Optimal. Leaf size=169 \[ \frac {64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (21 A+13 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 a (21 A+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac {4 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d} \]
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Rubi [A] time = 0.32, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4083, 4001, 3793, 3792} \[ \frac {16 a^2 (21 A+13 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a (21 A+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac {4 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d} \]
Antiderivative was successfully verified.
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Rule 3792
Rule 3793
Rule 4001
Rule 4083
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {2 \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (9 A+7 C)-a C \sec (c+d x)\right ) \, dx}{9 a}\\ &=-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{21} (21 A+13 C) \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac {2 a (21 A+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{105} (8 a (21 A+13 C)) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 (21 A+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{315} \left (32 a^2 (21 A+13 C)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (21 A+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}\\ \end {align*}
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Mathematica [A] time = 1.54, size = 125, normalized size = 0.74 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4(c+d x) \sqrt {a (\sec (c+d x)+1)} (4 (441 A+698 C) \cos (c+d x)+4 (966 A+803 C) \cos (2 (c+d x))+588 A \cos (3 (c+d x))+903 A \cos (4 (c+d x))+2961 A+584 C \cos (3 (c+d x))+584 C \cos (4 (c+d x))+2908 C)}{1260 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 130, normalized size = 0.77 \[ \frac {2 \, {\left ({\left (903 \, A + 584 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 2 \, {\left (147 \, A + 146 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 73 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 130 \, C a^{2} \cos \left (d x + c\right ) + 35 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.13, size = 261, normalized size = 1.54 \[ \frac {8 \, {\left ({\left ({\left (4 \, {\left (2 \, \sqrt {2} {\left (21 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \sqrt {2} {\left (21 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 63 \, \sqrt {2} {\left (21 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 210 \, \sqrt {2} {\left (5 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3 \, C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, \sqrt {2} {\left (A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.72, size = 132, normalized size = 0.78 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (903 A \left (\cos ^{4}\left (d x +c \right )\right )+584 C \left (\cos ^{4}\left (d x +c \right )\right )+294 A \left (\cos ^{3}\left (d x +c \right )\right )+292 C \left (\cos ^{3}\left (d x +c \right )\right )+63 A \left (\cos ^{2}\left (d x +c \right )\right )+219 C \left (\cos ^{2}\left (d x +c \right )\right )+130 C \cos \left (d x +c \right )+35 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )} \]
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 = 11.20, size = 766, normalized size = 4.53 \[ \text {result too large to display} \]
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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