Integrand size = 23, antiderivative size = 207 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=-\frac {2 \left (2 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b^2 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b d}+\frac {2 (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b d} \] Output:
-2/15*(2*a^2-9*b^2)*(a+b*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^ (1/2)*(b/(a+b))^(1/2))/b^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+4/15*a*(a^2-b^ 2)*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b /(a+b))^(1/2))/b^2/d/(a+b*cos(d*x+c))^(1/2)-4/15*a*(a+b*cos(d*x+c))^(1/2)* sin(d*x+c)/b/d+2/5*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d
Time = 1.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.87 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\frac {-2 \left (2 a^3+2 a^2 b-9 a b^2-9 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+b \left (2 a^2+3 b^2+8 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))\right ) \sin (c+d x)}{15 b^2 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]],x]
Output:
(-2*(2*a^3 + 2*a^2*b - 9*a*b^2 - 9*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)] *EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(2*a^2 + 3*b^2 + 8*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*b^2*d*Sq rt[a + b*Cos[c + d*x]])
Time = 1.05 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 3270, 27, 3042, 3232, 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.
| \(\displaystyle \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3270 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (3 b-2 a \cos (c+d x)) \sqrt {a+b \cos (c+d x)}dx}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (3 b-2 a \cos (c+d x)) \sqrt {a+b \cos (c+d x)}dx}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (3 b-2 a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {2}{3} \int \frac {7 a b-\left (2 a^2-9 b^2\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {7 a b-\left (2 a^2-9 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {7 a b+\left (9 b^2-2 a^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (2 a^2-9 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^2-9 b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (2 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^2-9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}\) |
Input:
Int[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]],x]
Output:
(2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d) + (((-2*(2*a^2 - 9*b^2 )*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqr t[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d* x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/3 - (4*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/(5*b)
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[((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]
Leaf count of result is larger than twice the leaf count of optimal. \(664\) vs. \(2(196)=392\).
Time = 8.02 (sec) , antiderivative size = 665, normalized size of antiderivative = 3.21
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (24 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{3}+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a \,b^{2}-48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b -24 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a \,b^{2}+30 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{3}+2 a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )-2 b^{2} a \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )-2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{3}+2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{2} b +9 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a \,b^{2}-9 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) b^{3}-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}-6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{15 b^{2} \sqrt {-2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}\, d}\) | \(665\) |
Input:
int(cos(d*x+c)^2*(a+cos(d*x+c)*b)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/15*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(24*cos( 1/2*d*x+1/2*c)^7*b^3+16*cos(1/2*d*x+1/2*c)^5*a*b^2-48*cos(1/2*d*x+1/2*c)^5 *b^3+2*cos(1/2*d*x+1/2*c)^3*a^2*b-24*cos(1/2*d*x+1/2*c)^3*a*b^2+30*cos(1/2 *d*x+1/2*c)^3*b^3+2*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2 *c)^2+a-b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-2 *b^2*a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b)) ^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-2*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticE(cos(1/ 2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b* cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/ (a-b))^(1/2))*a^2*b+9*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c )^2+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b ^2-9*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^( 1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3-2*cos(1/2*d*x+1/ 2*c)*a^2*b+8*cos(1/2*d*x+1/2*c)*a*b^2-6*cos(1/2*d*x+1/2*c)*b^3)/b^2/(-2*b* sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/ (-2*sin(1/2*d*x+1/2*c)^2*b+a+b)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.11 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=-\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (4 i \, a^{3} + 3 i \, a b^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right ) + \sqrt {\frac {1}{2}} {\left (-4 i \, a^{3} - 3 i \, a b^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (2 i \, a^{2} b - 9 i \, b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (-2 i \, a^{2} b + 9 i \, b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right )\right ) - 3 \, {\left (3 \, b^{3} \cos \left (d x + c\right ) + a b^{2}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )\right )}}{45 \, b^{3} d} \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
-2/45*(sqrt(1/2)*(4*I*a^3 + 3*I*a*b^2)*sqrt(b)*weierstrassPInverse(4/3*(4* a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I *b*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(-4*I*a^3 - 3*I*a*b^2)*sqrt(b)*weier strassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*( 3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*sqrt(1/2)*(2*I*a^2*b - 9*I*b^3)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 3*sq rt(1/2)*(-2*I*a^2*b + 9*I*b^3)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2) /b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2) /b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*(3*b^3*cos(d*x + c) + a*b^2)*sqrt(b*cos(d*x + c) + a)*s in(d*x + c))/(b^3*d)
\[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\int \sqrt {a + b \cos {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )}\, dx \] Input:
integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**(1/2),x)
Output:
Integral(sqrt(a + b*cos(c + d*x))*cos(c + d*x)**2, x)
\[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\int { \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*cos(d*x + c) + a)*cos(d*x + c)^2, x)
\[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\int { \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*cos(d*x + c) + a)*cos(d*x + c)^2, x)
Timed out. \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \] Input:
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(1/2),x)
Output:
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(1/2), x)
\[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \] Input:
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2),x)
Output:
int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)