Optimal. Leaf size=90 \[ \frac{2 b (b B-a C) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{x (b B-a C)}{a^2}+\frac{B \sin (c+d x)}{a d} \]
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Rubi [A] time = 0.225905, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4072, 4034, 12, 3783, 2659, 208} \[ \frac{2 b (b B-a C) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{x (b B-a C)}{a^2}+\frac{B \sin (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 4034
Rule 12
Rule 3783
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx &=\int \frac{\cos (c+d x) (B+C \sec (c+d x))}{a+b \sec (c+d x)} \, dx\\ &=\frac{B \sin (c+d x)}{a d}-\frac{\int \frac{b B-a C}{a+b \sec (c+d x)} \, dx}{a}\\ &=\frac{B \sin (c+d x)}{a d}-\frac{(b B-a C) \int \frac{1}{a+b \sec (c+d x)} \, dx}{a}\\ &=-\frac{(b B-a C) x}{a^2}+\frac{B \sin (c+d x)}{a d}+\frac{(b B-a C) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{a^2}\\ &=-\frac{(b B-a C) x}{a^2}+\frac{B \sin (c+d x)}{a d}+\frac{(2 (b B-a C)) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 d}\\ &=-\frac{(b B-a C) x}{a^2}+\frac{2 b (b B-a C) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^2 \sqrt{a-b} \sqrt{a+b} d}+\frac{B \sin (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.203042, size = 85, normalized size = 0.94 \[ \frac{-\frac{2 b (b B-a C) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+(c+d x) (a C-b B)+a B \sin (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.114, size = 172, normalized size = 1.9 \begin{align*} 2\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) Bb}{d{a}^{2}}}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{ad}}+2\,{\frac{B{b}^{2}}{d{a}^{2}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-2\,{\frac{Cb}{ad\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \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 = 0.555837, size = 702, normalized size = 7.8 \begin{align*} \left [\frac{2 \,{\left (C a^{3} - B a^{2} b - C a b^{2} + B b^{3}\right )} d x -{\left (C a b - B b^{2}\right )} \sqrt{a^{2} - b^{2}} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) -{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + 2 \,{\left (B a^{3} - B a b^{2}\right )} \sin \left (d x + c\right )}{2 \,{\left (a^{4} - a^{2} b^{2}\right )} d}, \frac{{\left (C a^{3} - B a^{2} b - C a b^{2} + B b^{3}\right )} d x -{\left (C a b - B b^{2}\right )} \sqrt{-a^{2} + b^{2}} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) +{\left (B a^{3} - B a b^{2}\right )} \sin \left (d x + c\right )}{{\left (a^{4} - a^{2} b^{2}\right )} d}\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{\left (B + C \sec{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}}{a + b \sec{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19928, size = 190, normalized size = 2.11 \begin{align*} \frac{\frac{{\left (C a - B b\right )}{\left (d x + c\right )}}{a^{2}} + \frac{2 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} a} - \frac{2 \,{\left (C a b - B b^{2}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{-a^{2} + b^{2}}}\right )\right )}}{\sqrt{-a^{2} + b^{2}} a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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