\(\int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx\) [477]

Optimal result

Integrand size = 23, antiderivative size = 215 \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {4 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{15 b d}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}+\frac {4 \left (a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{15 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{15 b^2 d \sqrt {a+b \sin (c+d x)}} \] Output:

-4/15*a*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b/d+2/5*cos(d*x+c)*(a+b*sin(d*x+ 
c))^(3/2)/b/d-4/15*(a^2+3*b^2)*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^2/d/((a+b*sin(d*x+c))/(a+b))^(1 
/2)-4/15*a*(a^2-b^2)*InverseJacobiAM(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^2/d/(a+b*sin(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {-4 \left (a^3+a^2 b+3 a b^2+3 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+4 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (2 a^2+3 b^2-3 b^2 \cos (2 (c+d x))+8 a b \sin (c+d x)\right )}{15 b^2 d \sqrt {a+b \sin (c+d x)}} \] Input:

Integrate[Cos[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]
 

Output:

(-4*(a^3 + a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b) 
/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 4*a*(a^2 - b^2)*EllipticF[( 
-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 
b*Cos[c + d*x]*(2*a^2 + 3*b^2 - 3*b^2*Cos[2*(c + d*x)] + 8*a*b*Sin[c + d*x 
]))/(15*b^2*d*Sqrt[a + b*Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 3174, 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.

\(\displaystyle \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^2 \sqrt {a+b \sin (c+d x)}dx\)

\(\Big \downarrow \) 3174

\(\displaystyle \frac {2 \int (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}dx}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \int (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}dx}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {2 \left (\frac {2}{3} \int \frac {4 a b+\left (a^2+3 b^2\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {1}{3} \int \frac {4 a b+\left (a^2+3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {1}{3} \int \frac {4 a b+\left (a^2+3 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {\left (a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {\left (a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {\left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {\left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 \left (\frac {1}{3} \left (\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )-\frac {2 a \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )}{5 b}+\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}\)

Input:

Int[Cos[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]
 

Output:

(2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*b*d) + (2*((-2*a*Cos[c + d* 
x]*Sqrt[a + b*Sin[c + d*x]])/(3*d) + ((2*(a^2 + 3*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*a*(a^2 - b^2)*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]] 
))/3))/(5*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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]
 

rule 3134

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]
 

rule 3140

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]
 

rule 3142

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]
 

rule 3174

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 + p))), x] + Simp[g^2*((p - 1)/(b*(m + p)))   Int[(g*Cos 
[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; F 
reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 
 0] && IntegersQ[2*m, 2*p]
 

rule 3231

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]
 

rule 3232

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(791\) vs. \(2(200)=400\).

Time = 1.02 (sec) , antiderivative size = 792, normalized size of antiderivative = 3.68

method	result	size
default	\(\frac {\frac {4 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {b \left (\sin \left (d x +c \right )-1\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b}{15}+\frac {4 a^{2} \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {b \left (\sin \left (d x +c \right )-1\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{2}}{5}-\frac {4 a \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {b \left (\sin \left (d x +c \right )-1\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{3}}{15}-\frac {4 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {b \left (\sin \left (d x +c \right )-1\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{4}}{5}-\frac {4 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {b \left (\sin \left (d x +c \right )-1\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4}}{15}-\frac {8 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {b \left (\sin \left (d x +c \right )-1\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{2}}{15}+\frac {4 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {b \left (\sin \left (d x +c \right )-1\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sin \left (d x +c \right )\right )}{a -b}}\, \operatorname {EllipticE}\left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{4}}{5}-\frac {2 b^{4} \sin \left (d x +c \right )^{4}}{5}-\frac {8 a \,b^{3} \sin \left (d x +c \right )^{3}}{15}-\frac {2 a^{2} b^{2} \sin \left (d x +c \right )^{2}}{15}+\frac {2 b^{4} \sin \left (d x +c \right )^{2}}{5}+\frac {8 a \,b^{3} \sin \left (d x +c \right )}{15}+\frac {2 b^{2} a^{2}}{15}}{b^{3} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\)	\(792\)

Input:

int(cos(d*x+c)^2*(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/15*(2*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-b*(sin(d*x+c)-1)/(a+b))^(1/2)*(-b 
*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a- 
b)/(a+b))^(1/2))*a^3*b+6*a^2*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-b*(sin(d*x+c 
)-1)/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+ 
c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^2-2*a*((a+b*sin(d*x+c))/(a-b))^(1/ 
2)*(-b*(sin(d*x+c)-1)/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*Ellipti 
cF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^3-6*((a+b*sin(d*x 
+c))/(a-b))^(1/2)*(-b*(sin(d*x+c)-1)/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b) 
)^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^4- 
2*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-b*(sin(d*x+c)-1)/(a+b))^(1/2)*(-b*(1+si 
n(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+ 
b))^(1/2))*a^4-4*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-b*(sin(d*x+c)-1)/(a+b))^ 
(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^( 
1/2),((a-b)/(a+b))^(1/2))*a^2*b^2+6*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-b*(si 
n(d*x+c)-1)/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*s 
in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^4-3*b^4*sin(d*x+c)^4-4*a*b^ 
3*sin(d*x+c)^3-a^2*b^2*sin(d*x+c)^2+3*b^4*sin(d*x+c)^2+4*a*b^3*sin(d*x+c)+ 
b^2*a^2)/b^3/cos(d*x+c)/(a+b*sin(d*x+c))^(1/2)/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.03 \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {2 \, {\left (4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \sqrt {\frac {1}{2} i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, 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 i \, a}{3 \, b}\right ) + 4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \sqrt {-\frac {1}{2} i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, 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 i \, a}{3 \, b}\right ) + 6 \, {\left (i \, a^{2} b + 3 i \, b^{3}\right )} \sqrt {\frac {1}{2} i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, 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 i \, a^{3} - 9 i \, 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 i \, a}{3 \, b}\right )\right ) + 6 \, {\left (-i \, a^{2} b - 3 i \, b^{3}\right )} \sqrt {-\frac {1}{2} i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, 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 i \, a^{3} + 9 i \, 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 i \, a}{3 \, b}\right )\right ) - 3 \, {\left (3 \, b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a b^{2} \cos \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{45 \, b^{3} d} \] Input:

integrate(cos(d*x+c)^2*(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

-2/45*(4*(a^3 - 3*a*b^2)*sqrt(1/2*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*(a^3 - 3*a*b^2)*sqrt(-1/2*I*b)*weierstrassPIn 
verse(-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) + 6*(I*a^2*b + 3*I*b^3)*sqr 
t(1/2*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)) + 
 6*(-I*a^2*b - 3*I*b^3)*sqrt(-1/2*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^3*cos(d*x + c)*sin(d*x + c) + a*b^2* 
cos(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^3*d)
 

Sympy [F]

\[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int \sqrt {a + b \sin {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )}\, dx \] Input:

integrate(cos(d*x+c)**2*(a+b*sin(d*x+c))**(1/2),x)
 

Output:

Integral(sqrt(a + b*sin(c + d*x))*cos(c + d*x)**2, x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cos(d*x+c)^2*(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*sin(d*x + c) + a)*cos(d*x + c)^2, x)
 

Giac [F]

\[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \,d x } \] Input:

integrate(cos(d*x+c)^2*(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*sin(d*x + c) + a)*cos(d*x + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \] Input:

int(cos(c + d*x)^2*(a + b*sin(c + d*x))^(1/2),x)
 

Output:

int(cos(c + d*x)^2*(a + b*sin(c + d*x))^(1/2), x)
 

Reduce [F]

\[ \int \cos ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \] Input:

int(cos(d*x+c)^2*(a+b*sin(d*x+c))^(1/2),x)
 

Output:

int(sqrt(sin(c + d*x)*b + a)*cos(c + d*x)**2,x)