\(\int \cos (c+d x) \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [1015]

Optimal result

Integrand size = 41, antiderivative size = 321 \[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (14 a^2 b B-63 b^3 B-8 a^3 C-a b^2 (35 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (35 A b^2-14 a b B+8 a^2 C+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (35 A b^2-14 a b B+8 a^2 C+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}+\frac {2 (7 b B-4 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d} \] Output:

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

Mathematica [A] (verified)

Time = 1.78 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (35 A b^2+49 a b B+2 a^2 C+25 b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-14 a^2 b B+63 b^3 B+8 a^3 C+a b^2 (35 A+19 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (140 A b^2+28 a b B-16 a^2 C+115 b^2 C\right ) \sin (c+d x)+3 b (2 (7 b B+a C) \sin (2 (c+d x))+5 b C \sin (3 (c+d x)))\right )}{210 b^3 d \sqrt {a+b \cos (c+d x)}} \] Input:

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

Output:

(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(35*A*b^2 + 49*a*b*B + 2*a^2*C 
+ 25*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-14*a^2*b*B + 63*b^3* 
B + 8*a^3*C + a*b^2*(35*A + 19*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/( 
a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x 
])*((140*A*b^2 + 28*a*b*B - 16*a^2*C + 115*b^2*C)*Sin[c + d*x] + 3*b*(2*(7 
*b*B + a*C)*Sin[2*(c + d*x)] + 5*b*C*Sin[3*(c + d*x)])))/(210*b^3*d*Sqrt[a 
 + b*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {3042, 3528, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {2 \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (b (21 b B-2 a C)+\left (8 C a^2-14 b B a+35 A b^2+25 b^2 C\right ) \cos (c+d x)\right )dx}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \sqrt {a+b \cos (c+d x)} \left (b (21 b B-2 a C)+\left (8 C a^2-14 b B a+35 A b^2+25 b^2 C\right ) \cos (c+d x)\right )dx}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (21 b B-2 a C)+\left (8 C a^2-14 b B a+35 A b^2+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {\frac {2}{3} \int \frac {b \left (2 C a^2+49 b B a+35 A b^2+25 b^2 C\right )-\left (-8 C a^3+14 b B a^2-b^2 (35 A+19 C) a-63 b^3 B\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{3} \int \frac {b \left (2 C a^2+49 b B a+35 A b^2+25 b^2 C\right )-\left (-8 C a^3+14 b B a^2-b^2 (35 A+19 C) a-63 b^3 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {1}{3} \int \frac {b \left (2 C a^2+49 b B a+35 A b^2+25 b^2 C\right )+\left (8 C a^3-14 b B a^2+b^2 (35 A+19 C) a+63 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {1}{3} \left (-\frac {\left (a^2-b^2\right ) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {2 \sin (c+d x) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (-\frac {2 \left (a^2-b^2\right ) \left (8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )}{5 b}+\frac {2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

Input:

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

Output:

(2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d) + ((2*( 
7*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d) + (((-2*(1 
4*a^2*b*B - 63*b^3*B - 8*a^3*C - a*b^2*(35*A + 19*C))*Sqrt[a + b*Cos[c + d 
*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x]) 
/(a + b)]) - (2*(a^2 - b^2)*(35*A*b^2 - 14*a*b*B + 8*a^2*C + 25*b^2*C)*Sqr 
t[(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*(35*A*b^2 - 14*a*b*B + 8*a^2*C + 25*b^ 
2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[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 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]
 

rule 3528

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1634\) vs. \(2(306)=612\).

Time = 13.34 (sec) , antiderivative size = 1635, normalized size of antiderivative = 5.09

method	result	size
default	\(\text {Expression too large to display}\)	\(1635\)
parts	\(\text {Expression too large to display}\)	\(1946\)

Input:

int(cos(d*x+c)*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,me 
thod=_RETURNVERBOSE)
 

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*C* 
b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(-168*B*b^4-144*C*a*b^3-360*C* 
b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(140*A*b^4+112*B*a*b^3+168*B* 
b^4-4*C*a^2*b^2+144*C*a*b^3+280*C*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 
2*c)+(-70*A*a*b^3-70*A*b^4-14*B*a^2*b^2-56*B*a*b^3-42*B*b^4+8*C*a^3*b+2*C* 
a^2*b^2-86*C*a*b^3-80*C*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-35*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^2*b^2+35*A*b^4*( 
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))+35*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)*Ellipti 
cE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-35*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(co 
s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^3+14*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^3*b-14*B*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*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)*EllipticE(cos(1/2*d*x+1/2*c),(-2* 
b/(a-b))^(1/2))*a^3*b+14*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.88 \[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

-2/315*(sqrt(1/2)*(-16*I*C*a^4 + 28*I*B*a^3*b - 2*I*(35*A + 16*C)*a^2*b^2 
+ 21*I*B*a*b^3 + 15*I*(7*A + 5*C)*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*C*a^4 - 28*I*B*a^3*b + 2*I*(35 
*A + 16*C)*a^2*b^2 - 21*I*B*a*b^3 - 15*I*(7*A + 5*C)*b^4)*sqrt(b)*weierstr 
assPInverse(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*C*a^3*b + 
 14*I*B*a^2*b^2 - I*(35*A + 19*C)*a*b^3 - 63*I*B*b^4)*sqrt(b)*weierstrassZ 
eta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInve 
rse(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*C*a^3*b - 14*I*B* 
a^2*b^2 + I*(35*A + 19*C)*a*b^3 + 63*I*B*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*(15*C*b^4*cos(d*x + c)^2 - 4*C*a^2*b^2 + 
 7*B*a*b^3 + 5*(7*A + 5*C)*b^4 + 3*(C*a*b^3 + 7*B*b^4)*cos(d*x + c))*sqrt( 
b*cos(d*x + c) + a)*sin(d*x + c))/(b^4*d)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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