Integrand size = 23, antiderivative size = 229 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}+\frac {4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 b^3 d}+\frac {8 \left (4 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{5 b^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {32 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{5 b^4 d \sqrt {a+b \sin (c+d x)}} \]
-2*cos(d*x+c)^3/b/d/(a+b*sin(d*x+c))^(1/2)+4/5*cos(d*x+c)*(4*a-3*b*sin(d*x +c))*(a+b*sin(d*x+c))^(1/2)/b^3/d-8/5*(4*a^2-3*b^2)*(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^4/d/((a+b*sin(d*x+c))/ (a+b))^(1/2)+32/5*a*(a^2-b^2)*(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^4/d/(a+b*sin(d*x+c))^(1/2)
Time = 0.79 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {-8 \left (4 a^3+4 a^2 b-3 a b^2-3 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+32 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (16 a^2-11 b^2+b^2 \cos (2 (c+d x))+4 a b \sin (c+d x)\right )}{5 b^4 d \sqrt {a+b \sin (c+d x)}} \]
(-8*(4*a^3 + 4*a^2*b - 3*a*b^2 - 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, ( 2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 32*a*(a^2 - b^2)*Ellipt icF[(-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]*(16*a^2 - 11*b^2 + b^2*Cos[2*(c + d*x)] + 4*a*b*Sin[c + d*x]))/(5*b^4*d*Sqrt[a + b*Sin[c + d*x]])
Time = 1.11 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 3172, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{(a+b \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {6 \int \frac {\cos ^2(c+d x) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {6 \int \frac {\cos (c+d x)^2 \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {6 \left (\frac {4 \int -\frac {a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \int \frac {a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \int \frac {a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \left (\frac {\left (4 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \left (\frac {\left (4 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \left (\frac {\left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \left (\frac {\left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \left (\frac {2 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \left (\frac {2 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \left (\frac {2 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {6 \left (-\frac {2 \left (\frac {2 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} (4 a-3 b \sin (c+d x))}{15 b^2 d}\right )}{b}-\frac {2 \cos ^3(c+d x)}{b d \sqrt {a+b \sin (c+d x)}}\) |
(-2*Cos[c + d*x]^3)/(b*d*Sqrt[a + b*Sin[c + d*x]]) - (6*((-2*Cos[c + d*x]* (4*a - 3*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(15*b^2*d) - (2*((2*(4* a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(a^2 - b^2)*Elli pticF[(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)))/b
3.6.23.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + 1))), x] + Simp[g^2*((p - 1)/(b*(m + 1))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; Fre eQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && I ntegersQ[2*m, 2*p]
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. \(796\) vs. \(2(277)=554\).
Time = 1.76 (sec) , antiderivative size = 797, normalized size of antiderivative = 3.48
| method | result | size |
| default | \(\frac {\frac {32 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b}{5}-\frac {24 a^{2} \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{2}}{5}-\frac {32 a \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{3}}{5}+\frac {24 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, F\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{4}}{5}-\frac {32 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, E\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4}}{5}+\frac {56 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, E\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{2}}{5}-\frac {24 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, E\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{4}}{5}+\frac {2 b^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{5}-\frac {4 a \,b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{5}-\frac {16 a^{2} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{5}+\frac {8 b^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{5}+\frac {4 a \,b^{3} \sin \left (d x +c \right )}{5}+\frac {16 a^{2} b^{2}}{5}-2 b^{4}}{b^{5} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(797\) |
2/5*(16*((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-12*a^2*((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))*b^2-16*a*((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)*Ellip ticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^3+12*((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))*b ^4-16*((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+28*((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^2-12*((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))*b^4+b^4*sin(d*x+c)^4-2* a*b^3*sin(d*x+c)^3-8*a^2*b^2*sin(d*x+c)^2+4*b^4*sin(d*x+c)^2+2*a*b^3*sin(d *x+c)+8*a^2*b^2-5*b^4)/b^5/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.61 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (\sqrt {2} {\left (8 \, a^{3} b - 9 \, a b^{3}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (8 \, a^{4} - 9 \, a^{2} b^{2}\right )}\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 ) + 2 \, {\left (\sqrt {2} {\left (8 \, a^{3} b - 9 \, a b^{3}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (8 \, a^{4} - 9 \, a^{2} b^{2}\right )}\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 \, {\left (\sqrt {2} {\left (4 i \, a^{2} b^{2} - 3 i \, b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (4 i \, a^{3} b - 3 i \, a b^{3}\right )}\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 \, {\left (\sqrt {2} {\left (-4 i \, a^{2} b^{2} + 3 i \, b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-4 i \, a^{3} b + 3 i \, a b^{3}\right )}\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 (b^{4} \cos \left (d x + c\right )^{3} + 2 \, a b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, {\left (4 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{15 \, {\left (b^{6} d \sin \left (d x + c\right ) + a b^{5} d\right )}} \]
-2/15*(2*(sqrt(2)*(8*a^3*b - 9*a*b^3)*sin(d*x + c) + sqrt(2)*(8*a^4 - 9*a^ 2*b^2))*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) + 2*(sqrt(2)*(8*a^3*b - 9*a*b^3)*sin(d*x + c) + sqrt(2)*(8*a^4 - 9*a^2 *b^2))*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)*(4*I*a^2*b^2 - 3*I*b^4)*sin(d*x + c) + sqrt(2)*(4*I*a^3* b - 3*I*a*b^3))*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)*(-4*I*a^2*b^2 + 3*I*b^4)*sin(d*x + c) + sqrt(2) *(-4*I*a^3*b + 3*I*a*b^3))*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)) - 3*(b^4*cos(d*x + c)^3 + 2*a*b^3*cos(d*x + c) *sin(d*x + c) + 2*(4*a^2*b^2 - 3*b^4)*cos(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^6*d*sin(d*x + c) + a*b^5*d)
\[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]