Optimal. Leaf size=254 \[ \frac {2 \left (7 a^2 A+18 a b B+9 A b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+18 a b B+9 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (5 a (a B+2 A b)+7 b^2 B\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 \left (5 a (a B+2 A b)+7 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A] time = 0.34, 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 = {4024, 4047, 3769, 3771, 2639, 4045, 2641} \[ \frac {2 \left (7 a^2 A+18 a b B+9 A b^2\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+18 a b B+9 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (5 a (a B+2 A b)+7 b^2 B\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 \left (5 a (a B+2 A b)+7 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 4024
Rule 4045
Rule 4047
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 = 1.87, size = 189, normalized size = 0.74 \[ \frac {\sqrt {\sec (c+d x)} \left (\sin (2 (c+d x)) \left (7 \left (43 a^2 A+72 a b B+36 A b^2\right ) \cos (c+d x)+5 \left (7 a^2 A \cos (3 (c+d x))+78 a^2 B+18 a (a B+2 A b) \cos (2 (c+d x))+156 a A b+84 b^2 B\right )\right )+120 \left (5 a^2 B+10 a A b+7 b^2 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+168 \left (7 a^2 A+18 a b B+9 A b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{1260 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B b^{2} \sec \left (d x + c\right )^{3} + A a^{2} + {\left (2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.66, size = 610, normalized size = 2.40 \[ -\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 \,a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2240 a^{2} A +1440 A a b +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 a b -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 a b +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 a b -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 )-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}+150 A a 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 +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 )\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} \]
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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
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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