Optimal. Leaf size=37 \[ -\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f (a-b)} \]
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Rubi [A] time = 0.0465261, antiderivative size = 37, 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 = {3664, 264} \[ -\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f (a-b)} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 264
Rubi steps
\begin{align*} \int \frac{\sin (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{\cos (e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{(a-b) f}\\ \end{align*}
Mathematica [A] time = 0.603409, size = 52, normalized size = 1.41 \[ \frac{\cos (e+f x) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{\sqrt{2} f (b-a)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 78, normalized size = 2.1 \begin{align*} -{\frac{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b}{f\cos \left ( fx+e \right ) \left ( a-b \right ) }{\frac{1}{\sqrt{{\frac{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b}{ \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 [A] time = 0.98396, size = 47, normalized size = 1.27 \begin{align*} -\frac{\sqrt{a - b + \frac{b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{{\left (a - b\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76052, size = 104, normalized size = 2.81 \begin{align*} -\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 )}{{\left (a - b\right )} 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 \tan ^{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 = 1.7781, size = 119, normalized size = 3.22 \begin{align*} \frac{\sqrt{b} \mathrm{sgn}\left (f\right ) \mathrm{sgn}\left (\cos \left (f x + e\right )\right )}{a{\left | f \right |} - b{\left | f \right |}} - \frac{\sqrt{a \cos \left (f x + e\right )^{2} - b \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 ) - b{\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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