Integrand size = 23, antiderivative size = 105 \[ \int \cos ^2(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {21 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 d}-\frac {\sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{20 d}-\frac {\sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {(3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d} \]
1/10*(3+4*cos(d*x+c))^(3/2)*sin(d*x+c)/d+21/20*(cos(1/2*d*x+1/2*c)^2)^(1/2 )/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)- 1/20*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(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 = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {21 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )-\sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )+2 \sqrt {3+4 \cos (c+d x)} (\sin (c+d x)+2 \sin (2 (c+d x)))}{20 d} \]
(21*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] - Sqrt[7]*EllipticF[(c + d*x)/2, 8 /7] + 2*Sqrt[3 + 4*Cos[c + d*x]]*(Sin[c + d*x] + 2*Sin[2*(c + d*x)]))/(20* d)
Time = 0.58 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10, 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, 3132, 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.
| \(\displaystyle \int \cos ^2(c+d x) \sqrt {4 \cos (c+d x)+3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx\) |
| \(\Big \downarrow \) 3270 |
| \(\displaystyle \frac {1}{10} \int 3 (2-\cos (c+d x)) \sqrt {4 \cos (c+d x)+3}dx+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{10} \int (2-\cos (c+d x)) \sqrt {4 \cos (c+d x)+3}dx+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \int \left (2-\sin \left (c+d x+\frac {\pi }{2}\right )\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {3}{10} \left (\frac {2}{3} \int \frac {7 (3 \cos (c+d x)+2)}{2 \sqrt {4 \cos (c+d x)+3}}dx-\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\right )+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \int \frac {3 \cos (c+d x)+2}{\sqrt {4 \cos (c+d x)+3}}dx-\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\right )+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \int \frac {3 \sin \left (c+d x+\frac {\pi }{2}\right )+2}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx-\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\right )+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3}{4} \int \sqrt {4 \cos (c+d x)+3}dx-\frac {1}{4} \int \frac {1}{\sqrt {4 \cos (c+d x)+3}}dx\right )-\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\right )+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3}{4} \int \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx-\frac {1}{4} \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\right )-\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\right )+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {1}{4} \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\right )-\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\right )+\frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}+\frac {3}{10} \left (\frac {7}{3} \left (\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {\operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{2 \sqrt {7} d}\right )-\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\right )\) |
((3 + 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(10*d) + (3*((7*((3*Sqrt[7]*Elli pticE[(c + d*x)/2, 8/7])/(2*d) - EllipticF[(c + 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
3.6.10.3.1 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 = 3.36 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.41
| method | result | size |
| default | \(-\frac {\sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-256 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+384 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-140 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{20 \sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(253\) |
-1/20*((8*cos(1/2*d*x+1/2*c)^2-1)*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+384*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2 *c)-140*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-7*(sin(1/2*d*x+1/2*c)^2)^( 1/2)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2*2^(1/ 2))-21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*Ellip ticE(cos(1/2*d*x+1/2*c),2*2^(1/2)))/(-8*sin(1/2*d*x+1/2*c)^4+7*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)^2-1)^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.31 \[ \int \cos ^2(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {4 \, \sqrt {4 \, \cos \left (d x + c\right ) + 3} {\left (4 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 7 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 7 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 42 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 42 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{40 \, d} \]
1/40*(4*sqrt(4*cos(d*x + c) + 3)*(4*cos(d*x + c) + 1)*sin(d*x + c) - 7*I*s qrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2) + 7 *I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2) + 42*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2)) - 42*I*sqrt(2)*weierstrassZeta(-1, 1, weiers trassPInverse(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2)))/d
\[ \int \cos ^2(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\int \sqrt {4 \cos {\left (c + d x \right )} + 3} \cos ^{2}{\left (c + d x \right )}\, dx \]
\[ \int \cos ^2(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2} \,d x } \]
\[ \int \cos ^2(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^2(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sqrt {4\,\cos \left (c+d\,x\right )+3} \,d x \]