cos \left (f x + e\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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