Integrand size = 29, antiderivative size = 283 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}+\frac {8 \left (32 a^4-57 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{315 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{315 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{315 b^4 d} \]
-2/63*cos(d*x+c)^3*(8*a-7*b*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^2/d+4/315 *cos(d*x+c)*(a*(32*a^2-33*b^2)-3*b*(8*a^2-7*b^2)*sin(d*x+c))*(a+b*sin(d*x+ c))^(1/2)/b^4/d-8/315*(32*a^4-57*a^2*b^2+21*b^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)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2 ^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/b^5/d/((a+b*sin(d*x+c))/(a+ b))^(1/2)+8/315*a*(32*a^4-65*a^2*b^2+33*b^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)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/b^5/d/(a+b*sin(d*x+c))^ (1/2)
Time = 2.13 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {-32 \left (32 a^5+32 a^4 b-57 a^3 b^2-57 a^2 b^3+21 a b^4+21 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+32 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-b \cos (c+d x) \left (-512 a^4+880 a^2 b^2-203 b^4-8 \left (4 a^2 b^2-21 b^4\right ) \cos (2 (c+d x))+35 b^4 \cos (4 (c+d x))-128 a^3 b \sin (c+d x)+202 a b^3 \sin (c+d x)+10 a b^3 \sin (3 (c+d x))\right )}{1260 b^5 d \sqrt {a+b \sin (c+d x)}} \]
(-32*(32*a^5 + 32*a^4*b - 57*a^3*b^2 - 57*a^2*b^3 + 21*a*b^4 + 21*b^5)*Ell ipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 32*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/ 4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*Cos[c + d*x]*(-51 2*a^4 + 880*a^2*b^2 - 203*b^4 - 8*(4*a^2*b^2 - 21*b^4)*Cos[2*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)] - 128*a^3*b*Sin[c + d*x] + 202*a*b^3*Sin[c + d*x] + 10*a*b^3*Sin[3*(c + d*x)]))/(1260*b^5*d*Sqrt[a + b*Sin[c + d*x]])
Time = 1.35 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3344, 27, 3042, 3344, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {\sin (c+d x) \cos ^4(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x) \cos (c+d x)^4}{\sqrt {a+b \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {4 \int -\frac {\cos ^2(c+d x) \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {\cos ^2(c+d x) \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \int \frac {\cos (c+d x)^2 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {2 \left (\frac {4 \int -\frac {4 a b \left (2 a^2-3 b^2\right )+\left (32 a^4-57 b^2 a^2+21 b^4\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \int \frac {4 a b \left (2 a^2-3 b^2\right )+\left (32 a^4-57 b^2 a^2+21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \int \frac {4 a b \left (2 a^2-3 b^2\right )+\left (32 a^4-57 b^2 a^2+21 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {\left (32 a^4-57 a^2 b^2+21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {\left (32 a^4-57 a^2 b^2+21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {\left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {\left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (a \left (32 a^2-33 b^2\right )-3 b \left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{15 b^2 d}-\frac {2 \left (\frac {2 \left (32 a^4-57 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a \left (32 a^4-65 a^2 b^2+33 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{63 b^2 d}\) |
(-2*Cos[c + d*x]^3*(8*a - 7*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(63* b^2*d) - (2*((-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^2 - 33*b^2 ) - 3*b*(8*a^2 - 7*b^2)*Sin[c + d*x]))/(15*b^2*d) - (2*((2*(32*a^4 - 57*a^ 2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Si n[c + d*x]])/(b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*a*(32*a^4 - 65* a^2*b^2 + 33*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b *Sin[c + d*x])/(a + b)])/(b*d*Sqrt[a + b*Sin[c + d*x]])))/(15*b^2)))/(21*b ^2)
3.12.70.3.1 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[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(1189\) vs. \(2(329)=658\).
Time = 1.22 (sec) , antiderivative size = 1190, normalized size of antiderivative = 4.20
-2/315*(-35*b^6*sin(d*x+c)^6+5*a*b^5*sin(d*x+c)^5+128*((a+b*sin(d*x+c))/(a -b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2) *EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6-356*((a +b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c ))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^( 1/2))*a^4*b^2+312*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b)) ^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^ (1/2),((a-b)/(a+b))^(1/2))*a^2*b^4-84*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si n(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b *sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^6-128*((a+b*sin(d*x+c))/( a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2 )*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b+96*( (a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x +c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b)) ^(1/2))*a^4*b^2+260*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b ))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b) )^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^3-180*((a+b*sin(d*x+c))/(a-b))^(1/2)*(- (sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF((( a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4-132*((a+b*sin(d* x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.90 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=-\frac {2 \, {\left (4 \, \sqrt {2} {\left (32 \, a^{5} - 69 \, a^{3} b^{2} + 39 \, a b^{4}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 4 \, \sqrt {2} {\left (32 \, a^{5} - 69 \, a^{3} b^{2} + 39 \, a b^{4}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 6 \, \sqrt {2} {\left (32 i \, a^{4} b - 57 i \, a^{2} b^{3} + 21 i \, b^{5}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 6 \, \sqrt {2} {\left (-32 i \, a^{4} b + 57 i \, a^{2} b^{3} - 21 i \, b^{5}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 3 \, {\left (40 \, a b^{4} \cos \left (d x + c\right )^{3} - 2 \, {\left (32 \, a^{3} b^{2} - 33 \, a b^{4}\right )} \cos \left (d x + c\right ) - {\left (35 \, b^{5} \cos \left (d x + c\right )^{3} - 6 \, {\left (8 \, a^{2} b^{3} - 7 \, b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{945 \, b^{6} d} \]
-2/945*(4*sqrt(2)*(32*a^5 - 69*a^3*b^2 + 39*a*b^4)*sqrt(I*b)*weierstrassPI nverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b *cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b) + 4*sqrt(2)*(32*a^5 - 69*a^ 3*b^2 + 39*a*b^4)*sqrt(-I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) + 6*sqrt(2)*(32*I*a^4*b - 57*I*a^2*b^3 + 21*I*b^5)*sqrt(I* b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b ^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b ^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) + 6*sqrt( 2)*(-32*I*a^4*b + 57*I*a^2*b^3 - 21*I*b^5)*sqrt(-I*b)*weierstrassZeta(-4/3 *(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, weierstrassPInvers e(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos (d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) + 3*(40*a*b^4*cos(d*x + c)^3 - 2*(32*a^3*b^2 - 33*a*b^4)*cos(d*x + c) - (35*b^5*cos(d*x + c)^3 - 6*(8*a^ 2*b^3 - 7*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^6* d)
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )}{\sqrt {b \sin \left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]