Integrand size = 37, antiderivative size = 313 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)} \] Output:
2/9009*a^3*(2224*A+2717*C)*sin(d*x+c)/d/cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^ (1/2)+2/15015*a^3*(8368*A+10439*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*cos( d*x+c))^(1/2)+8/45045*a^3*(8368*A+10439*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)/( a+a*cos(d*x+c))^(1/2)+16/45045*a^3*(8368*A+10439*C)*sin(d*x+c)/d/cos(d*x+c )^(1/2)/(a+a*cos(d*x+c))^(1/2)+2/1287*a^2*(136*A+143*C)*(a+a*cos(d*x+c))^( 1/2)*sin(d*x+c)/d/cos(d*x+c)^(9/2)+10/143*a*A*(a+a*cos(d*x+c))^(3/2)*sin(d *x+c)/d/cos(d*x+c)^(11/2)+2/13*A*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d/cos(d *x+c)^(13/2)
Time = 0.56 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.55 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (343612 A+322751 C+1120 (347 A+286 C) \cos (c+d x)+14 (30334 A+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+141570 C \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+156585 C \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+20878 C \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+20878 C \cos (6 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{180180 d \cos ^{\frac {13}{2}}(c+d x)} \] Input:
Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x] ^(15/2),x]
Output:
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(343612*A + 322751*C + 1120*(347*A + 286*C )*Cos[c + d*x] + 14*(30334*A + 32747*C)*Cos[2*(c + d*x)] + 125520*A*Cos[3* (c + d*x)] + 141570*C*Cos[3*(c + d*x)] + 125520*A*Cos[4*(c + d*x)] + 15658 5*C*Cos[4*(c + d*x)] + 16736*A*Cos[5*(c + d*x)] + 20878*C*Cos[5*(c + d*x)] + 16736*A*Cos[6*(c + d*x)] + 20878*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/ (180180*d*Cos[c + d*x]^(13/2))
Time = 1.87 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.459, Rules used = {3042, 3523, 27, 3042, 3454, 27, 3042, 3454, 27, 3042, 3459, 3042, 3251, 3042, 3251, 3042, 3250}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{15/2}}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^{5/2} (5 a A+a (6 A+13 C) \cos (c+d x))}{2 \cos ^{\frac {13}{2}}(c+d x)}dx}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^{5/2} (5 a A+a (6 A+13 C) \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)}dx}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (5 a A+a (6 A+13 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{11} \int \frac {(\cos (c+d x) a+a)^{3/2} \left ((136 A+143 C) a^2+(96 A+143 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {11}{2}}(c+d x)}dx+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \int \frac {(\cos (c+d x) a+a)^{3/2} \left ((136 A+143 C) a^2+(96 A+143 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left ((136 A+143 C) a^2+(96 A+143 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((2224 A+2717 C) a^3+15 (112 A+143 C) \cos (c+d x) a^3\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((2224 A+2717 C) a^3+15 (112 A+143 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((2224 A+2717 C) a^3+15 (112 A+143 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+10439 C) \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a^4 (2224 A+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+10439 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a^4 (2224 A+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+10439 C) \left (\frac {4}{5} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (2224 A+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+10439 C) \left (\frac {4}{5} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (2224 A+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+10439 C) \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (2224 A+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {1}{9} \left (\frac {3}{7} a^3 (8368 A+10439 C) \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (2224 A+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}\right )+\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3250 |
| \(\displaystyle \frac {\frac {10 a^2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {1}{11} \left (\frac {2 a^3 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (\frac {2 a^4 (2224 A+2717 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {3}{7} a^3 (8368 A+10439 C) \left (\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4}{5} \left (\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )\right )\right )\right )}{13 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)}\) |
Input:
Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2 ),x]
Output:
(2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2)) + ((10*a^2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(1 1/2)) + ((2*a^3*(136*A + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9* d*Cos[c + d*x]^(9/2)) + ((2*a^4*(2224*A + 2717*C)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (3*a^3*(8368*A + 10439*C)*((2*a* Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*((2*a *Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*Si n[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])))/5))/7)/9)/ 11)/(13*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq rt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) *(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)) Int[Sqrt[a + b*Sin[e + f*x] ]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Time = 3.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.50
| method | result | size |
| default | \(\frac {2 a^{2} \sin \left (d x +c \right ) \left (\left (66944 \cos \left (d x +c \right )^{6}+33472 \cos \left (d x +c \right )^{5}+25104 \cos \left (d x +c \right )^{4}+20920 \cos \left (d x +c \right )^{3}+18305 \cos \left (d x +c \right )^{2}+11970 \cos \left (d x +c \right )+3465\right ) A +\cos \left (d x +c \right )^{2} \left (83512 \cos \left (d x +c \right )^{4}+41756 \cos \left (d x +c \right )^{3}+31317 \cos \left (d x +c \right )^{2}+18590 \cos \left (d x +c \right )+5005\right ) C \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{45045 d \cos \left (d x +c \right )^{\frac {13}{2}} \left (1+\cos \left (d x +c \right )\right )}\) | \(158\) |
| parts | \(\frac {2 A \sqrt {2}\, a^{2} \sin \left (d x +c \right ) \left (66944 \cos \left (d x +c \right )^{6}+33472 \cos \left (d x +c \right )^{5}+25104 \cos \left (d x +c \right )^{4}+20920 \cos \left (d x +c \right )^{3}+18305 \cos \left (d x +c \right )^{2}+11970 \cos \left (d x +c \right )+3465\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{45045 d \cos \left (d x +c \right )^{\frac {13}{2}} \left (1+\cos \left (d x +c \right )\right )}+\frac {2 C \,a^{2} \sin \left (d x +c \right ) \left (584 \cos \left (d x +c \right )^{4}+292 \cos \left (d x +c \right )^{3}+219 \cos \left (d x +c \right )^{2}+130 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{315 d \cos \left (d x +c \right )^{\frac {9}{2}} \left (1+\cos \left (d x +c \right )\right )}\) | \(198\) |
Input:
int((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x,method=_ RETURNVERBOSE)
Output:
2/45045/d*a^2*sin(d*x+c)*((66944*cos(d*x+c)^6+33472*cos(d*x+c)^5+25104*cos (d*x+c)^4+20920*cos(d*x+c)^3+18305*cos(d*x+c)^2+11970*cos(d*x+c)+3465)*A+c os(d*x+c)^2*(83512*cos(d*x+c)^4+41756*cos(d*x+c)^3+31317*cos(d*x+c)^2+1859 0*cos(d*x+c)+5005)*C)*(a*(1+cos(d*x+c)))^(1/2)/cos(d*x+c)^(13/2)/(1+cos(d* x+c))
Time = 0.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.54 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {2 \, {\left (8 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 4 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 3 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (2092 \, A + 1859 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (523 \, A + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 11970 \, A a^{2} \cos \left (d x + c\right ) + 3465 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{8} + d \cos \left (d x + c\right )^{7}\right )}} \] Input:
integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x, a lgorithm="fricas")
Output:
2/45045*(8*(8368*A + 10439*C)*a^2*cos(d*x + c)^6 + 4*(8368*A + 10439*C)*a^ 2*cos(d*x + c)^5 + 3*(8368*A + 10439*C)*a^2*cos(d*x + c)^4 + 10*(2092*A + 1859*C)*a^2*cos(d*x + c)^3 + 35*(523*A + 143*C)*a^2*cos(d*x + c)^2 + 11970 *A*a^2*cos(d*x + c) + 3465*A*a^2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*sin(d*x + c)/(d*cos(d*x + c)^8 + d*cos(d*x + c)^7)
Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**(5/2)*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(15/2),x )
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (271) = 542\).
Time = 0.22 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.14 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x, a lgorithm="maxima")
Output:
8/45045*(143*(315*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 945*sq rt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1449*sqrt(2)*a^(5/2)*s in(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1287*sqrt(2)*a^(5/2)*sin(d*x + c)^7/( cos(d*x + c) + 1)^7 + 572*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1 )^9 - 104*sqrt(2)*a^(5/2)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)*C*(sin(d* x + c)^2/(cos(d*x + c) + 1)^2 + 1)^3/((sin(d*x + c)/(cos(d*x + c) + 1) + 1 )^(11/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(3*sin(d*x + c)^2/( cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + sin(d*x + c) ^6/(cos(d*x + c) + 1)^6 + 1)) + (45045*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d *x + c) + 1) - 165165*sqrt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 414414*sqrt(2)*a^(5/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 604890*sqrt (2)*a^(5/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 522665*sqrt(2)*a^(5/2)*s in(d*x + c)^9/(cos(d*x + c) + 1)^9 - 289185*sqrt(2)*a^(5/2)*sin(d*x + c)^1 1/(cos(d*x + c) + 1)^11 + 88980*sqrt(2)*a^(5/2)*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 11864*sqrt(2)*a^(5/2)*sin(d*x + c)^15/(cos(d*x + c) + 1)^15) *A*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5/((sin(d*x + c)/(cos(d*x + c ) + 1) + 1)^(15/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(15/2)*(5*sin(d* x + c)^2/(cos(d*x + c) + 1)^2 + 10*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1 0*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*sin(d*x + c)^8/(cos(d*x + c) + 1 )^8 + sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1)))/d
Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x, a lgorithm="giac")
Output:
Timed out
Time = 4.14 (sec) , antiderivative size = 911, normalized size of antiderivative = 2.91 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Too large to display} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2))/cos(c + d*x)^(15/2 ),x)
Output:
((a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((a^2*(8368 *A + 10439*C)*16i)/(45045*d) - (a^2*exp(c*5i + d*x*5i)*(6*A + 23*C)*16i)/( 15*d) + (a^2*exp(c*8i + d*x*8i)*(6*A + 23*C)*16i)/(15*d) + (a^2*exp(c*6i + d*x*6i)*(348*A + 379*C)*16i)/(105*d) - (a^2*exp(c*7i + d*x*7i)*(348*A + 3 79*C)*16i)/(105*d) + (a^2*exp(c*4i + d*x*4i)*(523*A + 554*C)*32i)/(315*d) - (a^2*exp(c*9i + d*x*9i)*(523*A + 554*C)*32i)/(315*d) + (a^2*exp(c*2i + d *x*2i)*(8368*A + 10439*C)*8i)/(3465*d) - (a^2*exp(c*11i + d*x*11i)*(8368*A + 10439*C)*8i)/(3465*d) - (a^2*exp(c*13i + d*x*13i)*(8368*A + 10439*C)*16 i)/(45045*d) - (C*a^2*exp(c*3i + d*x*3i)*8i)/(3*d) + (C*a^2*exp(c*10i + d* x*10i)*8i)/(3*d)))/((exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + exp(c*1i + d*x*1i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 6*exp(c*2i + d*x*2i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1 /2) + 6*exp(c*3i + d*x*3i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2) ^(1/2) + 15*exp(c*4i + d*x*4i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i )/2)^(1/2) + 15*exp(c*5i + d*x*5i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d* x*1i)/2)^(1/2) + 20*exp(c*6i + d*x*6i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 20*exp(c*7i + d*x*7i)*(exp(- c*1i - d*x*1i)/2 + exp(c *1i + d*x*1i)/2)^(1/2) + 15*exp(c*8i + d*x*8i)*(exp(- c*1i - d*x*1i)/2 + e xp(c*1i + d*x*1i)/2)^(1/2) + 15*exp(c*9i + d*x*9i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 6*exp(c*10i + d*x*10i)*(exp(- c*1i - d...
\[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\sqrt {a}\, a^{2} \left (\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{8}}d x \right ) a +2 \left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{7}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) c +2 \left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) c \right ) \] Input:
int((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x)
Output:
sqrt(a)*a**2*(int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x) **8,x)*a + 2*int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)* *7,x)*a + int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**6, x)*a + int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**6,x)* c + 2*int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**5,x)*c + int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**4,x)*c)