Integrand size = 33, antiderivative size = 204 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {4 a^3 (21 A+17 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (13 A+11 B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d} \]
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Time = 0.67 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3055, 3047, 3102, 2827, 2719, 2715, 2720} \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {4 a^3 (13 A+11 B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (21 A+17 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (24 A+23 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {2 (9 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{63 d}+\frac {4 a^3 (13 A+11 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 a B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3055
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {2 a B \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2}{9} \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \left (\frac {3}{2} a (3 A+B)+\frac {1}{2} a (9 A+13 B) \cos (c+d x)\right ) \, dx \\ & = \frac {2 a B \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {4}{63} \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x)) \left (\frac {15}{2} a^2 (3 A+2 B)+\frac {3}{2} a^2 (24 A+23 B) \cos (c+d x)\right ) \, dx \\ & = \frac {2 a B \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {4}{63} \int \sqrt {\cos (c+d x)} \left (\frac {15}{2} a^3 (3 A+2 B)+\left (\frac {15}{2} a^3 (3 A+2 B)+\frac {3}{2} a^3 (24 A+23 B)\right ) \cos (c+d x)+\frac {3}{2} a^3 (24 A+23 B) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {4 a^3 (24 A+23 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {8}{315} \int \sqrt {\cos (c+d x)} \left (\frac {21}{4} a^3 (21 A+17 B)+\frac {45}{4} a^3 (13 A+11 B) \cos (c+d x)\right ) \, dx \\ & = \frac {4 a^3 (24 A+23 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{7} \left (2 a^3 (13 A+11 B)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {4 a^3 (21 A+17 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{21} \left (2 a^3 (13 A+11 B)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {4 a^3 (21 A+17 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (13 A+11 B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.62 (sec) , antiderivative size = 944, normalized size of antiderivative = 4.63 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-\frac {(21 A+17 B) \cot (c)}{30 d}+\frac {(107 A+97 B) \cos (d x) \sin (c)}{336 d}+\frac {(54 A+73 B) \cos (2 d x) \sin (2 c)}{720 d}+\frac {(A+3 B) \cos (3 d x) \sin (3 c)}{112 d}+\frac {B \cos (4 d x) \sin (4 c)}{288 d}+\frac {(107 A+97 B) \cos (c) \sin (d x)}{336 d}+\frac {(54 A+73 B) \cos (2 c) \sin (2 d x)}{720 d}+\frac {(A+3 B) \cos (3 c) \sin (3 d x)}{112 d}+\frac {B \cos (4 c) \sin (4 d x)}{288 d}\right )-\frac {13 A (a+a \cos (c+d x))^3 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{42 d \sqrt {1+\cot ^2(c)}}-\frac {11 B (a+a \cos (c+d x))^3 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{42 d \sqrt {1+\cot ^2(c)}}-\frac {7 A (a+a \cos (c+d x))^3 \csc (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{20 d}-\frac {17 B (a+a \cos (c+d x))^3 \csc (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{60 d} \]
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Time = 13.12 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.02
method | result | size |
default | \(-\frac {4 \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 )}\, a^{3} \left (-560 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (360 A +2200 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 (-1296 A -3412 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 (1806 A +2702 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 (-624 A -738 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 )+195 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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-441 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}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+165 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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-357 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}\, E\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}\) | \(413\) |
parts | \(\text {Expression too large to display}\) | \(967\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.09 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (13 \, A + 11 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (13 \, A + 11 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (21 \, A + 17 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (21 \, A + 17 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, B a^{3} \cos \left (d x + c\right )^{3} + 45 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 7 \, {\left (27 \, A + 34 \, B\right )} a^{3} \cos \left (d x + c\right ) + 30 \, {\left (13 \, A + 11 \, B\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d} \]
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Timed out. \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\text {Timed out} \]
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\[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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Time = 1.09 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.58 \[ \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {2\,\left (A\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {B\,a^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {6\,A\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,B\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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