Optimal. Leaf size=110 \[ -\frac{2 b^3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left (a^2+2 b^2\right )}{2 a^3}-\frac{b \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d} \]
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Rubi [A] time = 0.277757, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3853, 4104, 3919, 3831, 2659, 208} \[ -\frac{2 b^3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left (a^2+2 b^2\right )}{2 a^3}-\frac{b \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3853
Rule 4104
Rule 3919
Rule 3831
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{a+b \sec (c+d x)} \, dx &=\frac{\cos (c+d x) \sin (c+d x)}{2 a d}+\frac{\int \frac{\cos (c+d x) \left (-2 b+a \sec (c+d x)+b \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a}\\ &=-\frac{b \sin (c+d x)}{a^2 d}+\frac{\cos (c+d x) \sin (c+d x)}{2 a d}-\frac{\int \frac{-a^2-2 b^2-a b \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^2}\\ &=\frac{\left (a^2+2 b^2\right ) x}{2 a^3}-\frac{b \sin (c+d x)}{a^2 d}+\frac{\cos (c+d x) \sin (c+d x)}{2 a d}-\frac{b^3 \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^3}\\ &=\frac{\left (a^2+2 b^2\right ) x}{2 a^3}-\frac{b \sin (c+d x)}{a^2 d}+\frac{\cos (c+d x) \sin (c+d x)}{2 a d}-\frac{b^2 \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{a^3}\\ &=\frac{\left (a^2+2 b^2\right ) x}{2 a^3}-\frac{b \sin (c+d x)}{a^2 d}+\frac{\cos (c+d x) \sin (c+d x)}{2 a d}-\frac{\left (2 b^2\right ) \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^3 d}\\ &=\frac{\left (a^2+2 b^2\right ) x}{2 a^3}-\frac{2 b^3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^3 \sqrt{a-b} \sqrt{a+b} d}-\frac{b \sin (c+d x)}{a^2 d}+\frac{\cos (c+d x) \sin (c+d x)}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.229584, size = 97, normalized size = 0.88 \[ \frac{2 \left (a^2+2 b^2\right ) (c+d x)+\frac{8 b^3 \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+a^2 \sin (2 (c+d x))-4 a b \sin (c+d x)}{4 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.072, size = 222, normalized size = 2. \begin{align*} -{\frac{1}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}b}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) b}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{1}{da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ){b}^{2}}{d{a}^{3}}}-2\,{\frac{{b}^{3}}{d{a}^{3}\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 = 1.83928, size = 730, normalized size = 6.64 \begin{align*} \left [\frac{\sqrt{a^{2} - b^{2}} b^{3} \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 ) +{\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} d x -{\left (2 \, a^{3} b - 2 \, a b^{3} -{\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \,{\left (a^{5} - a^{3} b^{2}\right )} d}, -\frac{2 \, \sqrt{-a^{2} + b^{2}} b^{3} \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 (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} d x +{\left (2 \, a^{3} b - 2 \, a b^{3} -{\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \,{\left (a^{5} - a^{3} 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{\cos ^{2}{\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 = 1.24599, size = 240, normalized size = 2.18 \begin{align*} -\frac{\frac{4 \,{\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 )} b^{3}}{\sqrt{-a^{2} + b^{2}} a^{3}} - \frac{{\left (a^{2} + 2 \, b^{2}\right )}{\left (d x + c\right )}}{a^{3}} + \frac{2 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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