Integrand size = 45, antiderivative size = 705 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (40 A b^4+10 a b^3 (3 A-11 B)+15 a^2 b^2 (19 A-121 B+33 C)+3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 a^2 d}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]
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Time = 3.69 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4306, 3126, 3134, 3077, 2895, 3073} \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{231 d}+\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{3465 a d}+\frac {2 (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)+15 a^2 b^2 (19 A-121 B+33 C)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{3465 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{3465 a^2 d}+\frac {2 (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (11 a B+5 A b) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d} \]
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Rule 2895
Rule 3073
Rule 3077
Rule 3126
Rule 3134
Rule 4306
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx \\ & = \frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{11} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^{3/2} \left (\frac {1}{2} (5 A b+11 a B)+\frac {1}{2} (9 a A+11 b B+11 a C) \cos (c+d x)+\frac {1}{2} b (4 A+11 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{99} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {3}{4} \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right )+\frac {1}{4} \left (152 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \cos (c+d x)+\frac {1}{4} b (56 A b+44 a B+99 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{693} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right )+\frac {1}{8} \left (1507 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1531 A+2079 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (836 a b B+36 a^2 (9 A+11 C)+b^2 (452 A+693 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{16} \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right )+\frac {1}{16} a \left (1617 a^3 B+6655 a b^2 B+15 a^2 b (337 A+429 C)+5 b^3 (461 A+693 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a} \\ & = -\frac {2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 a^2 d}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{32} \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right )+\frac {3}{32} a \left (10 A b^4+2871 a^3 b B+1705 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (221 A+297 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{10395 a^2} \\ & = -\frac {2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 a^2 d}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (\left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2}+\frac {\left ((a-b) \left (40 A b^4+10 a b^3 (3 A-11 B)+15 a^2 b^2 (19 A-121 B+33 C)+3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2} \\ & = \frac {2 (a-b) \sqrt {a+b} \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (40 A b^4+10 a b^3 (3 A-11 B)+15 a^2 b^2 (19 A-121 B+33 C)+3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3465 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 a^2 d}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 a d}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 (5 A b+11 a B) (a+b \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+b \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(5243\) vs. \(2(705)=1410\).
Time = 28.78 (sec) , antiderivative size = 5243, normalized size of antiderivative = 7.44 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Result too large to show} \]
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hanged
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\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
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\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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