Integrand size = 25, antiderivative size = 210 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx=-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}+\frac {2 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e} \]
-22/315*a*b*(17*a^2+18*b^2)*(e*cos(d*x+c))^(3/2)/d/e-2/105*b*(41*a^2+14*b^ 2)*(e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))/d/e-10/21*a*b*(e*cos(d*x+c))^(3/2 )*(a+b*sin(d*x+c))^2/d/e-2/9*b*(e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^3/d/e +2/15*(15*a^4+36*a^2*b^2+4*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1 /2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/cos(d*x +c)^(1/2)
Time = 1.67 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.65 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx=\frac {\sqrt {e \cos (c+d x)} \left (84 \left (15 a^4+36 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )-b \cos ^{\frac {3}{2}}(c+d x) \left (-360 a b^2 \cos (2 (c+d x))+21 b \left (72 a^2+13 b^2\right ) \sin (c+d x)+5 \left (336 a^3+264 a b^2-7 b^3 \sin (3 (c+d x))\right )\right )\right )}{630 d \sqrt {\cos (c+d x)}} \]
(Sqrt[e*Cos[c + d*x]]*(84*(15*a^4 + 36*a^2*b^2 + 4*b^4)*EllipticE[(c + d*x )/2, 2] - b*Cos[c + d*x]^(3/2)*(-360*a*b^2*Cos[2*(c + d*x)] + 21*b*(72*a^2 + 13*b^2)*Sin[c + d*x] + 5*(336*a^3 + 264*a*b^2 - 7*b^3*Sin[3*(c + d*x)]) )))/(630*d*Sqrt[Cos[c + d*x]])
Time = 1.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3171, 27, 3042, 3341, 27, 3042, 3341, 27, 3042, 3148, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{9} \int \frac {3}{2} \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2 \left (3 a^2+5 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2 \left (3 a^2+5 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2 \left (3 a^2+5 b \sin (c+d x) a+2 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (a \left (21 a^2+34 b^2\right )+b \left (41 a^2+14 b^2\right ) \sin (c+d x)\right )dx-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (a \left (21 a^2+34 b^2\right )+b \left (41 a^2+14 b^2\right ) \sin (c+d x)\right )dx-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (a \left (21 a^2+34 b^2\right )+b \left (41 a^2+14 b^2\right ) \sin (c+d x)\right )dx-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {e \cos (c+d x)} \left (7 \left (15 a^4+36 b^2 a^2+4 b^4\right )+11 a b \left (17 a^2+18 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \sqrt {e \cos (c+d x)} \left (7 \left (15 a^4+36 b^2 a^2+4 b^4\right )+11 a b \left (17 a^2+18 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \sqrt {e \cos (c+d x)} \left (7 \left (15 a^4+36 b^2 a^2+4 b^4\right )+11 a b \left (17 a^2+18 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (7 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \int \sqrt {e \cos (c+d x)}dx-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (7 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {7 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)}}-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {7 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)}}-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {14 \left (15 a^4+36 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\) |
(-2*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3)/(9*d*e) + ((-10*a*b*( e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(7*d*e) + (((-22*a*b*(17*a^2 + 18*b^2)*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (14*(15*a^4 + 36*a^2*b^2 + 4* b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]] ))/5 - (2*b*(41*a^2 + 14*b^2)*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])) /(5*d*e))/7)/3
3.6.67.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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. \(524\) vs. \(2(214)=428\).
Time = 9.96 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.50
| method | result | size |
| default | \(\frac {2 e \left (1120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-2240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+2880 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-3024 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+1064 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-5760 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+3024 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+56 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-1680 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b +2640 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-756 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}-84 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+315 \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^{4}+756 \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^{2}+84 \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^{4}+1680 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b +240 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-420 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b -240 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{3}\right )}{315 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(525\) |
| parts | \(\frac {2 a^{4} \sqrt {e \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 )}\, e \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 )}{\sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {8 b^{4} \sqrt {e \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 )}\, e \left (40 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+118 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \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 )+3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {8 a \,b^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,e^{3}}+\frac {24 a^{2} b^{2} \sqrt {e \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 )}\, e \left (4 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\cos ^{3}\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 )-\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {8 a^{3} b \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3 e d}\) | \(630\) |
2/315/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e*(1120*cos(1 /2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^4-2240*cos(1/2*d*x+1/2*c)*sin(1/2*d* x+1/2*c)^8*b^4+2880*sin(1/2*d*x+1/2*c)^9*a*b^3-3024*cos(1/2*d*x+1/2*c)*sin (1/2*d*x+1/2*c)^6*a^2*b^2+1064*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*b^4 -5760*sin(1/2*d*x+1/2*c)^7*a*b^3+3024*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c )^4*a^2*b^2+56*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*b^4-1680*sin(1/2*d* x+1/2*c)^5*a^3*b+2640*sin(1/2*d*x+1/2*c)^5*a*b^3-756*cos(1/2*d*x+1/2*c)*si n(1/2*d*x+1/2*c)^2*a^2*b^2-84*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b^4+ 315*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elliptic E(cos(1/2*d*x+1/2*c),2^(1/2))*a^4+756*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin( 1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^2+84 *(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(c os(1/2*d*x+1/2*c),2^(1/2))*b^4+1680*sin(1/2*d*x+1/2*c)^3*a^3*b+240*sin(1/2 *d*x+1/2*c)^3*a*b^3-420*sin(1/2*d*x+1/2*c)*a^3*b-240*sin(1/2*d*x+1/2*c)*a* b^3)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.91 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx=-\frac {21 \, \sqrt {2} {\left (-15 i \, a^{4} - 36 i \, a^{2} b^{2} - 4 i \, b^{4}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, a^{4} + 36 i \, a^{2} b^{2} + 4 i \, b^{4}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (180 \, a b^{3} \cos \left (d x + c\right )^{3} - 420 \, {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right ) + 7 \, {\left (5 \, b^{4} \cos \left (d x + c\right )^{3} - {\left (54 \, a^{2} b^{2} + 11 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{315 \, d} \]
-1/315*(21*sqrt(2)*(-15*I*a^4 - 36*I*a^2*b^2 - 4*I*b^4)*sqrt(e)*weierstras sZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(15*I*a^4 + 36*I*a^2*b^2 + 4*I*b^4)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(180*a* b^3*cos(d*x + c)^3 - 420*(a^3*b + a*b^3)*cos(d*x + c) + 7*(5*b^4*cos(d*x + c)^3 - (54*a^2*b^2 + 11*b^4)*cos(d*x + c))*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]