Optimal. Leaf size=101 \[ \frac{2 (5 A-2 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a (5 A+7 B) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d} \]
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Rubi [A] time = 0.227687, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4010, 4001, 3792} \[ \frac{2 (5 A-2 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a (5 A+7 B) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d} \]
Antiderivative was successfully verified.
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Rule 4010
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx &=\frac{2 B (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}+\frac{2 \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{3 a B}{2}+\frac{1}{2} a (5 A-2 B) \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac{2 (5 A-2 B) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac{2 B (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}+\frac{1}{15} (5 A+7 B) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a (5 A+7 B) \tan (c+d x)}{15 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (5 A-2 B) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac{2 B (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.277759, size = 80, normalized size = 0.79 \[ \frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} ((5 A+4 B) \cos (c+d x)+(5 A+4 B) \cos (2 (c+d x))+5 A+7 B)}{15 d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.331, size = 94, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 10\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+8\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+5\,A\cos \left ( dx+c \right ) +4\,B\cos \left ( dx+c \right ) +3\,B \right ) }{15\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \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 = 0.464324, size = 217, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (2 \,{\left (5 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (5 \, A + 4 \, B\right )} \cos \left (d x + c\right ) + 3 \, B\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\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 (A + B \sec{\left (c + d x \right )}\right ) \sec ^{2}{\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 = 4.81692, size = 238, normalized size = 2.36 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{2} A a^{3} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 15 \, \sqrt{2} B a^{3} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (20 \, \sqrt{2} A a^{3} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 10 \, \sqrt{2} B a^{3} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (5 \, \sqrt{2} A a^{3} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 7 \, \sqrt{2} B a^{3} \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 )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{15 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} \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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