Integrand size = 25, antiderivative size = 242 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx=\frac {10 a \left (11 a^2+6 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e} \]
-2/77*a*(11*a^2+6*b^2)*e*cos(d*x+c)*(e*sin(d*x+c))^(5/2)/d+2/1287*b*(177*a ^2+44*b^2)*(e*sin(d*x+c))^(9/2)/d/e+34/143*a*b*(a+b*cos(d*x+c))*(e*sin(d*x +c))^(9/2)/d/e+2/13*b*(a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(9/2)/d/e-10/231*a *(11*a^2+6*b^2)*e^4*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1 /2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/d/(e *sin(d*x+c))^(1/2)-10/231*a*(11*a^2+6*b^2)*e^3*cos(d*x+c)*(e*sin(d*x+c))^( 1/2)/d
Time = 3.65 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.85 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx=\frac {\left (154 b \left (78 a^2+11 b^2\right ) \csc ^3(c+d x)+\frac {1}{3} \left (-156 a \left (506 a^2+213 b^2\right ) \cos (c+d x)-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))+234 a \left (44 a^2-39 b^2\right ) \cos (3 (c+d x))-154 b \left (-78 a^2+b^2\right ) \cos (4 (c+d x))+4914 a b^2 \cos (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right ) \csc ^3(c+d x)-\frac {2080 a \left (11 a^2+6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )}{\sin ^{\frac {7}{2}}(c+d x)}\right ) (e \sin (c+d x))^{7/2}}{48048 d} \]
((154*b*(78*a^2 + 11*b^2)*Csc[c + d*x]^3 + ((-156*a*(506*a^2 + 213*b^2)*Co s[c + d*x] - 77*b*(624*a^2 + 73*b^2)*Cos[2*(c + d*x)] + 234*a*(44*a^2 - 39 *b^2)*Cos[3*(c + d*x)] - 154*b*(-78*a^2 + b^2)*Cos[4*(c + d*x)] + 4914*a*b ^2*Cos[5*(c + d*x)] + 693*b^3*Cos[6*(c + d*x)])*Csc[c + d*x]^3)/3 - (2080* a*(11*a^2 + 6*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(7/2) )*(e*Sin[c + d*x])^(7/2))/(48048*d)
Time = 1.11 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.98, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3171, 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.
| \(\displaystyle \int (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{13} \int \frac {1}{2} (a+b \cos (c+d x)) \left (13 a^2+17 b \cos (c+d x) a+4 b^2\right ) (e \sin (c+d x))^{7/2}dx+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \int (a+b \cos (c+d x)) \left (13 a^2+17 b \cos (c+d x) a+4 b^2\right ) (e \sin (c+d x))^{7/2}dx+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \int \left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (13 a^2+17 b \sin \left (c+d x+\frac {\pi }{2}\right ) a+4 b^2\right )dx+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} \left (13 a \left (11 a^2+6 b^2\right )+b \left (177 a^2+44 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{7/2}dx+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \left (13 a \left (11 a^2+6 b^2\right )+b \left (177 a^2+44 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{7/2}dx+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2} \left (13 a \left (11 a^2+6 b^2\right )-b \left (177 a^2+44 b^2\right ) \sin \left (c+d x-\frac {\pi }{2}\right )\right )dx+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \int (e \sin (c+d x))^{7/2}dx+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \int (e \sin (c+d x))^{7/2}dx+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \int (e \sin (c+d x))^{3/2}dx-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \int (e \sin (c+d x))^{3/2}dx-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}\) |
(2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(9/2))/(13*d*e) + ((34*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(9/2))/(11*d*e) + ((2*b*(177*a^2 + 44* b^2)*(e*Sin[c + d*x])^(9/2))/(9*d*e) + 13*a*(11*a^2 + 6*b^2)*((-2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(7*d) + (5*e^2*((2*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (2*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d)))/7))/11)/13
3.1.49.3.1 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])
Time = 39.30 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(\frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}} \left (9 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )+39 a^{2}+4 b^{2}\right )}{117 e}-\frac {e^{4} a \left (-126 b^{2} \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )-66 a^{2} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+216 b^{2} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+55 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}+30 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}+176 a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-30 b^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )}{231 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(276\) |
| parts | \(-\frac {a^{3} e^{4} \left (-6 \left (\sin ^{5}\left (d x +c \right )\right )+5 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-4 \left (\sin ^{3}\left (d x +c \right )\right )+10 \sin \left (d x +c \right )\right )}{21 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b^{3} \left (\frac {\left (e \sin \left (d x +c \right )\right )^{\frac {13}{2}}}{13}-\frac {e^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}\right )}{d \,e^{3}}+\frac {2 a^{2} b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{3 d e}-\frac {2 a \,b^{2} e^{4} \left (21 \left (\sin ^{7}\left (d x +c \right )\right )-27 \left (\sin ^{5}\left (d x +c \right )\right )+5 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-4 \left (\sin ^{3}\left (d x +c \right )\right )+10 \sin \left (d x +c \right )\right )}{77 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(294\) |
(2/117/e*b*(e*sin(d*x+c))^(9/2)*(9*b^2*cos(d*x+c)^2+39*a^2+4*b^2)-1/231*e^ 4*a*(-126*b^2*cos(d*x+c)^6*sin(d*x+c)-66*a^2*cos(d*x+c)^4*sin(d*x+c)+216*b ^2*cos(d*x+c)^4*sin(d*x+c)+55*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)* sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2+30*(1-sin (d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d* x+c))^(1/2),1/2*2^(1/2))*b^2+176*a^2*cos(d*x+c)^2*sin(d*x+c)-30*b^2*cos(d* x+c)^2*sin(d*x+c))/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.05 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx=\frac {195 \, \sqrt {2} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} \sqrt {-i \, e} e^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 \, \sqrt {2} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} \sqrt {i \, e} e^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (693 \, b^{3} e^{3} \cos \left (d x + c\right )^{6} + 2457 \, a b^{2} e^{3} \cos \left (d x + c\right )^{5} + 77 \, {\left (39 \, a^{2} b - 14 \, b^{3}\right )} e^{3} \cos \left (d x + c\right )^{4} + 117 \, {\left (11 \, a^{3} - 36 \, a b^{2}\right )} e^{3} \cos \left (d x + c\right )^{3} - 77 \, {\left (78 \, a^{2} b - b^{3}\right )} e^{3} \cos \left (d x + c\right )^{2} - 39 \, {\left (88 \, a^{3} - 15 \, a b^{2}\right )} e^{3} \cos \left (d x + c\right ) + 77 \, {\left (39 \, a^{2} b + 4 \, b^{3}\right )} e^{3}\right )} \sqrt {e \sin \left (d x + c\right )}}{9009 \, d} \]
1/9009*(195*sqrt(2)*(11*a^3 + 6*a*b^2)*sqrt(-I*e)*e^3*weierstrassPInverse( 4, 0, cos(d*x + c) + I*sin(d*x + c)) + 195*sqrt(2)*(11*a^3 + 6*a*b^2)*sqrt (I*e)*e^3*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(69 3*b^3*e^3*cos(d*x + c)^6 + 2457*a*b^2*e^3*cos(d*x + c)^5 + 77*(39*a^2*b - 14*b^3)*e^3*cos(d*x + c)^4 + 117*(11*a^3 - 36*a*b^2)*e^3*cos(d*x + c)^3 - 77*(78*a^2*b - b^3)*e^3*cos(d*x + c)^2 - 39*(88*a^3 - 15*a*b^2)*e^3*cos(d* x + c) + 77*(39*a^2*b + 4*b^3)*e^3)*sqrt(e*sin(d*x + c)))/d
Timed out. \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \]
\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]