Integrand size = 23, antiderivative size = 113 \[ \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=-\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 d}+\frac {23 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{10 d}-\frac {\sqrt {3-4 \cos (c+d x)} \cos (c+d x) \sin (c+d x)}{10 d} \]
-23/140*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)*EllipticF(cos(1/2* d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)+9/20*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1 /2*d*x+1/2*c)*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)-1/10*si n(d*x+c)*(3-4*cos(d*x+c))^(1/2)/d-1/10*cos(d*x+c)*sin(d*x+c)*(3-4*cos(d*x+ c))^(1/2)/d
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {9 \sqrt {-3+4 \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )+23 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )-4 \sin (c+d x)+\sin (2 (c+d x))+2 \sin (3 (c+d x))}{20 d \sqrt {3-4 \cos (c+d x)}} \]
(9*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 23*Sqrt[-3 + 4*Co s[c + d*x]]*EllipticF[(c + d*x)/2, 8] - 4*Sin[c + d*x] + Sin[2*(c + d*x)] + 2*Sin[3*(c + d*x)])/(20*d*Sqrt[3 - 4*Cos[c + d*x]])
Time = 0.61 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 3272, 27, 3042, 3502, 27, 3042, 3231, 3042, 3133, 3141}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle -\frac {1}{10} \int \frac {3 \left (-2 \cos ^2(c+d x)-2 \cos (c+d x)+1\right )}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{10} \int \frac {-2 \cos ^2(c+d x)-2 \cos (c+d x)+1}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3}{10} \int \frac {-2 \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 \sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {3}{10} \left (\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}-\frac {1}{6} \int -\frac {2 (1-9 \cos (c+d x))}{\sqrt {3-4 \cos (c+d x)}}dx\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{10} \left (\frac {1}{3} \int \frac {1-9 \cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx+\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3}{10} \left (\frac {1}{3} \int \frac {1-9 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {3}{10} \left (\frac {1}{3} \left (\frac {9}{4} \int \sqrt {3-4 \cos (c+d x)}dx-\frac {23}{4} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}}dx\right )+\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3}{10} \left (\frac {1}{3} \left (\frac {9}{4} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {23}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 3133 |
| \(\displaystyle -\frac {3}{10} \left (\frac {1}{3} \left (\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}-\frac {23}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}\) |
| \(\Big \downarrow \) 3141 |
| \(\displaystyle -\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)} \cos (c+d x)}{10 d}-\frac {3}{10} \left (\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {9 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}-\frac {23 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{2 \sqrt {7} d}\right )\right )\) |
-1/10*(Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]*Sin[c + d*x])/d - (3*(((9*Sqr t[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(2*d) - (23*EllipticF[(c + Pi + d*x )/2, 8/7])/(2*Sqrt[7]*d))/3 + (Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d )))/10
3.6.54.3.1 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[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[((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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 4.36 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.25
| method | result | size |
| default | \(-\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-448 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+23 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-63 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )\right )}{140 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d}\) | \(254\) |
-1/140*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-448*cos( 1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+504*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1 /2*c)-56*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+23*(sin(1/2*d*x+1/2*c)^2) ^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2/7* 14^(1/2))-63*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2 )*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2)))/(8*sin(1/2*d*x+1/2*c)^4-sin( 1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-8*cos(1/2*d*x+1/2*c)^2+7)^(1/ 2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=-\frac {4 \, {\left (\cos \left (d x + c\right ) + 1\right )} \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) + 7 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 7 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) - 18 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) - 18 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{40 \, d} \]
-1/40*(4*(cos(d*x + c) + 1)*sqrt(-4*cos(d*x + c) + 3)*sin(d*x + c) + 7*sqr t(2)*weierstrassPInverse(-1, -1, cos(d*x + c) + I*sin(d*x + c) - 1/2) + 7* sqrt(2)*weierstrassPInverse(-1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2) - 18*sqrt(2)*weierstrassZeta(-1, -1, weierstrassPInverse(-1, -1, cos(d*x + c) + I*sin(d*x + c) - 1/2)) - 18*sqrt(2)*weierstrassZeta(-1, -1, weierstra ssPInverse(-1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2)))/d
Timed out. \[ \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{3}}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
\[ \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{3}}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3}{\sqrt {3-4\,\cos \left (c+d\,x\right )}} \,d x \]