Optimal. Leaf size=254 \[ \frac {2 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (7 b^2 B+5 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (2 A b+a B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 b^2 B+5 a (2 A b+a B)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 0.22, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4109, 4132,
3854, 3856, 2719, 4130, 2720} \begin {gather*} \frac {2 \left (7 a^2 A+18 a b B+9 A b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+18 a b B+9 A 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 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (5 a (a B+2 A b)+7 b^2 B\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 \left (5 a (a B+2 A b)+7 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3854
Rule 3856
Rule 4109
Rule 4130
Rule 4132
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int \frac {-\frac {9}{2} a (2 A b+a B)+\left (A \left (-\frac {7 a^2}{2}-\frac {9 b^2}{2}\right )-9 a b B\right ) \sec (c+d x)-\frac {9}{2} b^2 B \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int \frac {-\frac {9}{2} a (2 A b+a B)-\frac {9}{2} b^2 B \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx-\frac {1}{9} \left (-7 a^2 A-9 A b^2-18 a b B\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (2 A b+a B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {1}{15} \left (-7 a^2 A-9 A b^2-18 a b B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx-\frac {1}{7} \left (-7 b^2 B-5 a (2 A b+a B)\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (2 A b+a B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 b^2 B+5 a (2 A b+a B)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {1}{21} \left (-7 b^2 B-5 a (2 A b+a B)\right ) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{15} \left (\left (-7 a^2 A-9 A b^2-18 a b B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\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 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (2 A b+a B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 b^2 B+5 a (2 A b+a B)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {1}{21} \left (\left (-7 b^2 B-5 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\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 (7 b^2 B+5 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (2 A b+a B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 b^2 B+5 a (2 A b+a B)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 1.89, size = 189, normalized size = 0.74 \begin {gather*} \frac {\sqrt {\sec (c+d x)} \left (168 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+120 \left (10 a A b+5 a^2 B+7 b^2 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\left (7 \left (43 a^2 A+36 A b^2+72 a b B\right ) \cos (c+d x)+5 \left (156 a A b+78 a^2 B+84 b^2 B+18 a (2 A b+a B) \cos (2 (c+d x))+7 a^2 A \cos (3 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1260 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs.
\(2(278)=556\).
time = 1.82, size = 610, normalized size = 2.40
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 (-1120 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+\left (2240 a^{2} A +1440 A b a +720 a^{2} B \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 (-2072 a^{2} A -2160 A b a -504 A \,b^{2}-1080 a^{2} B -1008 B a b \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 (952 a^{2} A +1680 A b a +504 A \,b^{2}+840 a^{2} B +1008 B a b +420 b^{2} B \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 (-168 a^{2} A -480 A b a -126 A \,b^{2}-240 a^{2} B -252 B a b -210 b^{2} B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+150 A b a \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 )-147 A \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^{2}-189 A \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 ) b^{2}+75 a^{2} B \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 )+105 b^{2} B \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 )-378 B \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 \right )}{315 \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}\) | \(610\) |
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 = 0.92, size = 282, normalized size = 1.11 \begin {gather*} -\frac {15 \, \sqrt {2} {\left (5 i \, B a^{2} + 10 i \, A a b + 7 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{2} - 10 i \, A a b - 7 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-7 i \, A a^{2} - 18 i \, B a b - 9 i \, A b^{2}\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 ) + 21 \, \sqrt {2} {\left (7 i \, A a^{2} + 18 i \, B a b + 9 i \, A b^{2}\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 (35 \, A a^{2} \cos \left (d x + c\right )^{4} + 45 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (7 \, A a^{2} + 18 \, B a b + 9 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, B a^{2} + 10 \, A a b + 7 \, B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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