Optimal. Leaf size=45 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right )}{a f \sqrt{a+b}}+\frac{x}{a} \]
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Rubi [A] time = 0.0420105, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4127, 3181, 205} \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right )}{a f \sqrt{a+b}}+\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 4127
Rule 3181
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a+b \sec ^2(e+f x)} \, dx &=\frac{x}{a}-\frac{b \int \frac{1}{b+a \cos ^2(e+f x)} \, dx}{a}\\ &=\frac{x}{a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{b+(a+b) x^2} \, dx,x,\cot (e+f x)\right )}{a f}\\ &=\frac{x}{a}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (e+f x)}{\sqrt{b}}\right )}{a \sqrt{a+b} f}\\ \end{align*}
Mathematica [C] time = 0.318623, size = 182, normalized size = 4.04 \[ \frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (f x \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}+b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left (\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right )\right )}{2 a f \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4} \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 48, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{fa}}-{\frac{b}{fa}\arctan \left ({\tan \left ( fx+e \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \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 = 0.554304, size = 544, normalized size = 12.09 \begin{align*} \left [\frac{4 \, f x + \sqrt{-\frac{b}{a + b}} \log \left (\frac{{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \,{\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} -{\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{-\frac{b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right )}{4 \, a f}, \frac{2 \, f x + \sqrt{\frac{b}{a + b}} \arctan \left (\frac{{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt{\frac{b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right )}{2 \, a 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{1}{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 = 1.34351, size = 92, normalized size = 2.04 \begin{align*} -\frac{\frac{{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b + b^{2}}}\right )\right )} b}{\sqrt{a b + b^{2}} a} - \frac{f x + e}{a}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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