Optimal. Leaf size=36 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{\sqrt{a} f \sqrt{a+b}} \]
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Rubi [A] time = 0.042072, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4147, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{\sqrt{a} f \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 4147
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{\sqrt{a} \sqrt{a+b} f}\\ \end{align*}
Mathematica [A] time = 0.0795575, size = 36, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a+b}}\right )}{\sqrt{a} f \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 28, normalized size = 0.8 \begin{align*}{\frac{1}{f}{\it Artanh} \left ({a\sin \left ( fx+e \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \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 = 0.510653, size = 271, normalized size = 7.53 \begin{align*} \left [\frac{\log \left (-\frac{a \cos \left (f x + e\right )^{2} - 2 \, \sqrt{a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right )}{2 \, \sqrt{a^{2} + a b} f}, -\frac{\sqrt{-a^{2} - a b} \arctan \left (\frac{\sqrt{-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right )}{{\left (a^{2} + a b\right )} f}\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{\sec{\left (e + f x \right )}}{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 [A] time = 1.2437, size = 53, normalized size = 1.47 \begin{align*} -\frac{\arctan \left (\frac{a \sin \left (f x + e\right )}{\sqrt{-a^{2} - a b}}\right )}{\sqrt{-a^{2} - a b} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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