Optimal. Leaf size=98 \[ \frac{\sqrt{a} (A+2 B) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}-\frac{a (A-2 C) \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d} \]
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Rubi [A] time = 0.210891, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {4086, 3915, 3774, 203, 3792} \[ \frac{\sqrt{a} (A+2 B) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}-\frac{a (A-2 C) \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 3915
Rule 3774
Rule 203
Rule 3792
Rubi steps
\begin{align*} \int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{d}+\frac{\int \sqrt{a+a \sec (c+d x)} \left (\frac{1}{2} a (A+2 B)-\frac{1}{2} a (A-2 C) \sec (c+d x)\right ) \, dx}{a}\\ &=\frac{A \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{2} (A+2 B) \int \sqrt{a+a \sec (c+d x)} \, dx+\frac{1}{2} (-A+2 C) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{A \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{d}-\frac{a (A-2 C) \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{(a (A+2 B)) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{a} (A+2 B) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{A \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{d}-\frac{a (A-2 C) \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.395689, size = 94, normalized size = 0.96 \[ \frac{\sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sqrt{2} (A+2 B) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}+2 \sin \left (\frac{1}{2} (c+d x)\right ) (A \cos (c+d x)+2 C)\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.351, size = 210, normalized size = 2.1 \begin{align*} -{\frac{1}{2\,d\sin \left ( dx+c \right ) } \left ( A\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}\sin \left ( dx+c \right ) }{2\,\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \sin \left ( dx+c \right ) +2\,B\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) +2\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,A\cos \left ( dx+c \right ) +4\,C\cos \left ( dx+c \right ) -4\,C \right ) \sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.97047, size = 1268, normalized size = 12.94 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.667057, size = 717, normalized size = 7.32 \begin{align*} \left [\frac{{\left ({\left (A + 2 \, B\right )} \cos \left (d x + c\right ) + A + 2 \, B\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \,{\left (A \cos \left (d x + c\right ) + 2 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{2 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac{{\left ({\left (A + 2 \, B\right )} \cos \left (d x + c\right ) + A + 2 \, B\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right ) -{\left (A \cos \left (d x + c\right ) + 2 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) + d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 6.38662, size = 531, normalized size = 5.42 \begin{align*} -\frac{\frac{4 \, \sqrt{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} C a \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a} +{\left (A \sqrt{-a} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 2 \, B \sqrt{-a} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \log \left ({\left |{\left (\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} - a{\left (2 \, \sqrt{2} + 3\right )} \right |}\right ) -{\left (A \sqrt{-a} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 2 \, B \sqrt{-a} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \log \left ({\left |{\left (\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} + a{\left (2 \, \sqrt{2} - 3\right )} \right |}\right ) + \frac{4 \, \sqrt{2}{\left (3 \,{\left (\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} A \sqrt{-a} a \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) - A \sqrt{-a} a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )}}{{\left (\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{4} - 6 \,{\left (\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} a + a^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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