Optimal. Leaf size=138 \[ \frac{8 a^2 (21 B+19 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (7 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 B+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d} \]
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Rubi [A] time = 0.353351, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4072, 4010, 4001, 3793, 3792} \[ \frac{8 a^2 (21 B+19 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (7 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 B+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 4010
Rule 4001
Rule 3793
Rule 3792
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} (B+C \sec (c+d x)) \, dx\\ &=\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac{2 \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{5 a C}{2}+\frac{1}{2} a (7 B-2 C) \sec (c+d x)\right ) \, dx}{7 a}\\ &=\frac{2 (7 B-2 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac{1}{35} (21 B+19 C) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac{2 a (21 B+19 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 d}+\frac{2 (7 B-2 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac{1}{105} (4 a (21 B+19 C)) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{8 a^2 (21 B+19 C) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (21 B+19 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 d}+\frac{2 (7 B-2 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}\\ \end{align*}
Mathematica [A] time = 0.372721, size = 82, normalized size = 0.59 \[ \frac{2 a^2 \tan (c+d x) \left (3 (7 B+13 C) \sec ^2(c+d x)+(63 B+52 C) \sec (c+d x)+2 (63 B+52 C)+15 C \sec ^3(c+d x)\right )}{105 d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.265, size = 117, normalized size = 0.9 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 126\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+104\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+63\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+52\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+21\,B\cos \left ( dx+c \right ) +39\,C\cos \left ( dx+c \right ) +15\,C \right ) }{105\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}\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 = 0.510531, size = 279, normalized size = 2.02 \begin{align*} \frac{2 \,{\left (2 \,{\left (63 \, B + 52 \, C\right )} a \cos \left (d x + c\right )^{3} +{\left (63 \, B + 52 \, C\right )} a \cos \left (d x + c\right )^{2} + 3 \,{\left (7 \, B + 13 \, C\right )} a \cos \left (d x + c\right ) + 15 \, C a\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\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 [A] time = 4.67613, size = 300, normalized size = 2.17 \begin{align*} -\frac{4 \,{\left (105 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 105 \, \sqrt{2} C a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (210 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 140 \, \sqrt{2} C a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (147 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 133 \, \sqrt{2} C a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) - 2 \,{\left (21 \, \sqrt{2} B a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 19 \, \sqrt{2} C a^{5} \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 )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{105 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} \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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