Integrand size = 33, antiderivative size = 413 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (8 a^4 A b-15 a^2 A b^3+3 A b^5-16 a^5 B+28 a^3 b^2 B-8 a b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (8 a^3 A b-9 a A b^3-16 a^4 B+16 a^2 b^2 B+b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^4 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {2 a^2 \left (3 a^2 A b-7 A b^3-6 a^3 B+10 a b^2 B\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (a A b-2 a^2 B+b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d} \]
2/3*a*(A*b-B*a)*cos(d*x+c)^2*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^(3/ 2)-2/3*a^2*(3*A*a^2*b-7*A*b^3-6*B*a^3+10*B*a*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*a*b-2*B*a^2+B*b^2)*sin(d*x+c)*(a+b*cos( d*x+c))^(1/2)/b^3/(a^2-b^2)/d+2/3*(8*A*a^4*b-15*A*a^2*b^3+3*A*b^5-16*B*a^5 +28*B*a^3*b^2-8*B*a*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*E llipticE(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/(a^2-b^2)^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/3*(8*A*a^3*b-9*A*a*b^ 3-16*B*a^4+16*B*a^2*b^2+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/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)
Time = 2.48 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (2 a^3 A b-6 a A b^3-4 a^4 B+7 a^2 b^2 B+b^4 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-8 a^4 A b+15 a^2 A b^3-3 A b^5+16 a^5 B-28 a^3 b^2 B+8 a b^4 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}+\frac {b \left (-8 a^5 A b+16 a^3 A b^3+16 a^6 B-25 a^4 b^2 B+b^6 B+2 a b \left (-5 a^3 A b+9 a A b^3+10 a^4 B-16 a^2 b^2 B+2 b^4 B\right ) \cos (c+d x)+\left (-a^2 b+b^3\right )^2 B \cos (2 (c+d x))\right ) \sin (c+d x)}{2 \left (a^2-b^2\right )^2}\right )}{3 b^4 d (a+b \cos (c+d x))^{3/2}} \]
(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(2*a^3*A*b - 6*a*A*b^3 - 4* a^4*B + 7*a^2*b^2*B + b^4*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-8*a ^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*((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*(-8*a^5*A*b + 16*a^3*A*b^3 + 16*a^ 6*B - 25*a^4*b^2*B + b^6*B + 2*a*b*(-5*a^3*A*b + 9*a*A*b^3 + 10*a^4*B - 16 *a^2*b^2*B + 2*b^4*B)*Cos[c + d*x] + (-(a^2*b) + b^3)^2*B*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 = 2.23 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 3468, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {2 \int -\frac {\cos (c+d x) \left (-3 \left (-2 B a^2+A b a+b^2 B\right ) \cos ^2(c+d x)-3 b (A b-a B) \cos (c+d x)+4 a (A b-a B)\right )}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) \left (-3 \left (-2 B a^2+A b a+b^2 B\right ) \cos ^2(c+d x)-3 b (A b-a B) \cos (c+d x)+4 a (A b-a B)\right )}{(a+b \cos (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (-3 \left (-2 B a^2+A b a+b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-3 b (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a (A b-a B)\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {\frac {2 \int \frac {-3 b \left (a^2-b^2\right ) \left (-2 B a^2+A b a+b^2 B\right ) \cos ^2(c+d x)+\left (-12 B a^5+6 A b a^4+22 b^2 B a^3-13 A b^3 a^2-6 b^4 B a+3 A b^5\right ) \cos (c+d x)+a b \left (-6 B a^3+3 A b a^2+10 b^2 B a-7 A b^3\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-3 b \left (a^2-b^2\right ) \left (-2 B a^2+A b a+b^2 B\right ) \cos ^2(c+d x)+\left (-12 B a^5+6 A b a^4+22 b^2 B a^3-13 A b^3 a^2-6 b^4 B a+3 A b^5\right ) \cos (c+d x)+a b \left (-6 B a^3+3 A b a^2+10 b^2 B a-7 A b^3\right )}{\sqrt {a+b \cos (c+d x)}}dx}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-3 b \left (a^2-b^2\right ) \left (-2 B a^2+A b a+b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-12 B a^5+6 A b a^4+22 b^2 B a^3-13 A b^3 a^2-6 b^4 B a+3 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a b \left (-6 B a^3+3 A b a^2+10 b^2 B a-7 A b^3\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {3 \left (\left (-4 B a^4+2 A b a^3+7 b^2 B a^2-6 A b^3 a+b^4 B\right ) b^2+\left (-16 B a^5+8 A b a^4+28 b^2 B a^3-15 A b^3 a^2-8 b^4 B a+3 A b^5\right ) \cos (c+d x) b\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (-4 B a^4+2 A b a^3+7 b^2 B a^2-6 A b^3 a+b^4 B\right ) b^2+\left (-16 B a^5+8 A b a^4+28 b^2 B a^3-15 A b^3 a^2-8 b^4 B a+3 A b^5\right ) \cos (c+d x) b}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (-4 B a^4+2 A b a^3+7 b^2 B a^2-6 A b^3 a+b^4 B\right ) b^2+\left (-16 B a^5+8 A b a^4+28 b^2 B a^3-15 A b^3 a^2-8 b^4 B a+3 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {\left (-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \int \sqrt {a+b \cos (c+d x)}dx-\left (a^2-b^2\right ) \left (-16 a^4 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+b^4 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\left (a^2-b^2\right ) \left (-16 a^4 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+b^4 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (-16 a^4 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+b^4 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (-16 a^4 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+b^4 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (-16 a^4 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+b^4 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-16 a^4 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-16 a^4 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\frac {\frac {\frac {2 \left (-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (-16 a^4 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 \left (a^2-b^2\right ) \left (-2 a^2 B+a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{b^2 \left (a^2-b^2\right )}-\frac {2 a^2 \left (-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b \left (a^2-b^2\right )}\) |
(2*a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Co s[c + d*x])^(3/2)) + ((-2*a^2*(3*a^2*A*b - 7*A*b^3 - 6*a^3*B + 10*a*b^2*B) *Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (((2*(8*a^4* A*b - 15*a^2*A*b^3 + 3*A*b^5 - 16*a^5*B + 28*a^3*b^2*B - 8*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*C os[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(8*a^3*A*b - 9*a*A*b^3 - 16*a^4*B + 16*a^2*b^2*B + b^4*B)*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*A*b - 2*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/d)/(b^2*(a ^2 - b^2)))/(3*b*(a^2 - b^2))
3.4.34.3.1 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*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((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 - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1411\) vs. \(2(447)=894\).
Time = 27.65 (sec) , antiderivative size = 1412, normalized size of antiderivative = 3.42
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1412\) |
| parts | \(\text {Expression too large to display}\) | \(2208\) |
-(-(-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^2/b^4 *(3*A*b-4*B*a)/sin(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)-2*a^3*(A*b-B*a)/b^4*(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(cos(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)*(El lipticF(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))))-2/3/b^4/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d *x+1/2*c)^2)^(1/2)*(-4*B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*b^2+9*A*a *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)/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.23 (sec) , antiderivative size = 1348, normalized size of antiderivative = 3.26 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
1/9*(6*(8*B*a^6*b^2 - 4*A*a^5*b^3 - 13*B*a^4*b^4 + 8*A*a^3*b^5 + B*a^2*b^6 + (B*a^4*b^4 - 2*B*a^2*b^6 + B*b^8)*cos(d*x + c)^2 + (10*B*a^5*b^3 - 5*A* a^4*b^4 - 16*B*a^3*b^5 + 9*A*a^2*b^6 + 2*B*a*b^7)*cos(d*x + c))*sqrt(b*cos (d*x + c) + a)*sin(d*x + c) + (sqrt(2)*(-32*I*B*a^6*b^2 + 16*I*A*a^5*b^3 + 68*I*B*a^4*b^4 - 36*I*A*a^3*b^5 - 37*I*B*a^2*b^6 + 24*I*A*a*b^7 - 3*I*B*b ^8)*cos(d*x + c)^2 - 2*sqrt(2)*(32*I*B*a^7*b - 16*I*A*a^6*b^2 - 68*I*B*a^5 *b^3 + 36*I*A*a^4*b^4 + 37*I*B*a^3*b^5 - 24*I*A*a^2*b^6 + 3*I*B*a*b^7)*cos (d*x + c) + sqrt(2)*(-32*I*B*a^8 + 16*I*A*a^7*b + 68*I*B*a^6*b^2 - 36*I*A* a^5*b^3 - 37*I*B*a^4*b^4 + 24*I*A*a^3*b^5 - 3*I*B*a^2*b^6))*sqrt(b)*weiers trassPInverse(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)*(32*I*B*a^6*b^2 - 16*I*A*a^5*b^3 - 68*I*B*a^4*b^4 + 36*I*A*a^3*b^5 + 37*I*B*a^2*b^6 - 24*I *A*a*b^7 + 3*I*B*b^8)*cos(d*x + c)^2 - 2*sqrt(2)*(-32*I*B*a^7*b + 16*I*A*a ^6*b^2 + 68*I*B*a^5*b^3 - 36*I*A*a^4*b^4 - 37*I*B*a^3*b^5 + 24*I*A*a^2*b^6 - 3*I*B*a*b^7)*cos(d*x + c) + sqrt(2)*(32*I*B*a^8 - 16*I*A*a^7*b - 68*I*B *a^6*b^2 + 36*I*A*a^5*b^3 + 37*I*B*a^4*b^4 - 24*I*A*a^3*b^5 + 3*I*B*a^2*b^ 6))*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)*(16*I*B*a^5*b^3 - 8*I*A*a^4*b^4 - 28*I*B*a^3*b^5 + 15*I*A*a^2*b^6 + 8* I*B*a*b^7 - 3*I*A*b^8)*cos(d*x + c)^2 + 2*sqrt(2)*(16*I*B*a^6*b^2 - 8*I...
Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]