Optimal. Leaf size=282 \[ \frac {2 a^3 (299 B+280 C) \tan (c+d x) \sec ^4(c+d x)}{1287 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (4615 B+4184 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt {a \sec (c+d x)+a}}+\frac {4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (13 B+16 C) \tan (c+d x) \sec ^4(c+d x) \sqrt {a \sec (c+d x)+a}}{143 d}-\frac {8 a^2 (4615 B+4184 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{45045 d}+\frac {4 a (4615 B+4184 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac {2 a C \tan (c+d x) \sec ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{13 d} \]
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Rubi [A] time = 0.84, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4072, 4018, 4016, 3803, 3800, 4001, 3792} \[ \frac {2 a^2 (13 B+16 C) \tan (c+d x) \sec ^4(c+d x) \sqrt {a \sec (c+d x)+a}}{143 d}+\frac {2 a^3 (299 B+280 C) \tan (c+d x) \sec ^4(c+d x)}{1287 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (4615 B+4184 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt {a \sec (c+d x)+a}}-\frac {8 a^2 (4615 B+4184 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{45045 d}+\frac {4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt {a \sec (c+d x)+a}}+\frac {4 a (4615 B+4184 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac {2 a C \tan (c+d x) \sec ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{13 d} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3800
Rule 3803
Rule 4001
Rule 4016
Rule 4018
Rule 4072
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^4(c+d x) (a+a \sec (c+d x))^{5/2} (B+C \sec (c+d x)) \, dx\\ &=\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {2}{13} \int \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (13 B+8 C)+\frac {1}{2} a (13 B+16 C) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {4}{143} \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (247 B+216 C)+\frac {1}{4} a^2 (299 B+280 C) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {\left (a^2 (4615 B+4184 C)\right ) \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx}{1287}\\ &=\frac {2 a^3 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {\left (2 a^2 (4615 B+4184 C)\right ) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx}{3003}\\ &=\frac {2 a^3 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {4 a (4615 B+4184 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {(4 a (4615 B+4184 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 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}-\frac {8 a^2 (4615 B+4184 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {4 a (4615 B+4184 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {\left (2 a^2 (4615 B+4184 C)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx}{6435}\\ &=\frac {4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}-\frac {8 a^2 (4615 B+4184 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {4 a (4615 B+4184 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 131, normalized size = 0.46 \[ \frac {2 a^3 \tan (c+d x) \left (315 (13 B+38 C) \sec ^5(c+d x)+35 (416 B+523 C) \sec ^4(c+d x)+5 (4615 B+4184 C) \sec ^3(c+d x)+6 (4615 B+4184 C) \sec ^2(c+d x)+8 (4615 B+4184 C) \sec (c+d x)+73840 B+3465 C \sec ^6(c+d x)+66944 C\right )}{45045 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 177, normalized size = 0.63 \[ \frac {2 \, {\left (16 \, {\left (4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 8 \, {\left (4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 6 \, {\left (4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (416 \, B + 523 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 315 \, {\left (13 \, B + 38 \, C\right )} a^{2} \cos \left (d x + c\right ) + 3465 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.39, size = 351, normalized size = 1.24 \[ \frac {8 \, {\left ({\left ({\left ({\left ({\left (4 \, {\left (2 \, \sqrt {2} {\left (1625 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1483 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13 \, \sqrt {2} {\left (1625 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1483 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 143 \, \sqrt {2} {\left (1625 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1483 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 858 \, \sqrt {2} {\left (415 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 362 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6006 \, \sqrt {2} {\left (50 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 49 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30030 \, \sqrt {2} {\left (5 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 4 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45045 \, \sqrt {2} {\left (B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{45045 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{6} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.82, size = 185, normalized size = 0.66 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (73840 B \left (\cos ^{6}\left (d x +c \right )\right )+66944 C \left (\cos ^{6}\left (d x +c \right )\right )+36920 B \left (\cos ^{5}\left (d x +c \right )\right )+33472 C \left (\cos ^{5}\left (d x +c \right )\right )+27690 B \left (\cos ^{4}\left (d x +c \right )\right )+25104 C \left (\cos ^{4}\left (d x +c \right )\right )+23075 B \left (\cos ^{3}\left (d x +c \right )\right )+20920 C \left (\cos ^{3}\left (d x +c \right )\right )+14560 B \left (\cos ^{2}\left (d x +c \right )\right )+18305 C \left (\cos ^{2}\left (d x +c \right )\right )+4095 B \cos \left (d x +c \right )+11970 C \cos \left (d x +c \right )+3465 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{45045 d \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.05, size = 988, normalized size = 3.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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