Integrand size = 27, antiderivative size = 389 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=-\frac {2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^2 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {4 a \left (4 A b^2-\left (a^2-5 b^2\right ) C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{5 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {4 a \left (4 A b^2-a^2 C+5 b^2 C\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac {2 \left (2 a^4 C-3 b^4 (3 A+5 C)-a^2 b^2 (23 A+19 C)\right ) \sin (c+d x)}{15 b \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}} \]
-2/5*(A*b^2+C*a^2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^(5/2)-4/15*a* (4*A*b^2-C*a^2+5*C*b^2)*sin(d*x+c)/b/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(3/2)+ 2/15*(2*a^4*C-3*b^4*(3*A+5*C)-a^2*b^2*(23*A+19*C))*sin(d*x+c)/b/(a^2-b^2)^ 3/d/(a+b*cos(d*x+c))^(1/2)-2/15*(2*a^4*C-3*b^4*(3*A+5*C)-a^2*b^2*(23*A+19* C))*(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^2/(a^2-b^2)^3/d/( (a+b*cos(d*x+c))/(a+b))^(1/2)-4/15*a*(4*A*b^2-(a^2-5*b^2)*C)*(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^2/(a^2-b^2)^2/d/(a+b*cos(d *x+c))^(1/2)
Time = 2.97 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.81 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{5/2} \left (\left (-2 a^4 C+3 b^4 (3 A+5 C)+a^2 b^2 (23 A+19 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+2 a (a-b) \left (-4 A b^2+\left (a^2-5 b^2\right ) C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )}{(a-b)^3}+\frac {b \left (68 a^4 A b^2+13 a^2 A b^4+15 A b^6-2 a^6 C+48 a^4 b^2 C+35 a^2 b^4 C+15 b^6 C-4 a b \left (3 a^4 C-5 b^4 (A+2 C)-a^2 b^2 (27 A+25 C)\right ) \cos (c+d x)+\left (-2 a^4 b^2 C+3 b^6 (3 A+5 C)+a^2 b^4 (23 A+19 C)\right ) \cos (2 (c+d x))\right ) \sin (c+d x)}{2 \left (-a^2+b^2\right )^3}\right )}{15 b^2 d (a+b \cos (c+d x))^{5/2}} \]
(2*((((a + b*Cos[c + d*x])/(a + b))^(5/2)*((-2*a^4*C + 3*b^4*(3*A + 5*C) + a^2*b^2*(23*A + 19*C))*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*a*(a - b )*(-4*A*b^2 + (a^2 - 5*b^2)*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^3 + (b*(68*a^4*A*b^2 + 13*a^2*A*b^4 + 15*A*b^6 - 2*a^6*C + 48*a^4*b^2 *C + 35*a^2*b^4*C + 15*b^6*C - 4*a*b*(3*a^4*C - 5*b^4*(A + 2*C) - a^2*b^2* (27*A + 25*C))*Cos[c + d*x] + (-2*a^4*b^2*C + 3*b^6*(3*A + 5*C) + a^2*b^4* (23*A + 19*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(-a^2 + b^2)^3)))/(15*b^ 2*d*(a + b*Cos[c + d*x])^(5/2))
Time = 1.94 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3501, 27, 3042, 3233, 27, 3042, 3233, 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.
| \(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3501 |
| \(\displaystyle -\frac {2 \int -\frac {5 a b (A+C)-\left (-2 C a^2+3 A b^2+5 b^2 C\right ) \cos (c+d x)}{2 (a+b \cos (c+d x))^{5/2}}dx}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 a b (A+C)-\left (-2 C a^2+3 A b^2+5 b^2 C\right ) \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}}dx}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a b (A+C)+\left (2 C a^2-3 A b^2-5 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {3 b \left ((5 A+3 C) a^2+b^2 (3 A+5 C)\right )-2 a \left (-C a^2+4 A b^2+5 b^2 C\right ) \cos (c+d x)}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 b \left ((5 A+3 C) a^2+b^2 (3 A+5 C)\right )-2 a \left (-C a^2+4 A b^2+5 b^2 C\right ) \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 b \left ((5 A+3 C) a^2+b^2 (3 A+5 C)\right )-2 a \left (-C a^2+4 A b^2+5 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \int -\frac {a b \left ((15 A+7 C) a^2+b^2 (17 A+25 C)\right )-\left (2 C a^4-b^2 (23 A+19 C) a^2-3 b^4 (3 A+5 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a b \left ((15 A+7 C) a^2+b^2 (17 A+25 C)\right )-\left (2 C a^4-b^2 (23 A+19 C) a^2-3 b^4 (3 A+5 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a b \left ((15 A+7 C) a^2+b^2 (17 A+25 C)\right )+\left (-2 C a^4+b^2 (23 A+19 C) a^2+3 b^4 (3 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {2 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}+\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {-\frac {4 a \left (a^2-b^2\right ) \left (a^2 (-C)+4 A b^2+5 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {4 a \left (4 A b^2-C \left (a^2-5 b^2\right )\right ) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{5 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\) |
(-2*(A*b^2 + a^2*C)*Sin[c + d*x])/(5*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^ (5/2)) + ((-4*a*(4*A*b^2 - (a^2 - 5*b^2)*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d *(a + b*Cos[c + d*x])^(3/2)) + (((-2*(2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^ 2*(23*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)]) - (4*a*(a^2 - b^2)*(4*A*b^ 2 - a^2*C + 5*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]]))/(a^2 - b^2) + (2*(2*a ^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sin[c + d*x])/((a^2 - b^ 2)*d*Sqrt[a + b*Cos[c + d*x]]))/(3*(a^2 - b^2)))/(5*b*(a^2 - b^2))
3.7.69.3.1 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[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*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] && LtQ[m, -1] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*((a + b*S in[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^ 2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[a*b*(A + C)*(m + 1) - (A* b^2 + a^2*C + b^2*(A + C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1308\) vs. \(2(423)=846\).
Time = 60.93 (sec) , antiderivative size = 1309, normalized size of antiderivative = 3.37
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1309\) |
| parts | \(\text {Expression too large to display}\) | \(1917\) |
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*C/b^2/s in(1/2*d*x+1/2*c)^2/(2*b*sin(1/2*d*x+1/2*c)^2-a-b)/(a^2-b^2)*(-2*sin(1/2*d *x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)*sin( 1/2*d*x+1/2*c)^2*b+EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*(-2*b/ (a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2) *a-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*(-2*b/(a-b)*sin(1/2*d* x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*b)-4*a*C/b^2*(1 /6/b/(a-b)/(a+b)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1 /2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2*(a-b)/b)^2+8/3*sin(1/2*d* x+1/2*c)^2*b/(a-b)^2/(a+b)^2*cos(1/2*d*x+1/2*c)*a/(-(-2*b*cos(1/2*d*x+1/2* c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+(3*a-b)/(3*a^3+3*a^2*b-3*a*b^2-3*b^3 )*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2 )/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(c os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-4/3*a/(a-b)/(a+b)^2*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+ 1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c) ,(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))))+2* (A*b^2+C*a^2)/b^2*(1/20/b^2/(a-b)/(a+b)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x +1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2*(a -b)/b)^3+4/15*a/b/(a-b)^2/(a+b)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 1407, normalized size of antiderivative = 3.62 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
1/45*(6*(C*a^6*b^2 - (34*A + 25*C)*a^4*b^4 + (5*A - 8*C)*a^2*b^6 - 3*A*b^8 + (2*C*a^4*b^4 - (23*A + 19*C)*a^2*b^6 - 3*(3*A + 5*C)*b^8)*cos(d*x + c)^ 2 + 2*(3*C*a^5*b^3 - (27*A + 25*C)*a^3*b^5 - 5*(A + 2*C)*a*b^7)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c) - (sqrt(2)*(4*I*C*a^5*b^3 - I*(A + 17*C)*a^3*b^5 + 3*I*(11*A + 15*C)*a*b^7)*cos(d*x + c)^3 - 3*sqrt(2)*(-4 *I*C*a^6*b^2 + I*(A + 17*C)*a^4*b^4 - 3*I*(11*A + 15*C)*a^2*b^6)*cos(d*x + c)^2 - 3*sqrt(2)*(-4*I*C*a^7*b + I*(A + 17*C)*a^5*b^3 - 3*I*(11*A + 15*C) *a^3*b^5)*cos(d*x + c) + sqrt(2)*(4*I*C*a^8 - I*(A + 17*C)*a^6*b^2 + 3*I*( 11*A + 15*C)*a^4*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)*(-4*I*C*a^5*b^3 + I*(A + 17*C)*a^3*b^5 - 3*I*(11*A + 15*C)*a*b^7)*cos(d*x + c)^3 - 3*sqrt(2)*(4*I*C*a^6*b^2 - I*(A + 17*C)*a^4* b^4 + 3*I*(11*A + 15*C)*a^2*b^6)*cos(d*x + c)^2 - 3*sqrt(2)*(4*I*C*a^7*b - I*(A + 17*C)*a^5*b^3 + 3*I*(11*A + 15*C)*a^3*b^5)*cos(d*x + c) + sqrt(2)* (-4*I*C*a^8 + I*(A + 17*C)*a^6*b^2 - 3*I*(11*A + 15*C)*a^4*b^4))*sqrt(b)*w eierstrassPInverse(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)*(-2*I*C*a ^4*b^4 + I*(23*A + 19*C)*a^2*b^6 + 3*I*(3*A + 5*C)*b^8)*cos(d*x + c)^3 + 3 *sqrt(2)*(-2*I*C*a^5*b^3 + I*(23*A + 19*C)*a^3*b^5 + 3*I*(3*A + 5*C)*a*b^7 )*cos(d*x + c)^2 + 3*sqrt(2)*(-2*I*C*a^6*b^2 + I*(23*A + 19*C)*a^4*b^4 ...
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]