Optimal. Leaf size=289 \[ \frac {8 a b \left (7 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A]
time = 0.30, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3926, 4159,
4132, 3854, 3856, 2719, 4130, 2720} \begin {gather*} \frac {52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (7 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 3926
Rule 4130
Rule 4132
Rule 4159
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2}{11} \int \frac {(a+b \sec (c+d x)) \left (13 a^2 b+\frac {3}{2} a \left (3 a^2+11 b^2\right ) \sec (c+d x)+\frac {1}{2} b \left (5 a^2+11 b^2\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {4}{99} \int \frac {-\frac {9}{4} a^2 \left (9 a^2+59 b^2\right )-11 a b \left (7 a^2+9 b^2\right ) \sec (c+d x)-\frac {9}{4} b^2 \left (5 a^2+11 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {4}{99} \int \frac {-\frac {9}{4} a^2 \left (9 a^2+59 b^2\right )-\frac {9}{4} b^2 \left (5 a^2+11 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx+\frac {1}{9} \left (4 a b \left (7 a^2+9 b^2\right )\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {1}{15} \left (4 a b \left (7 a^2+9 b^2\right )\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{77} \left (-45 a^4-330 a^2 b^2-77 b^4\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {1}{231} \left (-45 a^4-330 a^2 b^2-77 b^4\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (4 a b \left (7 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {8 a b \left (7 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {1}{231} \left (\left (-45 a^4-330 a^2 b^2-77 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {8 a b \left (7 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a^2 \left (9 a^2+59 b^2\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (7 a^2+9 b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 1.23, size = 199, normalized size = 0.69 \begin {gather*} \frac {\sqrt {\sec (c+d x)} \left (14784 a b \left (7 a^2+9 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 \left (45 a^4+330 a^2 b^2+77 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\left (616 a b \left (43 a^2+36 b^2\right ) \cos (c+d x)+5 \left (1593 a^4+10296 a^2 b^2+1848 b^4+72 \left (8 a^4+33 a^2 b^2\right ) \cos (2 (c+d x))+616 a^3 b \cos (3 (c+d x))+63 a^4 \cos (4 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{27720 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 586, normalized size = 2.03
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (20160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}+\left (-50400 a^{4}-49280 b \,a^{3}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (56880 a^{4}+98560 b \,a^{3}+47520 b^{2} a^{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-34920 a^{4}-91168 b \,a^{3}-71280 b^{2} a^{2}-22176 b^{3} a \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (13860 a^{4}+41888 b \,a^{3}+55440 b^{2} a^{2}+22176 b^{3} a +4620 b^{4}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2790 a^{4}-7392 b \,a^{3}-15840 b^{2} a^{2}-5544 b^{3} a -2310 b^{4}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+675 a^{4} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+4950 b^{2} a^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1155 b^{4} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-6468 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3} b -8316 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{3}\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(586\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.90, size = 286, normalized size = 0.99 \begin {gather*} -\frac {15 \, \sqrt {2} {\left (45 i \, a^{4} + 330 i \, a^{2} b^{2} + 77 i \, b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-45 i \, a^{4} - 330 i \, a^{2} b^{2} - 77 i \, b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 924 \, \sqrt {2} {\left (-7 i \, a^{3} b - 9 i \, a b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 924 \, \sqrt {2} {\left (7 i \, a^{3} b + 9 i \, a b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, a^{4} \cos \left (d x + c\right )^{5} + 1540 \, a^{3} b \cos \left (d x + c\right )^{4} + 135 \, {\left (3 \, a^{4} + 22 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 308 \, {\left (7 \, a^{3} b + 9 \, a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (45 \, a^{4} + 330 \, a^{2} b^{2} + 77 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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