Integrand size = 33, antiderivative size = 254 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=-\frac {7 (17 A-33 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {(11 A-21 B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}+\frac {(11 A-21 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {7 (17 A-33 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 a^3 d}+\frac {(A-B) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(7 A-12 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {3 (11 A-21 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )} \]
[Out]
Time = 0.58 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3056, 2827, 2715, 2720, 2719} \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\frac {(11 A-21 B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}-\frac {7 (17 A-33 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {3 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{10 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac {7 (17 A-33 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{30 a^3 d}+\frac {(11 A-21 B) \sin (c+d x) \sqrt {\cos (c+d x)}}{2 a^3 d}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {(7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]
[In]
[Out]
Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 3056
Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (\frac {9}{2} a (A-B)-\frac {5}{2} a (A-3 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2} \\ & = \frac {(A-B) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(7 A-12 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7}{2} a^2 (7 A-12 B)-\frac {5}{2} a^2 (10 A-21 B) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4} \\ & = \frac {(A-B) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(7 A-12 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {3 (11 A-21 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {45}{4} a^3 (11 A-21 B)-\frac {35}{4} a^3 (17 A-33 B) \cos (c+d x)\right ) \, dx}{15 a^6} \\ & = \frac {(A-B) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(7 A-12 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {3 (11 A-21 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(7 (17 A-33 B)) \int \cos ^{\frac {5}{2}}(c+d x) \, dx}{12 a^3}+\frac {(3 (11 A-21 B)) \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{4 a^3} \\ & = \frac {(11 A-21 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {7 (17 A-33 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 a^3 d}+\frac {(A-B) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(7 A-12 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {3 (11 A-21 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {(7 (17 A-33 B)) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}+\frac {(11 A-21 B) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3} \\ & = -\frac {7 (17 A-33 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {(11 A-21 B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}+\frac {(11 A-21 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {7 (17 A-33 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 a^3 d}+\frac {(A-B) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(7 A-12 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {3 (11 A-21 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.26 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.90 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=-\frac {\sqrt {\cos (c+d x)} \csc ^5(c+d x) \left (-396 A+396 B-680 A \cos (c+d x)+680 B \cos (c+d x)+792 A \cos ^2(c+d x)-792 B \cos ^2(c+d x)+680 A \cos ^3(c+d x)-680 B \cos ^3(c+d x)-440 A \cos ^4(c+d x)+440 B \cos ^4(c+d x)-180 A \cos ^5(c+d x)+180 B \cos ^5(c+d x)+20 A \cos ^6(c+d x)-20 B \cos ^6(c+d x)+66 A \sin ^2(c+d x)+234 B \sin ^2(c+d x)+448 B \cos (c+d x) \sin ^2(c+d x)+40 B \cos ^4(c+d x) \sin ^2(c+d x)-12 B \cos ^5(c+d x) \sin ^2(c+d x)+165 A \sin ^4(c+d x)-315 B \sin ^4(c+d x)+15 (11 A-21 B) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cos ^2(c+d x)\right ) \sin ^4(c+d x) \sqrt {\sin ^2(c+d x)}-448 B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\cos ^2(c+d x)\right ) \sin ^4(c+d x) \sqrt {\sin ^2(c+d x)}+680 A \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {7}{4},\cos ^2(c+d x)\right ) \sin ^4(c+d x) \sqrt {\sin ^2(c+d x)}-680 B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {7}{4},\cos ^2(c+d x)\right ) \sin ^4(c+d x) \sqrt {\sin ^2(c+d x)}-90 B \sin ^2(2 (c+d x))-24 B \csc (c+d x) \sin ^3(2 (c+d x))\right )}{30 a^3 d} \]
[In]
[Out]
Time = 15.76 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {\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 (192 B \left (\cos ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 A \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-864 B \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+468 A \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+330 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+714 A \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-228 B \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-630 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1386 B \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )-1058 A \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1590 B \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+474 A \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-744 B \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-47 A \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+57 B \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 A -3 B \right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \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}\) | \(493\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.95 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\frac {2 \, {\left (12 \, B \cos \left (d x + c\right )^{4} + 4 \, {\left (5 \, A - 6 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (79 \, A - 147 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (188 \, A - 357 \, B\right )} \cos \left (d x + c\right ) + 165 \, A - 315 \, B\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, {\left (\sqrt {2} {\left (11 i \, A - 21 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (11 i \, A - 21 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (11 i \, A - 21 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (11 i \, A - 21 i \, B\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, {\left (\sqrt {2} {\left (-11 i \, A + 21 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-11 i \, A + 21 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-11 i \, A + 21 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-11 i \, A + 21 i \, B\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, {\left (\sqrt {2} {\left (17 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (17 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (17 i \, A - 33 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (17 i \, A - 33 i \, B\right )}\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 \, {\left (\sqrt {2} {\left (-17 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-17 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-17 i \, A + 33 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-17 i \, A + 33 i \, B\right )}\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 )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{9/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
[In]
[Out]