Optimal. Leaf size=273 \[ \frac{2 a^3 (2717 A+2224 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (143 A+136 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+8368 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}-\frac{4 a^2 (10439 A+8368 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+8368 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac{10 a C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d} \]
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Rubi [A] time = 0.864525, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4089, 4018, 4016, 3800, 4001, 3792} \[ \frac{2 a^3 (2717 A+2224 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (143 A+136 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+8368 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}-\frac{4 a^2 (10439 A+8368 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+8368 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac{10 a C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d} \]
Antiderivative was successfully verified.
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Rule 4089
Rule 4018
Rule 4016
Rule 3800
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{2 \int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac{1}{2} a (13 A+6 C)+\frac{5}{2} a C \sec (c+d x)\right ) \, dx}{13 a}\\ &=\frac{10 a C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{4 \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{1}{4} a^2 (143 A+96 C)+\frac{1}{4} a^2 (143 A+136 C) \sec (c+d x)\right ) \, dx}{143 a}\\ &=\frac{2 a^2 (143 A+136 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{10 a C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{8 \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{15}{8} a^3 (143 A+112 C)+\frac{1}{8} a^3 (2717 A+2224 C) \sec (c+d x)\right ) \, dx}{1287 a}\\ &=\frac{2 a^3 (2717 A+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (143 A+136 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{10 a C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{\left (a^2 (10439 A+8368 C)\right ) \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx}{3003}\\ &=\frac{2 a^3 (2717 A+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (143 A+136 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{2 a (10439 A+8368 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac{10 a C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{(2 a (10439 A+8368 C)) \int \sec (c+d x) \left (\frac{3 a}{2}-a \sec (c+d x)\right ) \sqrt{a+a \sec (c+d x)} \, dx}{15015}\\ &=\frac{2 a^3 (2717 A+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt{a+a \sec (c+d x)}}-\frac{4 a^2 (10439 A+8368 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac{2 a^2 (143 A+136 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{2 a (10439 A+8368 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac{10 a C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}+\frac{\left (a^2 (10439 A+8368 C)\right ) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx}{6435}\\ &=\frac{2 a^3 (10439 A+8368 C) \tan (c+d x)}{6435 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^3 (2717 A+2224 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt{a+a \sec (c+d x)}}-\frac{4 a^2 (10439 A+8368 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac{2 a^2 (143 A+136 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{1287 d}+\frac{2 a (10439 A+8368 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac{10 a C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{143 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{13 d}\\ \end{align*}
Mathematica [A] time = 1.92376, size = 169, normalized size = 0.62 \[ \frac{a^2 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6(c+d x) \sqrt{a (\sec (c+d x)+1)} (1120 (286 A+347 C) \cos (c+d x)+14 (32747 A+30334 C) \cos (2 (c+d x))+141570 A \cos (3 (c+d x))+156585 A \cos (4 (c+d x))+20878 A \cos (5 (c+d x))+20878 A \cos (6 (c+d x))+322751 A+125520 C \cos (3 (c+d x))+125520 C \cos (4 (c+d x))+16736 C \cos (5 (c+d x))+16736 C \cos (6 (c+d x))+343612 C)}{180180 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.359, size = 176, normalized size = 0.6 \begin{align*} -{\frac{2\,{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 83512\,A \left ( \cos \left ( dx+c \right ) \right ) ^{6}+66944\,C \left ( \cos \left ( dx+c \right ) \right ) ^{6}+41756\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+33472\,C \left ( \cos \left ( dx+c \right ) \right ) ^{5}+31317\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+25104\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+18590\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+20920\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+5005\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+18305\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+11970\,C\cos \left ( dx+c \right ) +3465\,C \right ) }{45045\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}\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 = 0.527034, size = 470, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (8 \,{\left (10439 \, A + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 4 \,{\left (10439 \, A + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 3 \,{\left (10439 \, A + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \,{\left (1859 \, A + 2092 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \,{\left (143 \, A + 523 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 11970 \, C a^{2} \cos \left (d x + c\right ) + 3465 \, C a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \,{\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\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 = 5.31685, size = 486, normalized size = 1.78 \begin{align*} \frac{8 \,{\left (45045 \, \sqrt{2} A a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 45045 \, \sqrt{2} C a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (180180 \, \sqrt{2} A a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 120120 \, \sqrt{2} C a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (342342 \, \sqrt{2} A a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 294294 \, \sqrt{2} C a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (391248 \, \sqrt{2} A a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 310596 \, \sqrt{2} C a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (265837 \, \sqrt{2} A a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 212069 \, \sqrt{2} C a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) - 4 \,{\left (24167 \, \sqrt{2} A a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 19279 \, \sqrt{2} C a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) - 2 \,{\left (1859 \, \sqrt{2} A a^{9} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 1483 \, \sqrt{2} C a^{9} \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 )^{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 )}{45045 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{6} \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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