Integrand size = 23, antiderivative size = 293 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}+\frac {16 a \left (32 a^2-29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{21 b^6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {16 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{21 b^6 d \sqrt {a+b \sin (c+d x)}}-\frac {20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {8 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{21 b^5 d} \]
-2/3*cos(d*x+c)^5/b/d/(a+b*sin(d*x+c))^(3/2)-20/21*cos(d*x+c)^3*(8*a+b*sin (d*x+c))/b^3/d/(a+b*sin(d*x+c))^(1/2)+8/21*cos(d*x+c)*(32*a^2-5*b^2-24*a*b *sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/b^5/d-16/21*a*(32*a^2-29*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^6/d/((a+ b*sin(d*x+c))/(a+b))^(1/2)+16/21*(32*a^4-37*a^2*b^2+5*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^6/d/(a+ b*sin(d*x+c))^(1/2)
Time = 0.81 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {-32 (a+b) \left (a \left (32 a^3+32 a^2 b-29 a b^2-29 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+\left (-32 a^4+37 a^2 b^2-5 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}+\frac {1}{2} b \cos (c+d x) \left (1024 a^4-736 a^2 b^2-111 b^4+\left (-64 a^2 b^2+52 b^4\right ) \cos (2 (c+d x))+3 b^4 \cos (4 (c+d x))+1280 a^3 b \sin (c+d x)-1076 a b^3 \sin (c+d x)+12 a b^3 \sin (3 (c+d x))\right )}{42 b^6 d (a+b \sin (c+d x))^{3/2}} \]
(-32*(a + b)*(a*(32*a^3 + 32*a^2*b - 29*a*b^2 - 29*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (-32*a^4 + 37*a^2*b^2 - 5*b^4)*EllipticF[( -2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2) + (b*Cos[c + d*x]*(1024*a^4 - 736*a^2*b^2 - 111*b^4 + (-64*a^2*b^2 + 52*b ^4)*Cos[2*(c + d*x)] + 3*b^4*Cos[4*(c + d*x)] + 1280*a^3*b*Sin[c + d*x] - 1076*a*b^3*Sin[c + d*x] + 12*a*b^3*Sin[3*(c + d*x)]))/2)/(42*b^6*d*(a + b* Sin[c + d*x])^(3/2))
Time = 1.50 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 3172, 3042, 3342, 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.
| \(\displaystyle \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{(a+b \sin (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {10 \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}}dx}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {10 \int \frac {\cos (c+d x)^4 \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}}dx}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {10 \left (\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {12 \int -\frac {\cos ^2(c+d x) (b+8 a \sin (c+d x))}{2 \sqrt {a+b \sin (c+d x)}}dx}{7 b^2}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {10 \left (\frac {6 \int \frac {\cos ^2(c+d x) (b+8 a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {10 \left (\frac {6 \int \frac {\cos (c+d x)^2 (b+8 a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (\frac {4 \int -\frac {b \left (8 a^2-5 b^2\right )+a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \int \frac {b \left (8 a^2-5 b^2\right )+a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \int \frac {b \left (8 a^2-5 b^2\right )+a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \left (\frac {a \left (32 a^2-29 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \left (\frac {a \left (32 a^2-29 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \left (\frac {a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \left (\frac {a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \left (\frac {2 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \left (\frac {2 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \left (\frac {2 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {10 \left (\frac {6 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{15 b^2 d}-\frac {2 \left (\frac {2 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{7 b^2}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}\right )}{3 b}-\frac {2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}\) |
(-2*Cos[c + d*x]^5)/(3*b*d*(a + b*Sin[c + d*x])^(3/2)) - (10*((2*Cos[c + d *x]^3*(8*a + b*Sin[c + d*x]))/(7*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (6*((-2 *Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 5*b^2 - 24*a*b*Sin[c + d* x]))/(15*b^2*d) - (2*((2*a*(32*a^2 - 29*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)]) - (2*(32*a^4 - 37*a^2*b^2 + 5*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)))/(7*b^2)))/(3*b)
3.6.34.3.1 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*C os[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d*p + b*d*(m + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Simp[g^2*(( p - 1)/(b^2*(m + 1)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin [e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x ], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && Lt Q[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
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. \(1641\) vs. \(2(335)=670\).
Time = 3.50 (sec) , antiderivative size = 1642, normalized size of antiderivative = 5.60
2/21*(3*cos(d*x+c)^6*b^6+6*sin(d*x+c)*cos(d*x+c)^4*a*b^5+(-16*a^2*b^4+10*b ^6)*cos(d*x+c)^4+(160*a^3*b^3-136*a*b^5)*cos(d*x+c)^2*sin(d*x+c)+(128*a^4* b^2-84*a^2*b^4-20*b^6)*cos(d*x+c)^2-8*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*( -b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*b*( 32*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5-6 1*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^ 2+29*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b ^4-32*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^ 4*b+24*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a ^3*b^2+37*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2) )*a^2*b^3-24*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1 /2))*a*b^4-5*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1 /2))*b^5)*sin(d*x+c)+256*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticF((b/( a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b /(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^5*b-192*(b/(a-b)*sin(d *x+c)+a/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/( a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a +b))^(1/2)*a^4*b^2-296*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticF((b/(a- b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/( a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^3*b^3+192*(b/(a-b)*si...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 874, normalized size of antiderivative = 2.98 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
-2/63*(4*(sqrt(2)*(64*a^4*b^2 - 82*a^2*b^4 + 15*b^6)*cos(d*x + c)^2 - 2*sq rt(2)*(64*a^5*b - 82*a^3*b^3 + 15*a*b^5)*sin(d*x + c) - sqrt(2)*(64*a^6 - 18*a^4*b^2 - 67*a^2*b^4 + 15*b^6))*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) + 4*(sqrt(2)*(64*a^4*b^2 - 82*a^2*b^4 + 15* b^6)*cos(d*x + c)^2 - 2*sqrt(2)*(64*a^5*b - 82*a^3*b^3 + 15*a*b^5)*sin(d*x + c) - sqrt(2)*(64*a^6 - 18*a^4*b^2 - 67*a^2*b^4 + 15*b^6))*sqrt(-I*b)*we ierstrassPInverse(-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) - 12*(sqrt(2)*( -32*I*a^3*b^3 + 29*I*a*b^5)*cos(d*x + c)^2 + 2*sqrt(2)*(32*I*a^4*b^2 - 29* I*a^2*b^4)*sin(d*x + c) + sqrt(2)*(32*I*a^5*b + 3*I*a^3*b^3 - 29*I*a*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)) - 12*(sqrt(2)*(32*I*a^3*b^3 - 29*I*a*b^5)*cos(d*x + c)^2 + 2*sqrt(2)*(-32*I *a^4*b^2 + 29*I*a^2*b^4)*sin(d*x + c) + sqrt(2)*(-32*I*a^5*b - 3*I*a^3*b^3 + 29*I*a*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)) + 3*(3*b^6*cos(d*x + c)^5 - 2*(8*a^2*b^4 - 5*b^6)*cos(...
Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{6}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^6}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]