Optimal. Leaf size=462 \[ -\frac {2 \left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d} \]
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Rubi [A]
time = 0.62, antiderivative size = 462, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3069, 3102,
2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {2 \left (-8 a^2 B+22 a A b-81 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac {2 \left (-40 a^3 B+110 a^2 A b-335 a b^2 B-539 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac {2 \left (-40 a^4 B+110 a^3 A b-285 a^2 b^2 B-1254 a A b^3-675 b^4 B\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 B+110 a^3 A b-285 a^2 b^2 B-1254 a A b^3-675 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\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 B+110 a^4 A b-255 a^3 b^2 B-3069 a^2 A b^3-3705 a b^4 B-1617 A b^5\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 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 3069
Rule 3102
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx &=\frac {2 B \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 B+\frac {9}{2} b B \cos (c+d x)+\frac {1}{2} (11 A b-4 a B) \cos ^2(c+d x)\right ) \, dx}{11 b}\\ &=\frac {2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 B \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 A b-10 a B)-\frac {1}{4} \left (22 a A b-8 a^2 B-81 b^2 B\right ) \cos (c+d x)\right ) \, dx}{99 b^2}\\ &=-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 B \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 A b-10 a^2 B+135 b^2 B\right )-\frac {1}{8} \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) \cos (c+d x)\right ) \, dx}{693 b^2}\\ &=-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 B \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 A b+539 A b^3-10 a^3 B+1010 a b^2 B\right )-\frac {3}{16} \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \cos (c+d x)\right ) \, dx}{3465 b^2}\\ &=-\frac {2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 B \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 A b+2871 a A b^3+10 a^4 B+3315 a^2 b^2 B+675 b^4 B\right )-\frac {3}{32} \left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{10395 b^2}\\ &=-\frac {2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 B \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 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3465 b^3}-\frac {\left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3465 b^3}\\ &=-\frac {2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 B \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 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\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 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\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 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 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 \left (a^2-b^2\right ) \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\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 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac {2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac {2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}\\ \end {align*}
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Mathematica [A]
time = 2.22, size = 357, normalized size = 0.77 \begin {gather*} \frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (1705 a^3 A b+2871 a A b^3+10 a^4 B+3315 a^2 b^2 B+675 b^4 B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+\left (-110 a^4 A b+3069 a^2 A b^3+1617 A b^5+40 a^5 B+255 a^3 b^2 B+3705 a b^4 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a F\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 A b+32868 a A b^3-320 a^4 B+18660 a^2 b^2 B+13050 b^4 B\right ) \sin (c+d x)+b \left (4 \left (1650 a^2 A b+1463 A b^3+30 a^3 B+3095 a b^2 B\right ) \sin (2 (c+d x))+5 b \left (\left (836 a A b+452 a^2 B+513 b^2 B\right ) \sin (3 (c+d x))+7 b ((22 A b+46 a B) \sin (4 (c+d x))+9 b B \sin (5 (c+d x)))\right )\right )\right )}{27720 b^3 d \sqrt {a+b \cos (c+d x)}} \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. \(1982\) vs.
\(2(488)=976\).
time = 0.48, size = 1983, normalized size = 4.29
method | result | size |
default | \(\text {Expression too large to display}\) | \(1983\) |
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 = 0.20, size = 726, normalized size = 1.57 \begin {gather*} \frac {\sqrt {2} {\left (80 i \, B a^{6} - 220 i \, A a^{5} b + 480 i \, B a^{4} b^{2} + 1023 i \, A a^{3} b^{3} - 2535 i \, B a^{2} b^{4} - 5379 i \, A a b^{5} - 2025 i \, B 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 \, B a^{6} + 220 i \, A a^{5} b - 480 i \, B a^{4} b^{2} - 1023 i \, A a^{3} b^{3} + 2535 i \, B a^{2} b^{4} + 5379 i \, A a b^{5} + 2025 i \, B 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 \, B a^{5} b + 110 i \, A a^{4} b^{2} - 255 i \, B a^{3} b^{3} - 3069 i \, A a^{2} b^{4} - 3705 i \, B a b^{5} - 1617 i \, A 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 \, B a^{5} b - 110 i \, A a^{4} b^{2} + 255 i \, B a^{3} b^{3} + 3069 i \, A a^{2} b^{4} + 3705 i \, B a b^{5} + 1617 i \, A 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 \, B b^{6} \cos \left (d x + c\right )^{4} - 20 \, B a^{4} b^{2} + 55 \, A a^{3} b^{3} + 1025 \, B a^{2} b^{4} + 1793 \, A a b^{5} + 675 \, B b^{6} + 35 \, {\left (23 \, B a b^{5} + 11 \, A b^{6}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (113 \, B a^{2} b^{4} + 209 \, A a b^{5} + 81 \, B b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (15 \, B a^{3} b^{3} + 825 \, A a^{2} b^{4} + 1145 \, B a b^{5} + 539 \, A 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} \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 {\cos \left (c+d\,x\right )}^2\,\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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