\(\int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx\) [297]

Optimal result

Integrand size = 33, antiderivative size = 303 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=-\frac {2 \left (14 a^2 A b-63 A b^3-8 a^3 B-19 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (14 a A b-8 a^2 B-25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (14 a A b-8 a^2 B-25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}+\frac {2 (7 A b-4 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d} \] Output:

-2/105*(14*A*a^2*b-63*A*b^3-8*B*a^3-19*B*a*b^2)*(a+b*cos(d*x+c))^(1/2)*Ell 
ipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b^3/d/((a+b*cos(d*x+c)) 
/(a+b))^(1/2)+2/105*(a^2-b^2)*(14*A*a*b-8*B*a^2-25*B*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^3/ 
d/(a+b*cos(d*x+c))^(1/2)-2/105*(14*A*a*b-8*B*a^2-25*B*b^2)*(a+b*cos(d*x+c) 
)^(1/2)*sin(d*x+c)/b^2/d+2/35*(7*A*b-4*B*a)*(a+b*cos(d*x+c))^(3/2)*sin(d*x 
+c)/b^2/d+2/7*B*cos(d*x+c)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\frac {4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (49 a A b+2 a^2 B+25 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-14 a^2 A b+63 A b^3+8 a^3 B+19 a b^2 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (28 a A b-16 a^2 B+115 b^2 B\right ) \sin (c+d x)+3 b (2 (7 A b+a B) \sin (2 (c+d x))+5 b B \sin (3 (c+d x)))\right )}{210 b^3 d \sqrt {a+b \cos (c+d x)}} \] Input:

Integrate[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]
 

Output:

(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(49*a*A*b + 2*a^2*B + 25*b^2*B) 
*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-14*a^2*A*b + 63*A*b^3 + 8*a^3*B 
 + 19*a*b^2*B)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*Elliptic 
F[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((28*a*A*b - 16*a 
^2*B + 115*b^2*B)*Sin[c + d*x] + 3*b*(2*(7*A*b + a*B)*Sin[2*(c + d*x)] + 5 
*b*B*Sin[3*(c + d*x)])))/(210*b^3*d*Sqrt[a + b*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 3469, 27, 3042, 3502, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\)

\(\Big \downarrow \) 3469

\(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left ((7 A b-4 a B) \cos ^2(c+d x)+5 b B \cos (c+d x)+2 a B\right )dx}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {a+b \cos (c+d x)} \left ((7 A b-4 a B) \cos ^2(c+d x)+5 b B \cos (c+d x)+2 a B\right )dx}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left ((7 A b-4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+5 b B \sin \left (c+d x+\frac {\pi }{2}\right )+2 a B\right )dx}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {2 \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (b (21 A b-2 a B)-\left (-8 B a^2+14 A b a-25 b^2 B\right ) \cos (c+d x)\right )dx}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \sqrt {a+b \cos (c+d x)} \left (b (21 A b-2 a B)-\left (-8 B a^2+14 A b a-25 b^2 B\right ) \cos (c+d x)\right )dx}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (21 A b-2 a B)+\left (8 B a^2-14 A b a+25 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {\frac {2}{3} \int \frac {b \left (2 B a^2+49 A b a+25 b^2 B\right )-\left (-8 B a^3+14 A b a^2-19 b^2 B a-63 A b^3\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{3} \int \frac {b \left (2 B a^2+49 A b a+25 b^2 B\right )-\left (-8 B a^3+14 A b a^2-19 b^2 B a-63 A b^3\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {1}{3} \int \frac {b \left (2 B a^2+49 A b a+25 b^2 B\right )+\left (8 B a^3-14 A b a^2+19 b^2 B a+63 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (-8 a^3 B+14 a^2 A b-19 a b^2 B-63 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^3 B+14 a^2 A b-19 a b^2 B-63 A b^3\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^3 B+14 a^2 A b-19 a b^2 B-63 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^3 B+14 a^2 A b-19 a b^2 B-63 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-8 a^3 B+14 a^2 A b-19 a b^2 B-63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^3 B+14 a^2 A b-19 a b^2 B-63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^3 B+14 a^2 A b-19 a b^2 B-63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^3 B+14 a^2 A b-19 a b^2 B-63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (-8 a^2 B+14 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

Input:

Int[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]
 

Output:

(2*B*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d) + ((2*( 
7*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d) + (((-2*(1 
4*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Elli 
pticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)] 
) + (2*(a^2 - b^2)*(14*a*A*b - 8*a^2*B - 25*b^2*B)*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 - (2*(14*a*A*b - 8*a^2*B - 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*S 
in[c + d*x])/(3*d))/(5*b))/(7*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 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]
 

rule 3469

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^( 
n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Sin[e + 
 f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*( 
m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m 
+ n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin 
[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c 
 - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !(IGt 
Q[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

rule 3502

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1304\) vs. \(2(288)=576\).

Time = 16.41 (sec) , antiderivative size = 1305, normalized size of antiderivative = 4.31

method	result	size
default	\(\text {Expression too large to display}\)	\(1305\)
parts	\(\text {Expression too large to display}\)	\(1494\)

Input:

int(cos(d*x+c)^2*(a+cos(d*x+c)*b)^(1/2)*(A+B*cos(d*x+c)),x,method=_RETURNV 
ERBOSE)
 

Output:

-2/105*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*B* 
cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8*b^4+(-168*A*b^4-144*B*a*b^3-360*B* 
b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(112*A*a*b^3+168*A*b^4-4*B*a^ 
2*b^2+144*B*a*b^3+280*B*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-14* 
A*a^2*b^2-56*A*a*b^3-42*A*b^4+8*B*a^3*b+2*B*a^2*b^2-86*B*a*b^3-80*B*b^4)*s 
in(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+14*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*( 
-2*b/(a-b)*sin(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))*a^3*b-14*A*a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/ 
(a-b)*sin(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))*b^3-14*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin 
(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*b+14*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(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^2+63*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(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^3-63*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(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^4-8*B 
*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(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))*a^4-17*B*(sin(1/2 
*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.85 \[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)),x, algorith 
m="fricas")
 

Output:

-2/315*(sqrt(1/2)*(-16*I*B*a^4 + 28*I*A*a^3*b - 32*I*B*a^2*b^2 + 21*I*A*a* 
b^3 + 75*I*B*b^4)*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)*(16*I*B*a^4 - 28*I*A*a^3*b + 32*I*B*a^2*b^2 - 21*I*A*a*b^ 
3 - 75*I*B*b^4)*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) + 3*sqrt(1/2)*(-8*I*B*a^3*b + 14*I*A*a^2*b^2 - 19*I*B*a*b^3 - 63*I*A*b^ 
4)*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*sqrt(1/2) 
*(8*I*B*a^3*b - 14*I*A*a^2*b^2 + 19*I*B*a*b^3 + 63*I*A*b^4)*sqrt(b)*weiers 
trassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstras 
sPInverse(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*c 
os(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*(15*B*b^4*cos(d*x + c)^2 - 
 4*B*a^2*b^2 + 7*A*a*b^3 + 25*B*b^4 + 3*(B*a*b^3 + 7*A*b^4)*cos(d*x + c))* 
sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^4*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

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

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a \] Input:

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

Output:

int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*b + int(sqrt(cos(c + d*x)* 
b + a)*cos(c + d*x)**2,x)*a