Integrand size = 23, antiderivative size = 425 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {2 b \sec ^3(c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {8 a b \sec ^3(c+d x)}{\left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 a \left (a^4-6 a^2 b^2-27 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right )^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 a^4-21 a^2 b^2-15 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{6 \left (a^2-b^2\right )^3 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^4 d} \]
2/3*b*sec(d*x+c)^3/(a^2-b^2)/d/(a+b*sin(d*x+c))^(3/2)+8*a*b*sec(d*x+c)^3/( a^2-b^2)^2/d/(a+b*sin(d*x+c))^(1/2)-1/3*sec(d*x+c)^3*(b*(29*a^2+3*b^2)-a*( a^2+31*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/(a^2-b^2)^3/d-1/6*sec(d*x+c )*(b*(a^4-114*a^2*b^2-15*b^4)-4*a*(a^4-6*a^2*b^2-27*b^4)*sin(d*x+c))*(a+b* sin(d*x+c))^(1/2)/(a^2-b^2)^4/d+2/3*a*(a^4-6*a^2*b^2-27*b^4)*(sin(1/2*c+1/ 4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*P i+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/(a^2-b^2)^4/d/( (a+b*sin(d*x+c))/(a+b))^(1/2)-1/6*(4*a^4-21*a^2*b^2-15*b^4)*(sin(1/2*c+1/4 *Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi +1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/(a^2-b^2 )^3/d/(a+b*sin(d*x+c))^(1/2)
Time = 1.69 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.80 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {\frac {\left (4 \left (a^5-6 a^3 b^2-27 a b^4\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+\left (-4 a^5+4 a^4 b+21 a^3 b^2-21 a^2 b^3+15 a b^4-15 b^5\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^4 (a+b)^2}+\frac {4 b^5 \left (a^2-b^2\right ) \cos (c+d x)+64 a b^5 \cos (c+d x) (a+b \sin (c+d x))+2 \left (a^2-b^2\right ) \sec ^3(c+d x) (a+b \sin (c+d x))^2 \left (-b \left (3 a^2+b^2\right )+a \left (a^2+3 b^2\right ) \sin (c+d x)\right )+\sec (c+d x) (a+b \sin (c+d x))^2 \left (-a^4 b+54 a^2 b^3+11 b^5+4 a \left (a^4-6 a^2 b^2-11 b^4\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^4}}{6 d (a+b \sin (c+d x))^{3/2}} \]
(((4*(a^5 - 6*a^3*b^2 - 27*a*b^4)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/( a + b)] + (-4*a^5 + 4*a^4*b + 21*a^3*b^2 - 21*a^2*b^3 + 15*a*b^4 - 15*b^5) *EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/((a - b)^4*(a + b)^2) + (4*b^5*(a^2 - b^2)*Cos[c + d*x] + 64 *a*b^5*Cos[c + d*x]*(a + b*Sin[c + d*x]) + 2*(a^2 - b^2)*Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*(-(b*(3*a^2 + b^2)) + a*(a^2 + 3*b^2)*Sin[c + d*x]) + Sec[c + d*x]*(a + b*Sin[c + d*x])^2*(-(a^4*b) + 54*a^2*b^3 + 11*b^5 + 4*a *(a^4 - 6*a^2*b^2 - 11*b^4)*Sin[c + d*x]))/(a^2 - b^2)^4)/(6*d*(a + b*Sin[ c + d*x])^(3/2))
Time = 2.41 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.09, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 3173, 27, 3042, 3343, 27, 3042, 3345, 27, 3042, 3345, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a+b \sin (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3173 |
| \(\displaystyle \frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}-\frac {2 \int -\frac {3 \sec ^4(c+d x) (a-3 b \sin (c+d x))}{2 (a+b \sin (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec ^4(c+d x) (a-3 b \sin (c+d x))}{(a+b \sin (c+d x))^{3/2}}dx}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-3 b \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))^{3/2}}dx}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}-\frac {2 \int -\frac {\sec ^4(c+d x) \left (a^2-28 b \sin (c+d x) a+3 b^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sec ^4(c+d x) \left (a^2-28 b \sin (c+d x) a+3 b^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-28 b \sin (c+d x) a+3 b^2}{\cos (c+d x)^4 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\sec ^2(c+d x) \left (4 a^4-21 b^2 a^2+3 b \left (a^2+31 b^2\right ) \sin (c+d x) a-15 b^4\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sec ^2(c+d x) \left (4 a^4-21 b^2 a^2+3 b \left (a^2+31 b^2\right ) \sin (c+d x) a-15 b^4\right )}{\sqrt {a+b \sin (c+d x)}}dx}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {4 a^4-21 b^2 a^2+3 b \left (a^2+31 b^2\right ) \sin (c+d x) a-15 b^4}{\cos (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {\left (a^4-114 b^2 a^2-15 b^4\right ) b^2+4 a \left (a^4-6 b^2 a^2-27 b^4\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {\left (a^4-114 b^2 a^2-15 b^4\right ) b^2+4 a \left (a^4-6 b^2 a^2-27 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int \frac {\left (a^4-114 b^2 a^2-15 b^4\right ) b^2+4 a \left (a^4-6 b^2 a^2-27 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {-\frac {4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {8 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {8 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\frac {8 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}+\frac {\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}+\frac {\frac {-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\frac {8 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}}{a^2-b^2}\) |
(2*b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + ((8*a* b*Sec[c + d*x]^3)/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (-1/3*(Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(b*(29*a^2 + 3*b^2) - a*(a^2 + 31*b^2)*S in[c + d*x]))/((a^2 - b^2)*d) + (-((Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]* (b*(a^4 - 114*a^2*b^2 - 15*b^4) - 4*a*(a^4 - 6*a^2*b^2 - 27*b^4)*Sin[c + d *x]))/((a^2 - b^2)*d)) - ((8*a*(a^4 - 6*a^2*b^2 - 27*b^4)*EllipticE[(c - P i/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[ c + d*x])/(a + b)]) - (2*(4*a^6 - 25*a^4*b^2 + 6*a^2*b^4 + 15*b^6)*Ellipti cF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/ (d*Sqrt[a + b*Sin[c + d*x]]))/(2*(a^2 - b^2)))/(6*(a^2 - b^2)))/(a^2 - b^2 ))/(a^2 - b^2)
3.6.38.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b ^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p *(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ [a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(2584\) vs. \(2(465)=930\).
Time = 4.39 (sec) , antiderivative size = 2585, normalized size of antiderivative = 6.08
(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*(1/4/(a+b)^2*(1/3/(a+b)*(-(-b*sin( d*x+c)-a)*cos(d*x+c)^2)^(1/2)/(sin(d*x+c)-1)^2-1/3*(-sin(d*x+c)^2*b-a*sin( d*x+c)-b*sin(d*x+c)-a)/(a+b)^2*(a+3*b)/((1+sin(d*x+c))*(sin(d*x+c)-1)*(-b* sin(d*x+c)-a))^(1/2)+2*b^2/(3*a^2+6*a*b+3*b^2)*(a/b-1)*((a+b*sin(d*x+c))/( a-b))^(1/2)*(b*(1-sin(d*x+c))/(a+b))^(1/2)*((-1-sin(d*x+c))*b/(a-b))^(1/2) /(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b) )^(1/2),((a-b)/(a+b))^(1/2))-1/3*b*(a+3*b)/(a+b)^2*(a/b-1)*((a+b*sin(d*x+c ))/(a-b))^(1/2)*(b*(1-sin(d*x+c))/(a+b))^(1/2)*((-1-sin(d*x+c))*b/(a-b))^( 1/2)/(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*((-a/b-1)*EllipticE(((a+b*sin (d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))+EllipticF(((a+b*sin(d*x+c))/(a- b))^(1/2),((a-b)/(a+b))^(1/2))))+1/4/(a-b)^2*(-1/3/(a-b)*(-(-b*sin(d*x+c)- a)*cos(d*x+c)^2)^(1/2)/(1+sin(d*x+c))^2-1/3*(-sin(d*x+c)^2*b-a*sin(d*x+c)+ b*sin(d*x+c)+a)/(a-b)^2*(a-3*b)/((1+sin(d*x+c))*(sin(d*x+c)-1)*(-b*sin(d*x +c)-a))^(1/2)+2*b^2/(3*a^2-6*a*b+3*b^2)*(a/b-1)*((a+b*sin(d*x+c))/(a-b))^( 1/2)*(b*(1-sin(d*x+c))/(a+b))^(1/2)*((-1-sin(d*x+c))*b/(a-b))^(1/2)/(-(-b* sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2) ,((a-b)/(a+b))^(1/2))-1/3*b*(a-3*b)/(a-b)^2*(a/b-1)*((a+b*sin(d*x+c))/(a-b ))^(1/2)*(b*(1-sin(d*x+c))/(a+b))^(1/2)*((-1-sin(d*x+c))*b/(a-b))^(1/2)/(- (-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*((-a/b-1)*EllipticE(((a+b*sin(d*x+c) )/(a-b))^(1/2),((a-b)/(a+b))^(1/2))+EllipticF(((a+b*sin(d*x+c))/(a-b))^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.22 (sec) , antiderivative size = 1242, normalized size of antiderivative = 2.92 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
1/36*((sqrt(2)*(8*a^6*b^2 - 51*a^4*b^4 + 126*a^2*b^6 + 45*b^8)*cos(d*x + c )^5 - 2*sqrt(2)*(8*a^7*b - 51*a^5*b^3 + 126*a^3*b^5 + 45*a*b^7)*cos(d*x + c)^3*sin(d*x + c) - sqrt(2)*(8*a^8 - 43*a^6*b^2 + 75*a^4*b^4 + 171*a^2*b^6 + 45*b^8)*cos(d*x + c)^3)*sqrt(I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b ^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*si n(d*x + c) - 2*I*a)/b) + (sqrt(2)*(8*a^6*b^2 - 51*a^4*b^4 + 126*a^2*b^6 + 45*b^8)*cos(d*x + c)^5 - 2*sqrt(2)*(8*a^7*b - 51*a^5*b^3 + 126*a^3*b^5 + 4 5*a*b^7)*cos(d*x + c)^3*sin(d*x + c) - sqrt(2)*(8*a^8 - 43*a^6*b^2 + 75*a^ 4*b^4 + 171*a^2*b^6 + 45*b^8)*cos(d*x + c)^3)*sqrt(-I*b)*weierstrassPInver se(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*co s(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) + 12*(sqrt(2)*(I*a^5*b^3 - 6*I *a^3*b^5 - 27*I*a*b^7)*cos(d*x + c)^5 + 2*sqrt(2)*(-I*a^6*b^2 + 6*I*a^4*b^ 4 + 27*I*a^2*b^6)*cos(d*x + c)^3*sin(d*x + c) + sqrt(2)*(-I*a^7*b + 5*I*a^ 5*b^3 + 33*I*a^3*b^5 + 27*I*a*b^7)*cos(d*x + c)^3)*sqrt(I*b)*weierstrassZe ta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, weierstrassP Inverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3* b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) + 12*(sqrt(2)*(-I*a^5*b^3 + 6*I*a^3*b^5 + 27*I*a*b^7)*cos(d*x + c)^5 + 2*sqrt(2)*(I*a^6*b^2 - 6*I*a ^4*b^4 - 27*I*a^2*b^6)*cos(d*x + c)^3*sin(d*x + c) + sqrt(2)*(I*a^7*b - 5* I*a^5*b^3 - 33*I*a^3*b^5 - 27*I*a*b^7)*cos(d*x + c)^3)*sqrt(-I*b)*weier...
\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]