Optimal. Leaf size=169 \[ \frac{a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac{a^2 (9 A+10 C) \sin (c+d x) \cos ^2(c+d x)}{30 d}+\frac{a^2 (3 A+4 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{10 d}+\frac{1}{4} a^2 x (3 A+4 C)+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d} \]
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Rubi [A] time = 0.387358, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4087, 4017, 3996, 3787, 2635, 8, 2637} \[ \frac{a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac{a^2 (9 A+10 C) \sin (c+d x) \cos ^2(c+d x)}{30 d}+\frac{a^2 (3 A+4 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{10 d}+\frac{1}{4} a^2 x (3 A+4 C)+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 4087
Rule 4017
Rule 3996
Rule 3787
Rule 2635
Rule 8
Rule 2637
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 (2 a A+a (2 A+5 C) \sec (c+d x)) \, dx}{5 a}\\ &=\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}+\frac{\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (2 a^2 (9 A+10 C)+4 a^2 (3 A+5 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}-\frac{\int \cos ^2(c+d x) \left (-30 a^3 (3 A+4 C)-4 a^3 (18 A+25 C) \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac{a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}+\frac{1}{2} \left (a^2 (3 A+4 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{15} \left (a^2 (18 A+25 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac{a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac{a^2 (3 A+4 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac{a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}+\frac{1}{4} \left (a^2 (3 A+4 C)\right ) \int 1 \, dx\\ &=\frac{1}{4} a^2 (3 A+4 C) x+\frac{a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac{a^2 (3 A+4 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac{a^2 (9 A+10 C) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{A \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{10 d}\\ \end{align*}
Mathematica [A] time = 0.396537, size = 97, normalized size = 0.57 \[ \frac{a^2 (30 (11 A+14 C) \sin (c+d x)+120 (A+C) \sin (2 (c+d x))+45 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+3 A \sin (5 (c+d x))+120 A c+180 A d x+20 C \sin (3 (c+d x))+240 C d x)}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.111, size = 160, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2}A\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+{\frac{{a}^{2}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,{a}^{2}A \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +2\,{a}^{2}C \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{\frac{{a}^{2}A \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{a}^{2}C\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.941552, size = 211, normalized size = 1.25 \begin{align*} \frac{16 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} - 80 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 80 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 240 \, C a^{2} \sin \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.498604, size = 258, normalized size = 1.53 \begin{align*} \frac{15 \,{\left (3 \, A + 4 \, C\right )} a^{2} d x +{\left (12 \, A a^{2} \cos \left (d x + c\right )^{4} + 30 \, A a^{2} \cos \left (d x + c\right )^{3} + 4 \,{\left (9 \, A + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \,{\left (3 \, A + 4 \, C\right )} a^{2} \cos \left (d x + c\right ) + 4 \,{\left (18 \, A + 25 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{60 \, d} \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 = 1.18922, size = 284, normalized size = 1.68 \begin{align*} \frac{15 \,{\left (3 \, A a^{2} + 4 \, C a^{2}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (45 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 60 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 210 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 280 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 432 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 560 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 270 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 520 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 195 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 180 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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