Integrand size = 40, antiderivative size = 462 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d} \]
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Time = 1.15 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {3108, 3069, 3102, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (-8 a^2 C+22 a b B-81 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac {2 \left (-40 a^3 C+110 a^2 b B-335 a b^2 C-539 b^3 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac {2 \left (-40 a^4 C+110 a^3 b B-285 a^2 b^2 C-1254 a b^3 B-675 b^4 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 b^2 d}+\frac {2 \left (a^2-b^2\right ) \left (-40 a^4 C+110 a^3 b B-285 a^2 b^2 C-1254 a b^3 B-675 b^4 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-40 a^5 C+110 a^4 b B-255 a^3 b^2 C-3069 a^2 b^3 B-3705 a b^4 C-1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 3069
Rule 3102
Rule 3108
Rubi steps \begin{align*} \text {integral}& = \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x)) \, dx \\ & = \frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {2 \int (a+b \cos (c+d x))^{5/2} \left (a C+\frac {9}{2} b C \cos (c+d x)+\frac {1}{2} (11 b B-4 a C) \cos ^2(c+d x)\right ) \, dx}{11 b} \\ & = \frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {4 \int (a+b \cos (c+d x))^{5/2} \left (\frac {1}{4} b (77 b B-10 a C)-\frac {1}{4} \left (22 a b B-8 a^2 C-81 b^2 C\right ) \cos (c+d x)\right ) \, dx}{99 b^2} \\ & = -\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {8 \int (a+b \cos (c+d x))^{3/2} \left (\frac {3}{8} b \left (143 a b B-10 a^2 C+135 b^2 C\right )-\frac {1}{8} \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) \cos (c+d x)\right ) \, dx}{693 b^2} \\ & = -\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {16 \int \sqrt {a+b \cos (c+d x)} \left (\frac {3}{16} b \left (605 a^2 b B+539 b^3 B-10 a^3 C+1010 a b^2 C\right )-\frac {3}{16} \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \cos (c+d x)\right ) \, dx}{3465 b^2} \\ & = -\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {32 \int \frac {\frac {3}{32} b \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+3315 a^2 b^2 C+675 b^4 C\right )-\frac {3}{32} \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{10395 b^2} \\ & = -\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac {\left (\left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3465 b^3}-\frac {\left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3465 b^3} \\ & = -\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}-\frac {\left (\left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{3465 b^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (\left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{3465 b^3 \sqrt {a+b \cos (c+d x)}} \\ & = -\frac {2 \left (110 a^4 b B-3069 a^2 b^3 B-1617 b^5 B-40 a^5 C-255 a^3 b^2 C-3705 a b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (110 a^3 b B-1254 a b^3 B-40 a^4 C-285 a^2 b^2 C-675 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 b B-539 b^3 B-40 a^3 C-335 a b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a b B-8 a^2 C-81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 b B-4 a C) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d} \\ \end{align*}
Time = 3.52 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.77 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (1705 a^3 b B+2871 a b^3 B+10 a^4 C+3315 a^2 b^2 C+675 b^4 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-110 a^4 b B+3069 a^2 b^3 B+1617 b^5 B+40 a^5 C+255 a^3 b^2 C+3705 a b^4 C\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (880 a^3 b B+32868 a b^3 B-320 a^4 C+18660 a^2 b^2 C+13050 b^4 C\right ) \sin (c+d x)+b \left (4 \left (1650 a^2 b B+1463 b^3 B+30 a^3 C+3095 a b^2 C\right ) \sin (2 (c+d x))+5 b \left (\left (836 a b B+452 a^2 C+513 b^2 C\right ) \sin (3 (c+d x))+7 b ((22 b B+46 a C) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )\right )\right )}{27720 b^3 d \sqrt {a+b \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1982\) vs. \(2(488)=976\).
Time = 29.41 (sec) , antiderivative size = 1983, normalized size of antiderivative = 4.29
method | result | size |
default | \(\text {Expression too large to display}\) | \(1983\) |
parts | \(\text {Expression too large to display}\) | \(2137\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.57 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (80 i \, C a^{6} - 220 i \, B a^{5} b + 480 i \, C a^{4} b^{2} + 1023 i \, B a^{3} b^{3} - 2535 i \, C a^{2} b^{4} - 5379 i \, B a b^{5} - 2025 i \, C b^{6}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (-80 i \, C a^{6} + 220 i \, B a^{5} b - 480 i \, C a^{4} b^{2} - 1023 i \, B a^{3} b^{3} + 2535 i \, C a^{2} b^{4} + 5379 i \, B a b^{5} + 2025 i \, C b^{6}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-40 i \, C a^{5} b + 110 i \, B a^{4} b^{2} - 255 i \, C a^{3} b^{3} - 3069 i \, B a^{2} b^{4} - 3705 i \, C a b^{5} - 1617 i \, B b^{6}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (40 i \, C a^{5} b - 110 i \, B a^{4} b^{2} + 255 i \, C a^{3} b^{3} + 3069 i \, B a^{2} b^{4} + 3705 i \, C a b^{5} + 1617 i \, B b^{6}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (315 \, C b^{6} \cos \left (d x + c\right )^{4} - 20 \, C a^{4} b^{2} + 55 \, B a^{3} b^{3} + 1025 \, C a^{2} b^{4} + 1793 \, B a b^{5} + 675 \, C b^{6} + 35 \, {\left (23 \, C a b^{5} + 11 \, B b^{6}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (113 \, C a^{2} b^{4} + 209 \, B a b^{5} + 81 \, C b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (15 \, C a^{3} b^{3} + 825 \, B a^{2} b^{4} + 1145 \, C a b^{5} + 539 \, B b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{10395 \, b^{4} d} \]
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Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
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\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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