Optimal. Leaf size=110 \[ \frac{\sqrt{a} (3 A+8 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}+\frac{a A \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d} \]
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Rubi [A] time = 0.250998, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {4087, 4015, 3774, 203} \[ \frac{\sqrt{a} (3 A+8 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{4 d}+\frac{a A \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d} \]
Antiderivative was successfully verified.
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Rule 4087
Rule 4015
Rule 3774
Rule 203
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac{\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{a A}{2}+\frac{1}{2} a (A+4 C) \sec (c+d x)\right ) \, dx}{2 a}\\ &=\frac{a A \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}+\frac{1}{8} (3 A+8 C) \int \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a A \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}-\frac{(a (3 A+8 C)) \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 )}{4 d}\\ &=\frac{\sqrt{a} (3 A+8 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{4 d}+\frac{a A \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos (c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.413832, size = 108, normalized size = 0.98 \[ \frac{\sqrt{\cos (c+d x)} \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sqrt{2} (3 A+8 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )+2 A \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{3}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.376, size = 376, normalized size = 3.4 \begin{align*} -{\frac{1}{16\,d\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) } \left ( -3\,A\sqrt{2}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ){\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 ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{3/2}-8\,C\cos \left ( dx+c \right ) \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 ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{3/2}\sin \left ( dx+c \right ) -3\,A\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 ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{3/2}\sin \left ( dx+c \right ) -8\,C{\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} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{3/2}\sin \left ( dx+c \right ) +8\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}-12\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2} \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 = 2.08601, size = 1629, normalized size = 14.81 \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 = 0.637843, size = 779, normalized size = 7.08 \begin{align*} \left [\frac{{\left ({\left (3 \, A + 8 \, C\right )} \cos \left (d x + c\right ) + 3 \, A + 8 \, C\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 (2 \, A \cos \left (d x + c\right )^{2} + 3 \, A \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{8 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac{{\left ({\left (3 \, A + 8 \, C\right )} \cos \left (d x + c\right ) + 3 \, A + 8 \, C\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 (2 \, A \cos \left (d x + c\right )^{2} + 3 \, A \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{4 \,{\left (d \cos \left (d x + c\right ) + d\right )}}\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 = 6.441, size = 602, normalized size = 5.47 \begin{align*} -\frac{{\left (3 \, A \sqrt{-a} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 8 \, C \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 (3 \, A \sqrt{-a} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 8 \, C \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 (5 \,{\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 )}^{6} A \sqrt{-a} a \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 19 \,{\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} A \sqrt{-a} a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) - 17 \,{\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^{3} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + A \sqrt{-a} a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )}}{{\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 )}^{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}\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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