Integrand size = 25, antiderivative size = 474 \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=-\frac {2 b \left (17 a^4+116 a^2 b^2-128 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \]
2/3*b^2*sin(d*x+c)/a/(a^2-b^2)/d/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^(3/2)+8 /3*b^2*(3*a^2-2*b^2)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)/(a+b*se c(d*x+c))^(1/2)-2/15*b*(17*a^4+116*a^2*b^2-128*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)*(a/(a+b))^( 1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/a^5/(a^2-b^2)/d/(a+b *sec(d*x+c))^(1/2)+2/15*(3*a^4-71*a^2*b^2+48*b^4)*sin(d*x+c)*(a+b*sec(d*x+ c))^(1/2)/a^3/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)-4/15*b*(7*a^4-49*a^2*b^2+32*b ^4)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^4/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)+2 /15*(9*a^6+55*a^4*b^2-212*a^2*b^4+128*b^6)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/co s(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+ b*sec(d*x+c))^(1/2)/a^5/(a^2-b^2)^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d *x+c)^(1/2)
Time = 1.87 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\frac {(b+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x) \left (\frac {2 \left (\frac {b+a \cos (c+d x)}{a+b}\right )^{3/2} \left (\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+b \left (-17 a^5+17 a^4 b-116 a^3 b^2+116 a^2 b^3+128 a b^4-128 b^5\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )\right )}{(a-b)^2}+a \left (\frac {10 b^5 \sin (c+d x)}{-a^2+b^2}-\frac {10 b^4 \left (-15 a^2+11 b^2\right ) (b+a \cos (c+d x)) \sin (c+d x)}{\left (a^2-b^2\right )^2}-28 b (b+a \cos (c+d x))^2 \sin (c+d x)+3 a (b+a \cos (c+d x))^2 \sin (2 (c+d x))\right )\right )}{15 a^5 d (a+b \sec (c+d x))^{5/2}} \]
((b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((2*((b + a*Cos[c + d*x])/(a + b) )^(3/2)*((9*a^6 + 55*a^4*b^2 - 212*a^2*b^4 + 128*b^6)*EllipticE[(c + d*x)/ 2, (2*a)/(a + b)] + b*(-17*a^5 + 17*a^4*b - 116*a^3*b^2 + 116*a^2*b^3 + 12 8*a*b^4 - 128*b^5)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a - b)^2 + a*( (10*b^5*Sin[c + d*x])/(-a^2 + b^2) - (10*b^4*(-15*a^2 + 11*b^2)*(b + a*Cos [c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 - 28*b*(b + a*Cos[c + d*x])^2*Sin[c + d*x] + 3*a*(b + a*Cos[c + d*x])^2*Sin[2*(c + d*x)])))/(15*a^5*d*(a + b* Sec[c + d*x])^(5/2))
Time = 3.78 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.01, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4334, 27, 3042, 4588, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 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.
| \(\displaystyle \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4334 |
| \(\displaystyle \frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \int -\frac {3 a^2-3 b \sec (c+d x) a-8 b^2+6 b^2 \sec ^2(c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 a^2-3 b \sec (c+d x) a-8 b^2+6 b^2 \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a^2-3 b \csc \left (c+d x+\frac {\pi }{2}\right ) a-8 b^2+6 b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {2 \int -\frac {3 a^4-71 b^2 a^2-2 b \left (3 a^2-b^2\right ) \sec (c+d x) a+48 b^4+16 b^2 \left (3 a^2-2 b^2\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 a^4-71 b^2 a^2-2 b \left (3 a^2-b^2\right ) \sec (c+d x) a+48 b^4+16 b^2 \left (3 a^2-2 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 a^4-71 b^2 a^2-2 b \left (3 a^2-b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+48 b^4+16 b^2 \left (3 a^2-2 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {-2 b \left (3 a^4-71 b^2 a^2+48 b^4\right ) \sec ^2(c+d x)-a \left (9 a^4+27 b^2 a^2-16 b^4\right ) \sec (c+d x)+6 b \left (7 a^4-49 b^2 a^2+32 b^4\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (3 a^4-71 b^2 a^2+48 b^4\right ) \sec ^2(c+d x)-a \left (9 a^4+27 b^2 a^2-16 b^4\right ) \sec (c+d x)+6 b \left (7 a^4-49 b^2 a^2+32 b^4\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (3 a^4-71 b^2 a^2+48 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (9 a^4+27 b^2 a^2-16 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+6 b \left (7 a^4-49 b^2 a^2+32 b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {3 \left (9 a^6+55 b^2 a^4-212 b^4 a^2-4 b \left (2 a^4+11 b^2 a^2-8 b^4\right ) \sec (c+d x) a+128 b^6\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {9 a^6+55 b^2 a^4-212 b^4 a^2-4 b \left (2 a^4+11 b^2 a^2-8 b^4\right ) \sec (c+d x) a+128 b^6}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {9 a^6+55 b^2 a^4-212 b^4 a^2-4 b \left (2 a^4+11 b^2 a^2-8 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+128 b^6}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {2 b \left (17 a^6+99 a^4 b^2-244 a^2 b^4+128 b^6\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
(2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) + ((8*b^2*(3*a^2 - 2*b^2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[ c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + ((2*(3*a^4 - 71*a^2*b^2 + 48*b^ 4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (-( ((-2*b*(17*a^6 + 99*a^4*b^2 - 244*a^2*b^4 + 128*b^6)*Sqrt[(b + a*Cos[c + d *x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a *d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^6 + 55*a^4*b^2 - 212*a^2*b^4 + 128* b^6)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d* Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]))/a) + (4*b*(7*a^4 - 49*a^2*b^2 + 32*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d*Sqrt[Sec [c + d*x]]))/(5*a))/(a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
3.7.68.3.1 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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* ((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a^2*(m + 1) - b^2*(m + n + 1) - a*b*(m + 1)*Csc[e + f*x] + b^2*(m + n + 2)*Csc[e + f*x ]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc [e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f *x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x ] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(5655\) vs. \(2(492)=984\).
Time = 11.09 (sec) , antiderivative size = 5656, normalized size of antiderivative = 11.93
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 1036, normalized size of antiderivative = 2.19 \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
-1/45*(2*sqrt(2)*(-21*I*a^6*b^3 - 121*I*a^4*b^5 + 260*I*a^2*b^7 - 128*I*b^ 9 + (-21*I*a^8*b - 121*I*a^6*b^3 + 260*I*a^4*b^5 - 128*I*a^2*b^7)*cos(d*x + c)^2 + 2*(-21*I*a^7*b^2 - 121*I*a^5*b^4 + 260*I*a^3*b^6 - 128*I*a*b^8)*c os(d*x + c))*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9 *a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) + 2*sqrt(2)*(21*I*a^6*b^3 + 121*I*a^4*b^5 - 260*I*a^2*b^7 + 128*I*b^9 + (2 1*I*a^8*b + 121*I*a^6*b^3 - 260*I*a^4*b^5 + 128*I*a^2*b^7)*cos(d*x + c)^2 + 2*(21*I*a^7*b^2 + 121*I*a^5*b^4 - 260*I*a^3*b^6 + 128*I*a*b^8)*cos(d*x + c))*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) + 3*sqrt (2)*(-9*I*a^7*b^2 - 55*I*a^5*b^4 + 212*I*a^3*b^6 - 128*I*a*b^8 + (-9*I*a^9 - 55*I*a^7*b^2 + 212*I*a^5*b^4 - 128*I*a^3*b^6)*cos(d*x + c)^2 + 2*(-9*I* a^8*b - 55*I*a^6*b^3 + 212*I*a^4*b^5 - 128*I*a^2*b^7)*cos(d*x + c))*sqrt(a )*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, we ierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/ 3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a)) + 3*sqrt(2)*(9*I*a^7*b ^2 + 55*I*a^5*b^4 - 212*I*a^3*b^6 + 128*I*a*b^8 + (9*I*a^9 + 55*I*a^7*b^2 - 212*I*a^5*b^4 + 128*I*a^3*b^6)*cos(d*x + c)^2 + 2*(9*I*a^8*b + 55*I*a^6* b^3 - 212*I*a^4*b^5 + 128*I*a^2*b^7)*cos(d*x + c))*sqrt(a)*weierstrassZeta (-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInve...
Timed out. \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]