Integrand size = 23, antiderivative size = 107 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\frac {21 \sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 d}-\frac {\sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{20 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d} \]
-1/10*(3-4*cos(d*x+c))^(3/2)*sin(d*x+c)/d-21/20*(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) +1/20*(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)+1/5*sin(d*x+c)*(3-4*cos(d*x+c))^(1/2)/d
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=-\frac {21 \sqrt {-3+4 \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )+7 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )+14 \sin (c+d x)-16 \sin (2 (c+d x))+8 \sin (3 (c+d x))}{20 d \sqrt {3-4 \cos (c+d x)}} \]
-1/20*(21*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 7*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8] + 14*Sin[c + d*x] - 16*Sin[2*( c + d*x)] + 8*Sin[3*(c + d*x)])/(d*Sqrt[3 - 4*Cos[c + d*x]])
Time = 0.56 (sec) , antiderivative size = 117, 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, 3270, 27, 3042, 3232, 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.
| \(\displaystyle \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 3270 |
| \(\displaystyle -\frac {1}{10} \int -3 \sqrt {3-4 \cos (c+d x)} (\cos (c+d x)+2)dx-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{10} \int \sqrt {3-4 \cos (c+d x)} (\cos (c+d x)+2)dx-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right )+2\right )dx-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {3}{10} \left (\frac {2}{3} \int \frac {7 (2-3 \cos (c+d x))}{2 \sqrt {3-4 \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \int \frac {2-3 \cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \int \frac {2-3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3}{4} \int \sqrt {3-4 \cos (c+d x)}dx-\frac {1}{4} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3}{4} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {1}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3133 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}-\frac {1}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3141 |
| \(\displaystyle \frac {3}{10} \left (\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}+\frac {7}{3} \left (\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}-\frac {\operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{2 \sqrt {7} d}\right )\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\) |
-1/10*((3 - 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/d + (3*((7*((3*Sqrt[7]*Ell ipticE[(c + Pi + d*x)/2, 8/7])/(2*d) - EllipticF[(c + Pi + d*x)/2, 8/7]/(2 *Sqrt[7]*d)))/3 + (2*Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/10
3.6.17.3.1 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_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(-d^2)*Cos[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[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x ], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && Ne Q[a^2 - b^2, 0] && !LtQ[m, -1]
Time = 5.15 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.36
| method | result | size |
| default | \(\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-256 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )\right )}{20 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d}\) | \(253\) |
1/20*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-256*cos(1/ 2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+128*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2 *c)-12*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+(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))-21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*Ell ipticE(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 = 138, normalized size of antiderivative = 1.29 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\frac {4 \, {\left (4 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) - 7 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) - 7 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) - 42 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) - 42 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{40 \, d} \]
1/40*(4*(4*cos(d*x + c) - 1)*sqrt(-4*cos(d*x + c) + 3)*sin(d*x + c) - 7*sq rt(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) - 42*sqrt(2)*weierstrassZeta(-1, -1, weierstrassPInverse(-1, -1, cos(d*x + c) + I*sin(d*x + c) - 1/2)) - 42*sqrt(2)*weierstrassZeta(-1, -1, weierstr assPInverse(-1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2)))/d
\[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )}\, dx \]
\[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2} \,d x } \]
\[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sqrt {3-4\,\cos \left (c+d\,x\right )} \,d x \]