Integrand size = 33, antiderivative size = 320 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \left (56 a^2 A b+63 A b^3-48 a^3 B-44 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (56 a^3 A b+49 a A b^3-48 a^4 B-32 a^2 b^2 B-25 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^4 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (28 a A b-24 a^2 B-25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^3 d}+\frac {2 (7 A b-6 a B) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^2 d}+\frac {2 B \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b d} \]
-2/105*(28*A*a*b-24*B*a^2-25*B*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^3/ d+2/35*(7*A*b-6*B*a)*cos(d*x+c)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2/d+2/ 7*B*cos(d*x+c)^2*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b/d+2/105*(56*A*a^2*b+6 3*A*b^3-48*B*a^3-44*B*a*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^4/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/105*(56*A*a^3*b+49*A*a*b^3-48 *B*a^4-32*B*a^2*b^2-25*B*b^4)*(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^4/d/(a+b*cos(d*x+c))^(1/2)
Time = 1.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (14 a A b-12 a^2 B+25 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-56 a^2 A b-63 A b^3+48 a^3 B+44 a b^2 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+2 b (a+b \cos (c+d x)) \left (-56 a A b+48 a^2 B+65 b^2 B+6 b (7 A b-6 a B) \cos (c+d x)+15 b^2 B \cos (2 (c+d x))\right ) \sin (c+d x)}{210 b^4 d \sqrt {a+b \cos (c+d x)}} \]
(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(14*a*A*b - 12*a^2*B + 25*b^2*B )*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-56*a^2*A*b - 63*A*b^3 + 48*a^3 *B + 44*a*b^2*B)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*Ellipt icF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(-56*a*A*b + 48*a^2*B + 65*b^2*B + 6*b*(7*A*b - 6*a*B)*Cos[c + d*x] + 15*b^2*B*Cos[2*(c + d*x)])*Sin[c + d*x])/(210*b^4*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.72 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.06, 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, 3528, 27, 3042, 3502, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3469 |
| \(\displaystyle \frac {2 \int \frac {\cos (c+d x) \left ((7 A b-6 a B) \cos ^2(c+d x)+5 b B \cos (c+d x)+4 a B\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) \left ((7 A b-6 a B) \cos ^2(c+d x)+5 b B \cos (c+d x)+4 a B\right )}{\sqrt {a+b \cos (c+d x)}}dx}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left ((7 A b-6 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+5 b B \sin \left (c+d x+\frac {\pi }{2}\right )+4 a B\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int \frac {-\left (\left (-24 B a^2+28 A b a-25 b^2 B\right ) \cos ^2(c+d x)\right )+b (21 A b+2 a B) \cos (c+d x)+2 a (7 A b-6 a B)}{2 \sqrt {a+b \cos (c+d x)}}dx}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-\left (\left (-24 B a^2+28 A b a-25 b^2 B\right ) \cos ^2(c+d x)\right )+b (21 A b+2 a B) \cos (c+d x)+2 a (7 A b-6 a B)}{\sqrt {a+b \cos (c+d x)}}dx}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (24 B a^2-28 A b a+25 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (21 A b+2 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a (7 A b-6 a B)}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {b \left (-12 B a^2+14 A b a+25 b^2 B\right )+\left (-48 B a^3+56 A b a^2-44 b^2 B a+63 A b^3\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b \left (-12 B a^2+14 A b a+25 b^2 B\right )+\left (-48 B a^3+56 A b a^2-44 b^2 B a+63 A b^3\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b \left (-12 B a^2+14 A b a+25 b^2 B\right )+\left (-48 B a^3+56 A b a^2-44 b^2 B a+63 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (-48 a^3 B+56 a^2 A b-44 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (-24 a^2 B+28 a A b-25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}+\frac {2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}+\frac {2 B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}\) |
(2*B*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d) + ((2*( 7*A*b - 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d) + (((2*(56*a^2*A*b + 63*A*b^3 - 48*a^3*B - 44*a*b^2*B)*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*(56*a^3*A*b + 49*a*A*b^3 - 48*a^4*B - 32*a^2*b^2*B - 25* b^4*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*b) - (2*(28*a*A*b - 24*a^2*B - 2 5*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d))/(5*b))/(7*b)
3.4.19.3.1 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_)*((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])))
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_.) + (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])))
Leaf count of result is larger than twice the leaf count of optimal. \(1304\) vs. \(2(354)=708\).
Time = 13.14 (sec) , antiderivative size = 1305, normalized size of antiderivative = 4.08
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1305\) |
| parts | \(\text {Expression too large to display}\) | \(1494\) |
-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+24*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)+(-28*A*a*b^3+168*A*b^4+24*B*a^ 2*b^2-24*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)+(56*A* a^2*b^2+14*A*a*b^3-42*A*b^4-48*B*a^3*b-12*B*a^2*b^2-44*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)-56*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-49*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+56*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-56*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+48* 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+32*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/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.76 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {\sqrt {2} {\left (-96 i \, B a^{4} + 112 i \, A a^{3} b - 52 i \, B a^{2} b^{2} + 84 i \, A a b^{3} - 75 i \, B 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 (96 i \, B a^{4} - 112 i \, A a^{3} b + 52 i \, B a^{2} b^{2} - 84 i \, A a b^{3} + 75 i \, B 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 (48 i \, B a^{3} b - 56 i \, A a^{2} b^{2} + 44 i \, B a b^{3} - 63 i \, A b^{4}\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 (-48 i \, B a^{3} b + 56 i \, A a^{2} b^{2} - 44 i \, B a b^{3} + 63 i \, A b^{4}\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 \, B b^{4} \cos \left (d x + c\right )^{2} + 24 \, B a^{2} b^{2} - 28 \, A a b^{3} + 25 \, B b^{4} - 3 \, {\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, b^{5} d} \]
1/315*(sqrt(2)*(-96*I*B*a^4 + 112*I*A*a^3*b - 52*I*B*a^2*b^2 + 84*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(2)*(96*I*B*a^4 - 112*I*A*a^3*b + 52*I*B*a^2*b^2 - 84*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(2)*(48*I*B*a^3*b - 56*I*A*a^2*b^2 + 44*I*B*a*b^3 - 63*I*A*b^4)*sq rt(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)*(-48*I *B*a^3*b + 56*I*A*a^2*b^2 - 44*I*B*a*b^3 + 63*I*A*b^4)*sqrt(b)*weierstrass Zeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInv erse(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*B*b^4*cos(d*x + c)^2 + 24*B *a^2*b^2 - 28*A*a*b^3 + 25*B*b^4 - 3*(6*B*a*b^3 - 7*A*b^4)*cos(d*x + c))*s qrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^5*d)
Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]