Integrand size = 33, antiderivative size = 205 \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b^2 (7 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d} \]
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Time = 0.50 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3069, 3112, 3102, 2827, 2720, 2719} \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{35 d}+\frac {2 b B \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d} \]
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Rule 2719
Rule 2720
Rule 2827
Rule 3069
Rule 3102
Rule 3112
Rubi steps \begin{align*} \text {integral}& = \frac {2 b B \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{2} a (7 a A+b B)+\frac {1}{2} \left (5 b^2 B+7 a (2 A b+a B)\right ) \cos (c+d x)+\frac {1}{2} b (7 A b+11 a B) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 b^2 (7 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {4}{35} \int \frac {\frac {5}{4} a^2 (7 a A+b B)+\frac {7}{4} \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \cos (c+d x)+\frac {5}{4} b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b^2 (7 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {8}{105} \int \frac {\frac {5}{8} \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right )+\frac {21}{8} \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b^2 (7 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{5} \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b^2 (7 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d} \\ \end{align*}
Time = 2.38 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {42 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+b \sqrt {\cos (c+d x)} \left (42 b (A b+3 a B) \cos (c+d x)+5 \left (42 a A b+42 a^2 B+13 b^2 B+3 b^2 B \cos (2 (c+d x))\right )\right ) \sin (c+d x)}{105 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(663\) vs. \(2(241)=482\).
Time = 13.34 (sec) , antiderivative size = 664, normalized size of antiderivative = 3.24
method | result | size |
default | \(-\frac {2 \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 )}\, \left (240 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+\left (-168 A \,b^{3}-504 B a \,b^{2}-360 B \,b^{3}\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 (420 A a \,b^{2}+168 A \,b^{3}+420 B \,a^{2} b +504 B a \,b^{2}+280 B \,b^{3}\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 (-210 A a \,b^{2}-42 A \,b^{3}-210 B \,a^{2} b -126 B a \,b^{2}-80 B \,b^{3}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+105 A \,a^{3} \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 )+105 A a \,b^{2} \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 )-315 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 ) a^{2} b -63 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 ) b^{3}+105 B \,a^{2} 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 )+25 B \,b^{3} \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 )-105 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 ) a^{3}-189 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 ) a \,b^{2}\right )}{105 \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}\) | \(664\) |
parts | \(\text {Expression too large to display}\) | \(777\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \, {\left (15 \, B b^{3} \cos \left (d x + c\right )^{2} + 105 \, B a^{2} b + 105 \, A a b^{2} + 25 \, B b^{3} + 21 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, \sqrt {2} {\left (21 i \, A a^{3} + 21 i \, B a^{2} b + 21 i \, A a b^{2} + 5 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 \, \sqrt {2} {\left (-21 i \, A a^{3} - 21 i \, B a^{2} b - 21 i \, A a b^{2} - 5 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-5 i \, B a^{3} - 15 i \, A a^{2} b - 9 i \, B a b^{2} - 3 i \, A b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (5 i \, B a^{3} + 15 i \, A a^{2} b + 9 i \, B a b^{2} + 3 i \, A b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{105 \, d} \]
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Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
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Time = 1.59 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx=\frac {2\,\left (B\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+B\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+B\,a^2\,b\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {2\,A\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,A\,a^2\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {3\,A\,a\,b^2\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,b^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,B\,a\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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