Optimal. Leaf size=334 \[ \frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)} \]
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Rubi [A]
time = 0.65, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4171, 4102,
4100, 3890, 3889} \begin {gather*} \frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{45045 d \sqrt {a \sec (c+d x)+a}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d \sec ^{\frac {11}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3889
Rule 3890
Rule 4100
Rule 4102
Rule 4171
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (5 A+13 B)+\frac {1}{2} a (6 A+13 C) \sec (c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx}{13 a}\\ &=\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (136 A+182 B+143 C)+\frac {1}{4} a^2 (96 A+78 B+143 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{143 a}\\ &=\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {8 \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (2224 A+2522 B+2717 C)+\frac {3}{8} a^3 (560 A+598 B+715 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{1287 a}\\ &=\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {\left (a^2 (8368 A+9230 B+10439 C)\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{3003}\\ &=\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {\left (4 a^2 (8368 A+9230 B+10439 C)\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{15015}\\ &=\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}+\frac {\left (8 a^2 (8368 A+9230 B+10439 C)\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{45045}\\ &=\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+13 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 1.84, size = 190, normalized size = 0.57 \begin {gather*} \frac {a^2 (2798182 A+2980640 B+3233516 C+4 (453146 A+454285 B+445588 C) \cos (c+d x)+(746519 A+676000 B+581152 C) \cos (2 (c+d x))+287060 A \cos (3 (c+d x))+225550 B \cos (3 (c+d x))+148720 C \cos (3 (c+d x))+94010 A \cos (4 (c+d x))+58240 B \cos (4 (c+d x))+20020 C \cos (4 (c+d x))+23940 A \cos (5 (c+d x))+8190 B \cos (5 (c+d x))+3465 A \cos (6 (c+d x))) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{720720 d \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 232, normalized size = 0.69
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (3465 A \left (\cos ^{6}\left (d x +c \right )\right )+11970 A \left (\cos ^{5}\left (d x +c \right )\right )+4095 B \left (\cos ^{5}\left (d x +c \right )\right )+18305 A \left (\cos ^{4}\left (d x +c \right )\right )+14560 B \left (\cos ^{4}\left (d x +c \right )\right )+5005 C \left (\cos ^{4}\left (d x +c \right )\right )+20920 A \left (\cos ^{3}\left (d x +c \right )\right )+23075 B \left (\cos ^{3}\left (d x +c \right )\right )+18590 C \left (\cos ^{3}\left (d x +c \right )\right )+25104 A \left (\cos ^{2}\left (d x +c \right )\right )+27690 B \left (\cos ^{2}\left (d x +c \right )\right )+31317 C \left (\cos ^{2}\left (d x +c \right )\right )+33472 A \cos \left (d x +c \right )+36920 B \cos \left (d x +c \right )+41756 C \cos \left (d x +c \right )+66944 A +73840 B +83512 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\cos ^{7}\left (d x +c \right )\right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {13}{2}} a^{2}}{45045 d \sin \left (d x +c \right )}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1562 vs.
\(2 (292) = 584\).
time = 0.77, size = 1562, normalized size = 4.68 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.18, size = 197, normalized size = 0.59 \begin {gather*} \frac {2 \, {\left (3465 \, A a^{2} \cos \left (d x + c\right )^{7} + 315 \, {\left (38 \, A + 13 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} + 35 \, {\left (523 \, A + 416 \, B + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 5 \, {\left (4184 \, A + 4615 \, B + 3718 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 3 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 4 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )\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 ) + d\right )} \sqrt {\cos \left (d x + c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.74, size = 458, normalized size = 1.37 \begin {gather*} \frac {\sqrt {a-\frac {a}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}}\,\left (2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}-1\right )\,\left (\frac {A\,a^2\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{208\,d}+\frac {a^2\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (7\,A+5\,B+2\,C\right )}{72\,d}+\frac {a^2\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (15\,A+13\,B+10\,C\right )}{56\,d}+\frac {a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (21\,A+23\,B+26\,C\right )}{4\,d}+\frac {a^2\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (51\,A+50\,B+48\,C\right )}{80\,d}+\frac {a^2\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (71\,A+76\,B+80\,C\right )}{48\,d}+\frac {a^2\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,\left (5\,A+2\,B\right )\,\left (-2\,{\sin \left (\frac {13\,c}{4}+\frac {13\,d\,x}{4}\right )}^2+\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{176\,d}\right )}{2\,\sqrt {-\frac {1}{2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1}}\,\left (2\,{\sin \left (\frac {c}{4}+\frac {d\,x}{4}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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