Integrand size = 25, antiderivative size = 305 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx=-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{6435 d e}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{385 d}-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e} \]
-34/6435*a*b*(53*a^2+38*b^2)*(e*cos(d*x+c))^(9/2)/d/e+2/385*(55*a^4+60*a^2 *b^2+4*b^4)*e*(e*cos(d*x+c))^(5/2)*sin(d*x+c)/d-2/715*b*(93*a^2+26*b^2)*(e *cos(d*x+c))^(9/2)*(a+b*sin(d*x+c))/d/e-14/65*a*b*(e*cos(d*x+c))^(9/2)*(a+ b*sin(d*x+c))^2/d/e-2/15*b*(e*cos(d*x+c))^(9/2)*(a+b*sin(d*x+c))^3/d/e+2/2 31*(55*a^4+60*a^2*b^2+4*b^4)*e^4*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+ 1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/d/(e*cos(d*x +c))^(1/2)+2/231*(55*a^4+60*a^2*b^2+4*b^4)*e^3*sin(d*x+c)*(e*cos(d*x+c))^( 1/2)/d
Time = 4.41 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.82 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx=\frac {(e \cos (c+d x))^{7/2} \left (-154 a b \left (26 a^2+11 b^2\right ) \sqrt {\cos (c+d x)}+104 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{120} \sqrt {\cos (c+d x)} \left (156 \left (5720 a^4+2460 a^2 b^2+87 b^4\right ) \sin (c+d x)+462 b^3 \cos (6 (c+d x)) (60 a+13 b \sin (c+d x))-28 b \cos (4 (c+d x)) \left (220 a \left (26 a^2-b^2\right )+39 b \left (180 a^2+b^2\right ) \sin (c+d x)\right )+\cos (2 (c+d x)) \left (-3080 \left (208 a^3 b+73 a b^3\right )+78 \left (2640 a^4-7200 a^2 b^2-557 b^4\right ) \sin (c+d x)\right )\right )\right )}{12012 d \cos ^{\frac {7}{2}}(c+d x)} \]
((e*Cos[c + d*x])^(7/2)*(-154*a*b*(26*a^2 + 11*b^2)*Sqrt[Cos[c + d*x]] + 1 04*(55*a^4 + 60*a^2*b^2 + 4*b^4)*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(156*(5720*a^4 + 2460*a^2*b^2 + 87*b^4)*Sin[c + d*x] + 462*b^3*Cos[ 6*(c + d*x)]*(60*a + 13*b*Sin[c + d*x]) - 28*b*Cos[4*(c + d*x)]*(220*a*(26 *a^2 - b^2) + 39*b*(180*a^2 + b^2)*Sin[c + d*x]) + Cos[2*(c + d*x)]*(-3080 *(208*a^3*b + 73*a*b^3) + 78*(2640*a^4 - 7200*a^2*b^2 - 557*b^4)*Sin[c + d *x])))/120))/(12012*d*Cos[c + d*x]^(7/2))
Time = 1.47 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.95, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 3171, 27, 3042, 3341, 27, 3042, 3341, 27, 3042, 3148, 3042, 3115, 3042, 3115, 3042, 3121, 3042, 3120}
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.
| \(\displaystyle \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{15} \int \frac {3}{2} (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \left (5 a^2+7 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \left (5 a^2+7 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \left (5 a^2+7 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{13} \int \frac {1}{2} (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (a \left (65 a^2+54 b^2\right )+b \left (93 a^2+26 b^2\right ) \sin (c+d x)\right )dx-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (a \left (65 a^2+54 b^2\right )+b \left (93 a^2+26 b^2\right ) \sin (c+d x)\right )dx-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (a \left (65 a^2+54 b^2\right )+b \left (93 a^2+26 b^2\right ) \sin (c+d x)\right )dx-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} (e \cos (c+d x))^{7/2} \left (13 \left (55 a^4+60 b^2 a^2+4 b^4\right )+17 a b \left (53 a^2+38 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \int (e \cos (c+d x))^{7/2} \left (13 \left (55 a^4+60 b^2 a^2+4 b^4\right )+17 a b \left (53 a^2+38 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \int (e \cos (c+d x))^{7/2} \left (13 \left (55 a^4+60 b^2 a^2+4 b^4\right )+17 a b \left (53 a^2+38 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \int (e \cos (c+d x))^{7/2}dx-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \left (\frac {5}{7} e^2 \int (e \cos (c+d x))^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \left (\frac {5}{7} e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{13} \left (\frac {1}{11} \left (13 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\) |
(-2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^3)/(15*d*e) + ((-14*a*b* (e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^2)/(13*d*e) + ((-2*b*(93*a^2 + 26*b^2)*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(11*d*e) + ((-34*a*b *(53*a^2 + 38*b^2)*(e*Cos[c + d*x])^(9/2))/(9*d*e) + 13*(55*a^4 + 60*a^2*b ^2 + 4*b^4)*((2*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (5*e^2*((2* e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x] ]) + (2*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/7))/11)/13)/5
3.6.64.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Leaf count of result is larger than twice the leaf count of optimal. \(780\) vs. \(2(301)=602\).
Time = 153.52 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.56
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(781\) |
| default | \(\text {Expression too large to display}\) | \(863\) |
-2/21*a^4*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^4*(4 8*cos(1/2*d*x+1/2*c)^9-120*cos(1/2*d*x+1/2*c)^7+128*cos(1/2*d*x+1/2*c)^5-7 2*cos(1/2*d*x+1/2*c)^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2* c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+16*cos(1/2*d*x+1/2*c)) /(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/sin(1/2*d*x+1/2* c)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d-8/1155*b^4*(e*(2*cos(1/2*d*x+1/2 *c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^4*(4928*cos(1/2*d*x+1/2*c)^17-22176 *cos(1/2*d*x+1/2*c)^15+41216*cos(1/2*d*x+1/2*c)^13-40768*cos(1/2*d*x+1/2*c )^11+22868*cos(1/2*d*x+1/2*c)^9-6994*cos(1/2*d*x+1/2*c)^7+926*cos(1/2*d*x+ 1/2*c)^5+5*cos(1/2*d*x+1/2*c)^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2 *d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-5*cos(1/2*d*x +1/2*c))/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/sin(1/2* d*x+1/2*c)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d+8*a*b^3/d/e^3*(1/13*(e*c os(d*x+c))^(13/2)-1/9*e^2*(e*cos(d*x+c))^(9/2))+8/77*a^2*b^2*(e*(2*cos(1/2 *d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^4*(672*cos(1/2*d*x+1/2*c)^1 3-2352*cos(1/2*d*x+1/2*c)^11+3312*cos(1/2*d*x+1/2*c)^9-2400*cos(1/2*d*x+1/ 2*c)^7+922*cos(1/2*d*x+1/2*c)^5-159*cos(1/2*d*x+1/2*c)^3-5*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2* c),2^(1/2))+5*cos(1/2*d*x+1/2*c))/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+ 1/2*c)^2))^(1/2)/sin(1/2*d*x+1/2*c)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.85 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx=\frac {-195 i \, \sqrt {2} {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (13860 \, a b^{3} e^{3} \cos \left (d x + c\right )^{6} - 20020 \, {\left (a^{3} b + a b^{3}\right )} e^{3} \cos \left (d x + c\right )^{4} + 39 \, {\left (77 \, b^{4} e^{3} \cos \left (d x + c\right )^{6} - 7 \, {\left (90 \, a^{2} b^{2} + 17 \, b^{4}\right )} e^{3} \cos \left (d x + c\right )^{4} + 3 \, {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{3} \cos \left (d x + c\right )^{2} + 5 \, {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{45045 \, d} \]
1/45045*(-195*I*sqrt(2)*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^(7/2)*weierstrassP Inverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 195*I*sqrt(2)*(55*a^4 + 60 *a^2*b^2 + 4*b^4)*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c)) + 2*(13860*a*b^3*e^3*cos(d*x + c)^6 - 20020*(a^3*b + a*b^3)*e^3* cos(d*x + c)^4 + 39*(77*b^4*e^3*cos(d*x + c)^6 - 7*(90*a^2*b^2 + 17*b^4)*e ^3*cos(d*x + c)^4 + 3*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^3*cos(d*x + c)^2 + 5 *(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^3)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]