Optimal. Leaf size=69 \[ \frac{a \tanh ^{-1}(\sin (e+f x))}{d f}-\frac{2 a \sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{d f \sqrt{c+d}} \]
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Rubi [A] time = 0.144089, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3998, 3770, 3831, 2659, 208} \[ \frac{a \tanh ^{-1}(\sin (e+f x))}{d f}-\frac{2 a \sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{d f \sqrt{c+d}} \]
Antiderivative was successfully verified.
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Rule 3998
Rule 3770
Rule 3831
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))}{c+d \sec (e+f x)} \, dx &=\frac{a \int \sec (e+f x) \, dx}{d}+\frac{(-a c+a d) \int \frac{\sec (e+f x)}{c+d \sec (e+f x)} \, dx}{d}\\ &=\frac{a \tanh ^{-1}(\sin (e+f x))}{d f}-\frac{(a (c-d)) \int \frac{1}{1+\frac{c \cos (e+f x)}{d}} \, dx}{d^2}\\ &=\frac{a \tanh ^{-1}(\sin (e+f x))}{d f}-\frac{(2 a (c-d)) \operatorname{Subst}\left (\int \frac{1}{1+\frac{c}{d}+\left (1-\frac{c}{d}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{d^2 f}\\ &=\frac{a \tanh ^{-1}(\sin (e+f x))}{d f}-\frac{2 a \sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{d \sqrt{c+d} f}\\ \end{align*}
Mathematica [A] time = 0.184981, size = 107, normalized size = 1.55 \[ \frac{a \left (\frac{2 (c-d) \tanh ^{-1}\left (\frac{(d-c) \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{\sqrt{c^2-d^2}}-\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{d f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.078, size = 135, normalized size = 2. \begin{align*}{\frac{a}{fd}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) }-{\frac{a}{fd}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) }-2\,{\frac{ac}{fd\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}{\it Artanh} \left ({\frac{\tan \left ( 1/2\,fx+e/2 \right ) \left ( c-d \right ) }{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}} \right ) }+2\,{\frac{a}{f\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}{\it Artanh} \left ({\frac{\tan \left ( 1/2\,fx+e/2 \right ) \left ( c-d \right ) }{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}} \right ) } \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 = 0.64202, size = 622, normalized size = 9.01 \begin{align*} \left [\frac{a \sqrt{\frac{c - d}{c + d}} \log \left (\frac{2 \, c d \cos \left (f x + e\right ) -{\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \,{\left (c^{2} + c d +{\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{c - d}{c + d}} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + a \log \left (\sin \left (f x + e\right ) + 1\right ) - a \log \left (-\sin \left (f x + e\right ) + 1\right )}{2 \, d f}, -\frac{2 \, a \sqrt{-\frac{c - d}{c + d}} \arctan \left (-\frac{{\left (d \cos \left (f x + e\right ) + c\right )} \sqrt{-\frac{c - d}{c + d}}}{{\left (c - d\right )} \sin \left (f x + e\right )}\right ) - a \log \left (\sin \left (f x + e\right ) + 1\right ) + a \log \left (-\sin \left (f x + e\right ) + 1\right )}{2 \, d 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*} a \left (\int \frac{\sec{\left (e + f x \right )}}{c + d \sec{\left (e + f x \right )}}\, dx + \int \frac{\sec ^{2}{\left (e + f x \right )}}{c + d \sec{\left (e + f x \right )}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.03586, size = 178, normalized size = 2.58 \begin{align*} \frac{\frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{d} - \frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{d} + \frac{2 \,{\left (\pi \left \lfloor \frac{f x + e}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{\sqrt{-c^{2} + d^{2}}}\right )\right )}{\left (a c - a d\right )}}{\sqrt{-c^{2} + d^{2}} d}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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