Optimal. Leaf size=62 \[ \frac{2 a (3 B+C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d} \]
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Rubi [A] time = 0.0842717, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {4054, 12, 3792} \[ \frac{2 a (3 B+C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d} \]
Antiderivative was successfully verified.
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Rule 4054
Rule 12
Rule 3792
Rubi steps
\begin{align*} \int \sqrt{a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}+\frac{2 \int \frac{1}{2} a (3 B+C) \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx}{3 a}\\ &=\frac{2 C \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}+\frac{1}{3} (3 B+C) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a (3 B+C) \tan (c+d x)}{3 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.159898, size = 43, normalized size = 0.69 \[ \frac{2 a \tan (c+d x) (3 B+C \sec (c+d x)+2 C)}{3 d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.275, size = 70, normalized size = 1.1 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 3\,B\cos \left ( dx+c \right ) +2\,C\cos \left ( dx+c \right ) +C \right ) }{3\,d\sin \left ( dx+c \right ) \cos \left ( dx+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 [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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Fricas [A] time = 0.496403, size = 169, normalized size = 2.73 \begin{align*} \frac{2 \,{\left ({\left (3 \, B + 2 \, C\right )} \cos \left (d x + c\right ) + C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\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 \sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )} \left (B + C \sec{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 4.31345, size = 174, normalized size = 2.81 \begin{align*} -\frac{2 \,{\left (3 \, \sqrt{2} B a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 3 \, \sqrt{2} C a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (3 \, \sqrt{2} B a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + \sqrt{2} C a^{2} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{3 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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