Integrand size = 25, antiderivative size = 559 \[ \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {2 b (5 a+4 b) \cos ^4(e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}-\frac {b \cos ^6(e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b-a \sin ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{15 a^4 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (3 a^2+61 a b+48 b^2\right ) \cos ^2(e+f x) \sin (e+f x) \left (a+b-a \sin ^2(e+f x)\right )}{15 a^3 (a+b)^2 f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\right ) E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \left (a+b-a \sin ^2(e+f x)\right )}{15 a^5 (a+b)^2 f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b \left (4 a^3-9 a^2 b+120 a b^2+128 b^3\right ) \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{15 a^5 (a+b) f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]
-2/3*b*(5*a+4*b)*cos(f*x+e)^4*sin(f*x+e)/a^2/(a+b)^2/f/(sec(f*x+e)^2*(a+b- a*sin(f*x+e)^2))^(1/2)-1/3*b*cos(f*x+e)^6*sin(f*x+e)/a/(a+b)/f/(a+b-a*sin( f*x+e)^2)/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)+2/15*(2*a^3-3*a^2*b-42 *a*b^2-32*b^3)*sin(f*x+e)*(a+b-a*sin(f*x+e)^2)/a^4/(a+b)^2/f/(sec(f*x+e)^2 *(a+b-a*sin(f*x+e)^2))^(1/2)+1/15*(3*a^2+61*a*b+48*b^2)*cos(f*x+e)^2*sin(f *x+e)*(a+b-a*sin(f*x+e)^2)/a^3/(a+b)^2/f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2 ))^(1/2)+1/15*(8*a^4-11*a^3*b+27*a^2*b^2+184*a*b^3+128*b^4)*EllipticE(sin( f*x+e),(a/(a+b))^(1/2))*(a+b-a*sin(f*x+e)^2)/a^5/(a+b)^2/f/(cos(f*x+e)^2)^ (1/2)/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/(1-a*sin(f*x+e)^2/(a+b))^( 1/2)-1/15*b*(4*a^3-9*a^2*b+120*a*b^2+128*b^3)*EllipticF(sin(f*x+e),(a/(a+b ))^(1/2))*(1-a*sin(f*x+e)^2/(a+b))^(1/2)/a^5/(a+b)/f/(cos(f*x+e)^2)^(1/2)/ (sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)
\[ \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \]
Time = 0.82 (sec) , antiderivative size = 531, normalized size of antiderivative = 0.95, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4636, 2057, 2058, 315, 25, 401, 403, 27, 403, 25, 399, 323, 321, 330, 327}
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.
| \(\displaystyle \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (e+f x)^5 \left (a+b \sec (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4636 |
| \(\displaystyle \frac {\int \frac {\left (1-\sin ^2(e+f x)\right )^2}{\left (a+\frac {b}{1-\sin ^2(e+f x)}\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \frac {\int \frac {\left (1-\sin ^2(e+f x)\right )^2}{\left (\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\left (1-\sin ^2(e+f x)\right )^{9/2}}{\left (-a \sin ^2(e+f x)+a+b\right )^{5/2}}d\sin (e+f x)}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (-\frac {\int -\frac {\left (1-\sin ^2(e+f x)\right )^{5/2} \left (-\left ((3 a+8 b) \sin ^2(e+f x)\right )+3 a+b\right )}{\left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\int \frac {\left (1-\sin ^2(e+f x)\right )^{5/2} \left (-\left ((3 a+8 b) \sin ^2(e+f x)\right )+3 a+b\right )}{\left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\int \frac {\left (1-\sin ^2(e+f x)\right )^{3/2} \left ((a+b) (3 a+8 b)-\left (3 a^2+61 b a+48 b^2\right ) \sin ^2(e+f x)\right )}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}-\frac {\int -\frac {3 \sqrt {1-\sin ^2(e+f x)} \left ((a+b) \left (4 a^2-7 b a-16 b^2\right )-2 \left (2 a^3-3 b a^2-42 b^2 a-32 b^3\right ) \sin ^2(e+f x)\right )}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {3 \int \frac {\sqrt {1-\sin ^2(e+f x)} \left ((a+b) \left (4 a^2-7 b a-16 b^2\right )-2 \left (2 a^3-3 b a^2-42 b^2 a-32 b^3\right ) \sin ^2(e+f x)\right )}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{5 a}+\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {3 \left (\frac {2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sin (e+f x) \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}-\frac {\int -\frac {(a+b) \left (8 a^3-15 b a^2+36 b^2 a+64 b^3\right )-\left (8 a^4-11 b a^3+27 b^2 a^2+184 b^3 a+128 b^4\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 a}\right )}{5 a}+\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {3 \left (\frac {\int \frac {(a+b) \left (8 a^3-15 b a^2+36 b^2 a+64 b^3\right )-\left (8 a^4-11 b a^3+27 b^2 a^2+184 b^3 a+128 b^4\right ) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{3 a}+\frac {2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}\right )}{5 a}+\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {3 \left (\frac {\frac {\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \left (4 a^3-9 a^2 b+120 a b^2+128 b^3\right ) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}}{3 a}+\frac {2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}\right )}{5 a}+\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {3 \left (\frac {\frac {\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \left (4 a^3-9 a^2 b+120 a b^2+128 b^3\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a}+\frac {2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}\right )}{5 a}+\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {3 \left (\frac {\frac {\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\right ) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \left (4 a^3-9 a^2 b+120 a b^2+128 b^3\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a}+\frac {2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}\right )}{5 a}+\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {3 \left (\frac {\frac {\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\right ) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) \left (4 a^3-9 a^2 b+120 a b^2+128 b^3\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a}+\frac {2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}\right )}{5 a}+\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\frac {\frac {\left (3 a^2+61 a b+48 b^2\right ) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}{5 a}+\frac {3 \left (\frac {2 \left (2 a^3-3 a^2 b-42 a b^2-32 b^3\right ) \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{3 a}+\frac {\frac {\left (8 a^4-11 a^3 b+27 a^2 b^2+184 a b^3+128 b^4\right ) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) \left (4 a^3-9 a^2 b+120 a b^2+128 b^3\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a}\right )}{5 a}}{a (a+b)}-\frac {2 b (5 a+4 b) \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{5/2}}{a (a+b) \sqrt {-a \sin ^2(e+f x)+a+b}}}{3 a (a+b)}-\frac {b \sin (e+f x) \left (1-\sin ^2(e+f x)\right )^{7/2}}{3 a (a+b) \left (-a \sin ^2(e+f x)+a+b\right )^{3/2}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
(Sqrt[a + b - a*Sin[e + f*x]^2]*(-1/3*(b*Sin[e + f*x]*(1 - Sin[e + f*x]^2) ^(7/2))/(a*(a + b)*(a + b - a*Sin[e + f*x]^2)^(3/2)) + ((-2*b*(5*a + 4*b)* Sin[e + f*x]*(1 - Sin[e + f*x]^2)^(5/2))/(a*(a + b)*Sqrt[a + b - a*Sin[e + f*x]^2]) + (((3*a^2 + 61*a*b + 48*b^2)*Sin[e + f*x]*(1 - Sin[e + f*x]^2)^ (3/2)*Sqrt[a + b - a*Sin[e + f*x]^2])/(5*a) + (3*((2*(2*a^3 - 3*a^2*b - 42 *a*b^2 - 32*b^3)*Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b - a*Sin[ e + f*x]^2])/(3*a) + (((8*a^4 - 11*a^3*b + 27*a^2*b^2 + 184*a*b^3 + 128*b^ 4)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^ 2])/(a*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b)*(4*a^3 - 9*a^2*b + 120*a*b^2 + 128*b^3)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(a*Sqrt[a + b - a*Sin[e + f*x]^2]))/(3*a))) /(5*a))/(a*(a + b)))/(3*a*(a + b))))/(f*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x]^2)])
3.3.88.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x], x , Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 13.44 (sec) , antiderivative size = 31577, normalized size of antiderivative = 56.49
\[ \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(b*sec(f*x + e)^2 + a)*cos(f*x + e)^5/(b^3*sec(f*x + e)^6 + 3 *a*b^2*sec(f*x + e)^4 + 3*a^2*b*sec(f*x + e)^2 + a^3), x)
Timed out. \[ \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^5}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \]