Optimal. Leaf size=144 \[ \frac{8 a^2 (35 A+21 B+19 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (35 A+21 B+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 (7 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d} \]
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Rubi [A] time = 0.283473, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {4082, 4001, 3793, 3792} \[ \frac{8 a^2 (35 A+21 B+19 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (35 A+21 B+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 (7 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d} \]
Antiderivative was successfully verified.
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Rule 4082
Rule 4001
Rule 3793
Rule 3792
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, 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{1}{2} a (7 A+5 C)+\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} (35 A+21 B+19 C) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac{2 a (35 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 (35 A+21 B+19 C)) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{8 a^2 (35 A+21 B+19 C) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (35 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 = 1.63287, size = 120, normalized size = 0.83 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} ((525 A+462 B+468 C) \cos (c+d x)+2 (35 A+63 B+52 C) \cos (2 (c+d x))+175 A \cos (3 (c+d x))+70 A+126 B \cos (3 (c+d x))+126 B+104 C \cos (3 (c+d x))+164 C)}{210 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.296, size = 139, normalized size = 1. \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 175\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+126\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+104\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+35\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+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 = 0.501427, size = 300, normalized size = 2.08 \begin{align*} \frac{2 \,{\left ({\left (175 \, A + 126 \, B + 104 \, C\right )} a \cos \left (d x + c\right )^{3} +{\left (35 \, A + 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 [B] time = 4.89244, size = 386, normalized size = 2.68 \begin{align*} -\frac{4 \,{\left (105 \, \sqrt{2} A a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 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 (280 \, \sqrt{2} A a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 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 (245 \, \sqrt{2} A a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 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 (35 \, \sqrt{2} A a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 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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