Optimal. Leaf size=622 \[ \frac {2 \left (81 a^2 B+209 a b C+113 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (539 a^3 C+1145 a^2 b B+825 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 B+1793 a^3 b C+1025 a^2 b^2 B+55 a b^3 C-20 b^4 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (3 a^4 (225 B-539 C)-6 a^3 b (505 B-209 C)+15 a^2 b^2 (19 B-121 C)+10 a b^3 (3 B-11 C)+40 b^4 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (1617 a^5 C+3705 a^4 b B+3069 a^3 b^2 C+255 a^2 b^3 B-110 a b^4 C+40 b^5 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^4 d}+\frac {2 a (11 a C+14 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
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Rubi [A] time = 2.75, antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {3029, 2989, 3047, 3055, 2998, 2816, 2994} \[ \frac {2 \left (1025 a^2 b^2 B+1793 a^3 b C+675 a^4 B+55 a b^3 C-20 b^4 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+539 a^3 C+825 a b^2 C+15 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+209 a b C+113 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (15 a^2 b^2 (19 B-121 C)-6 a^3 b (505 B-209 C)+3 a^4 (225 B-539 C)+10 a b^3 (3 B-11 C)+40 b^4 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (255 a^2 b^3 B+3069 a^3 b^2 C+3705 a^4 b B+1617 a^5 C-110 a b^4 C+40 b^5 B\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^4 d}+\frac {2 a (11 a C+14 b B) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2816
Rule 2989
Rule 2994
Rule 2998
Rule 3029
Rule 3047
Rule 3055
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^{5/2} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx &=\int \frac {(a+b \cos (c+d x))^{5/2} (B+C \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\\ &=\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{2} a (14 b B+11 a C)+\frac {1}{2} \left (9 a^2 B+11 b^2 B+22 a b C\right ) \cos (c+d x)+\frac {1}{2} b (6 a B+11 b C) \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4}{99} \int \frac {\frac {1}{4} a \left (81 a^2 B+113 b^2 B+209 a b C\right )+\frac {1}{4} \left (233 a^2 b B+99 b^3 B+77 a^3 C+297 a b^2 C\right ) \cos (c+d x)+\frac {3}{4} b \left (46 a b B+22 a^2 C+33 b^2 C\right ) \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {8 \int \frac {\frac {1}{8} a \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right )+\frac {1}{8} a \left (405 a^3 B+1531 a b^2 B+1507 a^2 b C+693 b^3 C\right ) \cos (c+d x)+\frac {1}{2} a b \left (81 a^2 B+113 b^2 B+209 a b C\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{693 a}\\ &=\frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {16 \int \frac {\frac {3}{16} a \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right )+\frac {1}{16} a^2 \left (5055 a^2 b B+2305 b^3 B+1617 a^3 C+6655 a b^2 C\right ) \cos (c+d x)+\frac {1}{8} a b \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2}\\ &=\frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {32 \int \frac {\frac {3}{32} a \left (3705 a^4 b B+255 a^2 b^3 B+40 b^5 B+1617 a^5 C+3069 a^3 b^2 C-110 a b^4 C\right )+\frac {3}{32} a^2 \left (675 a^4 B+3315 a^2 b^2 B+10 b^4 B+2871 a^3 b C+1705 a b^3 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{10395 a^3}\\ &=\frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {\left ((a-b) \left (40 b^4 B+3 a^4 (225 B-539 C)-6 a^3 b (505 B-209 C)+15 a^2 b^2 (19 B-121 C)+10 a b^3 (3 B-11 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2}+\frac {\left (3705 a^4 b B+255 a^2 b^3 B+40 b^5 B+1617 a^5 C+3069 a^3 b^2 C-110 a b^4 C\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 {2 (a-b) \sqrt {a+b} \left (3705 a^4 b B+255 a^2 b^3 B+40 b^5 B+1617 a^5 C+3069 a^3 b^2 C-110 a b^4 C\right ) \cot (c+d x) E\left (\sin ^{-1}\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}+\frac {2 (a-b) \sqrt {a+b} \left (40 b^4 B+3 a^4 (225 B-539 C)-6 a^3 b (505 B-209 C)+15 a^2 b^2 (19 B-121 C)+10 a b^3 (3 B-11 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\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}+\frac {2 a (14 b B+11 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 B+113 b^2 B+209 a b C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 b B+15 b^3 B+539 a^3 C+825 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 B+1025 a^2 b^2 B-20 b^4 B+1793 a^3 b C+55 a b^3 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a B (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] time = 6.88, size = 1640, normalized size = 2.64 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C b^{2} \cos \left (d x + c\right )^{3} + B a^{2} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {13}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.02, size = 5373, normalized size = 8.64 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {15}{2}}}\,{d x} \]
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 \[ \int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{15/2}} \,d x \]
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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