Integrand size = 33, antiderivative size = 291 \[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a \left (35 A b^2+8 a^2 C+19 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (35 A b^2+8 a^2 C+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d} \]
-8/35*a*C*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d+2/7*C*cos(d*x+c)*(a+b*co s(d*x+c))^(3/2)*sin(d*x+c)/b/d+2/105*(8*a^2*C+5*b^2*(7*A+5*C))*sin(d*x+c)* (a+b*cos(d*x+c))^(1/2)/b^2/d+2/105*a*(35*A*b^2+8*C*a^2+19*C*b^2)*(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^(1/2 )*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(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+8*C*a^2+25*C*b^2)*(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^(1/2)*(b/(a+b))^(1/2 ))*((a+b*cos(d*x+c))/(a+b))^(1/2)/b^3/d/(a+b*cos(d*x+c))^(1/2)
Time = 1.23 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.74 \[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (35 A b^3+2 a^2 b C+25 b^3 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+a \left (35 A b^2+8 a^2 C+19 b^2 C\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+2 b (a+b \cos (c+d x)) \left (70 A b^2-8 a^2 C+65 b^2 C+6 a b C \cos (c+d x)+15 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)}{210 b^3 d \sqrt {a+b \cos (c+d x)}} \]
(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(35*A*b^3 + 2*a^2*b*C + 25*b^3*C) *EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*(35*A*b^2 + 8*a^2*C + 19*b^2*C) *((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(70*A*b^2 - 8*a^2*C + 65*b^2* C + 6*a*b*C*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(210*b ^3*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.60 (sec) , antiderivative size = 305, 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, 3529, 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.
| \(\displaystyle \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (-4 a C \cos ^2(c+d x)+b (7 A+5 C) \cos (c+d x)+2 a C\right )dx}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {a+b \cos (c+d x)} \left (-4 a C \cos ^2(c+d x)+b (7 A+5 C) \cos (c+d x)+2 a C\right )dx}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (-4 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (7 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a C\right )dx}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int -\frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (2 a b C-\left (8 C a^2+5 b^2 (7 A+5 C)\right ) \cos (c+d x)\right )dx}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \sqrt {a+b \cos (c+d x)} \left (2 a b C-\left (8 C a^2+5 b^2 (7 A+5 C)\right ) \cos (c+d x)\right )dx}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (2 a b C+\left (-8 C a^2-5 b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {\frac {2}{3} \int -\frac {b \left (2 C a^2+35 A b^2+25 b^2 C\right )+a \left (8 C a^2+35 A b^2+19 b^2 C\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {1}{3} \int \frac {b \left (2 C a^2+35 A b^2+25 b^2 C\right )+a \left (8 C a^2+35 A b^2+19 b^2 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {1}{3} \int \frac {b \left (2 C a^2+35 A b^2+25 b^2 C\right )+a \left (8 C a^2+35 A b^2+19 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^2 C+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {a \left (8 a^2 C+35 A b^2+19 b^2 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^2 C+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^2 C+35 A b^2+19 b^2 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^2 C+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^2 C+35 A b^2+19 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^2 C+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^2 C+35 A b^2+19 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^2 C+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^2 C+35 A b^2+19 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^2 C+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^2 C+35 A b^2+19 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^2 C+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^2 C+35 A b^2+19 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (8 a^2 C+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^2 C+35 A b^2+19 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2 C+5 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\) |
(2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d) + ((-8* a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d) - (((-2*a*(35*A*b^2 + 8*a^2*C + 19*b^2*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 + 8*a^2*C + 25*b^2*C)*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*(8*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/(5*b)) /(7*b)
3.7.24.3.1 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[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_.)*((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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, 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])))
Leaf count of result is larger than twice the leaf count of optimal. \(1130\) vs. \(2(325)=650\).
Time = 18.86 (sec) , antiderivative size = 1131, normalized size of antiderivative = 3.89
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1131\) |
| parts | \(\text {Expression too large to display}\) | \(1281\) |
-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* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8*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^4-4*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+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+3 5*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 )*EllipticE(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)*Ell ipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^3-8*C*(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(c os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4-17*C*(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+25*C*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))+8*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*s in(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.82 \[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (16 i \, C a^{4} + 2 i \, {\left (35 \, A + 16 \, C\right )} a^{2} b^{2} - 15 i \, {\left (7 \, A + 5 \, C\right )} b^{4}\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 {2} {\left (-16 i \, C a^{4} - 2 i \, {\left (35 \, A + 16 \, C\right )} a^{2} b^{2} + 15 i \, {\left (7 \, A + 5 \, C\right )} b^{4}\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 {2} {\left (-8 i \, C a^{3} b - i \, {\left (35 \, A + 19 \, C\right )} a 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 {2} {\left (8 i \, C a^{3} b + i \, {\left (35 \, A + 19 \, C\right )} a 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 ) + 6 \, {\left (15 \, C b^{4} \cos \left (d x + c\right )^{2} + 3 \, C a b^{3} \cos \left (d x + c\right ) - 4 \, C a^{2} b^{2} + 5 \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, b^{4} d} \]
1/315*(sqrt(2)*(16*I*C*a^4 + 2*I*(35*A + 16*C)*a^2*b^2 - 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(2 )*(-16*I*C*a^4 - 2*I*(35*A + 16*C)*a^2*b^2 + 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) - 3*sqrt(2)*(-8*I*C* a^3*b - I*(35*A + 19*C)*a*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*sqrt(2)*(8*I*C*a^3*b + I*(35*A + 19*C)*a*b^3)*sqrt(b)*w eierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 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)) + 6*(15*C*b^4*cos(d*x + c )^2 + 3*C*a*b^3*cos(d*x + c) - 4*C*a^2*b^2 + 5*(7*A + 5*C)*b^4)*sqrt(b*cos (d*x + c) + a)*sin(d*x + c))/(b^4*d)
\[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sqrt {a + b \cos {\left (c + d x \right )}} \cos {\left (c + d x \right )}\, dx \]
\[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right ) \,d x } \]
\[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos (c+d x) \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \]