Optimal. Leaf size=75 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a^2 f \sqrt{a+b}}+\frac{x (a-2 b)}{2 a^2}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f} \]
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Rubi [A] time = 0.103376, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4146, 414, 522, 203, 205} \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a^2 f \sqrt{a+b}}+\frac{x (a-2 b)}{2 a^2}+\frac{\sin (e+f x) \cos (e+f x)}{2 a f} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 414
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\cos (e+f x) \sin (e+f x)}{2 a f}-\frac{\operatorname{Subst}\left (\int \frac{-a+b-b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 a f}\\ &=\frac{\cos (e+f x) \sin (e+f x)}{2 a f}+\frac{(a-2 b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{2 a^2 f}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{a^2 f}\\ &=\frac{(a-2 b) x}{2 a^2}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a^2 \sqrt{a+b} f}+\frac{\cos (e+f x) \sin (e+f x)}{2 a f}\\ \end{align*}
Mathematica [A] time = 0.252891, size = 67, normalized size = 0.89 \[ \frac{\frac{4 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{\sqrt{a+b}}+2 (a-2 b) (e+f x)+a \sin (2 (e+f x))}{4 a^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.101, size = 92, normalized size = 1.2 \begin{align*}{\frac{\tan \left ( fx+e \right ) }{2\,fa \left ( \left ( \tan \left ( fx+e \right ) \right ) ^{2}+1 \right ) }}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{2\,fa}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) b}{f{a}^{2}}}+{\frac{{b}^{2}}{f{a}^{2}}\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 = 0.566991, size = 663, normalized size = 8.84 \begin{align*} \left [\frac{2 \,{\left (a - 2 \, b\right )} f x + 2 \, a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \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^{2} f}, \frac{{\left (a - 2 \, b\right )} f x + a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - b \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^{2} 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{\cos ^{2}{\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 = 1.35891, size = 134, normalized size = 1.79 \begin{align*} \frac{\frac{2 \,{\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^{2}}{\sqrt{a b + b^{2}} a^{2}} + \frac{{\left (f x + e\right )}{\left (a - 2 \, b\right )}}{a^{2}} + \frac{\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )} a}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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