Integrand size = 35, antiderivative size = 392 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=-\frac {4 a \left (a^2 b^2 (A-14 C)-b^4 (3 A-4 C)+8 a^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (2 a^2 b^2 (A-8 C)+16 a^4 C-b^4 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^4 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {4 a \left (2 A b^4-3 a^4 C+5 a^2 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (A b^2+2 a^2 C-b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d} \] Output:
-4/3*a*(a^2*b^2*(A-14*C)-b^4*(3*A-4*C)+8*a^4*C)*(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^4/(a^2-b^2)^2/d/((a+b *cos(d*x+c))/(a+b))^(1/2)+2/3*(2*a^2*b^2*(A-8*C)+16*a^4*C-b^4*(3*A+C))*((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^4/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)-2/3*(A*b^2+C*a^2)*cos(d*x+ c)^2*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^(3/2)-4/3*a*(2*A*b^4-3*C*a^ 4+5*C*a^2*b^2)*sin(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)+2/3*(A* b^2+2*C*a^2-C*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^3/(a^2-b^2)/d
Time = 4.03 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b \left (-4 a^4 b C+b^5 (3 A+C)+a^2 b^3 (A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-2 a \left (a^2 b^2 (A-14 C)+8 a^4 C+b^4 (-3 A+4 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}+\frac {b \left (2 a^4 A b^2-10 a^2 A b^4+16 a^6 C-25 a^4 b^2 C+b^6 C+4 a b \left (a^2 b^2 (A-8 C)+5 a^4 C+b^4 (-3 A+C)\right ) \cos (c+d x)+\left (-a^2 b+b^3\right )^2 C \cos (2 (c+d x))\right ) \sin (c+d x)}{2 \left (a^2-b^2\right )^2}\right )}{3 b^4 d (a+b \cos (c+d x))^{3/2}} \] Input:
Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/ 2),x]
Output:
(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b*(-4*a^4*b*C + b^5*(3*A + C) + a^2*b^3*(A + 7*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(a^2*b^2*( A - 14*C) + 8*a^4*C + b^4*(-3*A + 4*C))*((a + b)*EllipticE[(c + d*x)/2, (2 *b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + (b*(2*a^4*A*b^2 - 10*a^2*A*b^4 + 16*a^6*C - 25*a^4*b^2*C + b^6*C + 4 *a*b*(a^2*b^2*(A - 8*C) + 5*a^4*C + b^4*(-3*A + C))*Cos[c + d*x] + (-(a^2* b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(a^2 - b^2)^2)))/(3*b^4*d *(a + b*Cos[c + d*x])^(3/2))
Time = 2.24 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {3042, 3527, 27, 3042, 3510, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle -\frac {2 \int \frac {\cos (c+d x) \left (-3 \left (2 C a^2+A b^2-b^2 C\right ) \cos ^2(c+d x)-3 a b (A+C) \cos (c+d x)+4 \left (C a^2+A b^2\right )\right )}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x) \left (-3 \left (2 C a^2+A b^2-b^2 C\right ) \cos ^2(c+d x)-3 a b (A+C) \cos (c+d x)+4 \left (C a^2+A b^2\right )\right )}{(a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (-3 \left (2 C a^2+A b^2-b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-3 a b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 \left (C a^2+A b^2\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {\frac {2 \int -\frac {3 b \left (a^2-b^2\right ) \left (2 C a^2+A b^2-b^2 C\right ) \cos ^2(c+d x)+2 a \left (2 A b^4-\left (6 a^4-11 b^2 a^2+3 b^4\right ) C\right ) \cos (c+d x)+2 b \left (-3 C a^4+5 b^2 C a^2+2 A b^4\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{b^2 \left (a^2-b^2\right )}+\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {3 b \left (a^2-b^2\right ) \left (2 C a^2+A b^2-b^2 C\right ) \cos ^2(c+d x)+2 a \left (2 A b^4-\left (6 a^4-11 b^2 a^2+3 b^4\right ) C\right ) \cos (c+d x)+2 b \left (-3 C a^4+5 b^2 C a^2+2 A b^4\right )}{\sqrt {a+b \cos (c+d x)}}dx}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\int \frac {3 b \left (a^2-b^2\right ) \left (2 C a^2+A b^2-b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (2 A b^4-\left (6 a^4-11 b^2 a^2+3 b^4\right ) C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 b \left (-3 C a^4+5 b^2 C a^2+2 A b^4\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \int -\frac {3 \left (\left (4 C a^4-b^2 (A+7 C) a^2-b^4 (3 A+C)\right ) b^2+2 a \left (8 C a^4+b^2 (A-14 C) a^2-b^4 (3 A-4 C)\right ) \cos (c+d x) b\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}+\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {\int \frac {\left (4 C a^4-b^2 (A+7 C) a^2-b^4 (3 A+C)\right ) b^2+2 a \left (8 C a^4+b^2 (A-14 C) a^2-b^4 (3 A-4 C)\right ) \cos (c+d x) b}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {\int \frac {\left (4 C a^4-b^2 (A+7 C) a^2-b^4 (3 A+C)\right ) b^2+2 a \left (8 C a^4+b^2 (A-14 C) a^2-b^4 (3 A-4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {2 a \left (8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx-\left (a^2-b^2\right ) \left (16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {2 a \left (8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\left (a^2-b^2\right ) \left (16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {\frac {2 a \left (8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {\frac {2 a \left (8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {\frac {4 a \left (8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {\frac {4 a \left (8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {\frac {4 a \left (8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\frac {4 a \left (-3 a^4 C+5 a^2 b^2 C+2 A b^4\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \left (2 a^2 C+A b^2-b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}-\frac {\frac {4 a \left (8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}}{b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]
Output:
(-2*(A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b *Cos[c + d*x])^(3/2)) - ((4*a*(2*A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*Sin[c + d* x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (-(((4*a*(a^2*b^2*(A - 14*C) - b^4*(3*A - 4*C) + 8*a^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(2*a^2*b^2*(A - 8*C) + 16*a^4*C - b^4*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[ c + d*x]]))/b) + (2*(a^2 - b^2)*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/d)/(b^2*(a^2 - b^2)))/(3*b*(a^2 - b^2))
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_.)*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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 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^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1326\) vs. \(2(377)=754\).
Time = 13.54 (sec) , antiderivative size = 1327, normalized size of antiderivative = 3.39
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1327\) |
| parts | \(\text {Expression too large to display}\) | \(2147\) |
Input:
int(cos(d*x+c)^2*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x,method=_RETUR NVERBOSE)
Output:
-(-(-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*(A*b^2+3 *C*a^2+2*C*a*b+C*b^2)/b^4*(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*b*sin(1/2*d*x+1/2*c)^4+(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))+2*a^2*(A*b^2 +C*a^2)/b^4*(1/6/b/(a-b)/(a+b)*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c) ^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2/b*(a-b))^2+ 8/3*b*sin(1/2*d*x+1/2*c)^2/(a-b)^2/(a+b)^2*cos(1/2*d*x+1/2*c)*a/(-(-2*b*co s(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*b*sin(1/2*d*x+1/2*c)^4+(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))-4/3*a/(a-b)/(a+b)^2*(s in(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*b*sin(1/2*d*x+1/2*c)^4+(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))))+8*C/b^2*(-1/6/b*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+ (a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+1/6/b*(a-b)*(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*b*sin(1/2*d*x+1/2*c)^4+( 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))-1/12/b^2*(-2*a+6*b)*(a-b)*(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*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(...
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 1138, normalized size of antiderivative = 2.90 \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x, algori thm="fricas")
Output:
-2/9*(sqrt(1/2)*(32*I*C*a^8 + 4*I*(A - 17*C)*a^6*b^2 - I*(9*A - 37*C)*a^4* b^4 + 3*I*(3*A + C)*a^2*b^6 + (32*I*C*a^6*b^2 + 4*I*(A - 17*C)*a^4*b^4 - I *(9*A - 37*C)*a^2*b^6 + 3*I*(3*A + C)*b^8)*cos(d*x + c)^2 + 2*(32*I*C*a^7* b + 4*I*(A - 17*C)*a^5*b^3 - I*(9*A - 37*C)*a^3*b^5 + 3*I*(3*A + C)*a*b^7) *cos(d*x + c))*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)*(-32*I*C*a^8 - 4*I*(A - 17*C)*a^6*b^2 + I*(9*A - 37*C)*a^4*b ^4 - 3*I*(3*A + C)*a^2*b^6 + (-32*I*C*a^6*b^2 - 4*I*(A - 17*C)*a^4*b^4 + I *(9*A - 37*C)*a^2*b^6 - 3*I*(3*A + C)*b^8)*cos(d*x + c)^2 + 2*(-32*I*C*a^7 *b - 4*I*(A - 17*C)*a^5*b^3 + I*(9*A - 37*C)*a^3*b^5 - 3*I*(3*A + C)*a*b^7 )*cos(d*x + c))*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) + 6*sqrt(1/2)*(8*I*C*a^7*b + I*(A - 14*C)*a^5*b^3 - I*(3*A - 4*C)*a^3*b ^5 + (8*I*C*a^5*b^3 + I*(A - 14*C)*a^3*b^5 - I*(3*A - 4*C)*a*b^7)*cos(d*x + c)^2 + 2*(8*I*C*a^6*b^2 + I*(A - 14*C)*a^4*b^4 - I*(3*A - 4*C)*a^2*b^6)* cos(d*x + c))*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)) + 6*sqrt(1/2)*(-8*I*C*a^7*b - I*(A - 14*C)*a^5*b^3 + I*(3*A - 4*C)*a^3*b^5 + (-8*I*C*a^5*b^3 - I*(A - 14*C)*a^3*b^5 + I*(3*A - 4*C)*a*b^7)*cos(d*x ...
Timed out. \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*(A+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^2*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*cos(d*x + c)^2/(b*cos(d*x + c) + a)^(5/2) , x)
\[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^2*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*cos(d*x + c)^2/(b*cos(d*x + c) + a)^(5/2) , x)
Timed out. \[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int((cos(c + d*x)^2*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(5/2),x)
Output:
int((cos(c + d*x)^2*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(5/2), x)
\[ \int \frac {\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) a \] Input:
int(cos(d*x+c)^2*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4)/(cos(c + d*x)**3*b**3 + 3*c os(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*c + int((sqrt(cos (c + d*x)*b + a)*cos(c + d*x)**2)/(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)** 2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*a