Optimal. Leaf size=284 \[ \frac {2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {4 a^3 (2840 A+3212 B+3795 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (32 A+44 B+33 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+11 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A]
time = 0.59, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{3465 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+11 B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d \sec ^{\frac {9}{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 {11}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (5 A+11 B)+\frac {1}{2} a (4 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac {2 a (5 A+11 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {3}{4} a^2 (32 A+44 B+33 C)+\frac {1}{4} a^2 (56 A+44 B+99 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac {2 a^2 (32 A+44 B+33 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+11 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8 \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (1160 A+1364 B+1485 C)+\frac {1}{8} a^3 (776 A+836 B+1089 C) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac {2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (32 A+44 B+33 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+11 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {\left (a^2 (2840 A+3212 B+3795 C)\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{1155}\\ &=\frac {2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (32 A+44 B+33 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+11 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {\left (2 a^2 (2840 A+3212 B+3795 C)\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{3465}\\ &=\frac {2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {4 a^3 (2840 A+3212 B+3795 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (32 A+44 B+33 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a (5 A+11 B) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 1.70, size = 157, normalized size = 0.55 \begin {gather*} \frac {a^2 (114640 A+124366 B+137280 C+(69890 A+68552 B+66660 C) \cos (c+d x)+16 (1625 A+1397 B+990 C) \cos (2 (c+d x))+8675 A \cos (3 (c+d x))+5720 B \cos (3 (c+d x))+1980 C \cos (3 (c+d x))+2240 A \cos (4 (c+d x))+770 B \cos (4 (c+d x))+315 A \cos (5 (c+d x))) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{27720 d \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 199, normalized size = 0.70
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (315 A \left (\cos ^{5}\left (d x +c \right )\right )+1120 A \left (\cos ^{4}\left (d x +c \right )\right )+385 B \left (\cos ^{4}\left (d x +c \right )\right )+1775 A \left (\cos ^{3}\left (d x +c \right )\right )+1430 B \left (\cos ^{3}\left (d x +c \right )\right )+495 C \left (\cos ^{3}\left (d x +c \right )\right )+2130 A \left (\cos ^{2}\left (d x +c \right )\right )+2409 B \left (\cos ^{2}\left (d x +c \right )\right )+1980 C \left (\cos ^{2}\left (d x +c \right )\right )+2840 A \cos \left (d x +c \right )+3212 B \cos \left (d x +c \right )+3795 C \cos \left (d x +c \right )+5680 A +6424 B +7590 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\cos ^{6}\left (d x +c \right )\right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {11}{2}} a^{2}}{3465 d \sin \left (d x +c \right )}\) | \(199\) |
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. 1266 vs.
\(2 (248) = 496\).
time = 0.75, size = 1266, normalized size = 4.46 \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 = 4.21, size = 173, normalized size = 0.61 \begin {gather*} \frac {2 \, {\left (315 \, A a^{2} \cos \left (d x + c\right )^{6} + 35 \, {\left (32 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 5 \, {\left (355 \, A + 286 \, B + 99 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 3 \, {\left (710 \, A + 803 \, B + 660 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (2840 \, A + 3212 \, B + 3795 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, {\left (2840 \, A + 3212 \, B + 3795 \, 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 )}{3465 \, {\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 = 11.31, size = 404, normalized size = 1.42 \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 {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}-1\right )\,\left (\frac {A\,a^2\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{88\,d}+\frac {a^2\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (13\,A+10\,B+4\,C\right )}{56\,d}+\frac {a^2\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (19\,A+20\,B+22\,C\right )}{12\,d}+\frac {a^2\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (25\,A+24\,B+20\,C\right )}{40\,d}+\frac {a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )\,\left (23\,A+26\,B+30\,C\right )}{4\,d}+\frac {a^2\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,\left (5\,A+2\,B\right )\,\left (-2\,{\sin \left (\frac {11\,c}{4}+\frac {11\,d\,x}{4}\right )}^2+\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )}{72\,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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