Integrand size = 23, antiderivative size = 325 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {2 b \sec (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {16 a b \sec (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {a \left (3 a^2+29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (3 a^2+5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d} \]
2/3*b*sec(d*x+c)/(a^2-b^2)/d/(a+b*sin(d*x+c))^(3/2)+16/3*a*b*sec(d*x+c)/(a ^2-b^2)^2/d/(a+b*sin(d*x+c))^(1/2)-1/3*sec(d*x+c)*(b*(27*a^2+5*b^2)-a*(3*a ^2+29*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/(a^2-b^2)^3/d+1/3*a*(3*a^2+2 9*b^2)*(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)*Ellip ticE(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))^( 1/2)/(a^2-b^2)^3/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/3*(3*a^2+5*b^2)*(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)^2/d/(a+b*sin(d*x+c))^(1/2)
Time = 1.31 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.74 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {\frac {\left (\left (3 a^3+29 a b^2\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+\left (-3 a^3+3 a^2 b-5 a b^2+5 b^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^3 (a+b)}-\frac {2 b^3 \left (a^2-b^2\right ) \cos (c+d x)+20 a b^3 \cos (c+d x) (a+b \sin (c+d x))+3 \sec (c+d x) (a+b \sin (c+d x))^2 \left (3 a^2 b+b^3-a \left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^3}}{3 d (a+b \sin (c+d x))^{3/2}} \]
((((3*a^3 + 29*a*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (- 3*a^3 + 3*a^2*b - 5*a*b^2 + 5*b^3)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/ (a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/((a - b)^3*(a + b)) - (2*b ^3*(a^2 - b^2)*Cos[c + d*x] + 20*a*b^3*Cos[c + d*x]*(a + b*Sin[c + d*x]) + 3*Sec[c + d*x]*(a + b*Sin[c + d*x])^2*(3*a^2*b + b^3 - a*(a^2 + 3*b^2)*Si n[c + d*x]))/(a^2 - b^2)^3)/(3*d*(a + b*Sin[c + d*x])^(3/2))
Time = 1.79 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 3173, 27, 3042, 3343, 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.
| \(\displaystyle \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^2 (a+b \sin (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3173 |
| \(\displaystyle \frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}-\frac {2 \int -\frac {\sec ^2(c+d x) (3 a-5 b \sin (c+d x))}{2 (a+b \sin (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec ^2(c+d x) (3 a-5 b \sin (c+d x))}{(a+b \sin (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 a-5 b \sin (c+d x)}{\cos (c+d x)^2 (a+b \sin (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}-\frac {2 \int -\frac {\sec ^2(c+d x) \left (3 a^2-24 b \sin (c+d x) a+5 b^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sec ^2(c+d x) \left (3 a^2-24 b \sin (c+d x) a+5 b^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 a^2-24 b \sin (c+d x) a+5 b^2}{\cos (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\left (27 a^2+5 b^2\right ) b^2+a \left (3 a^2+29 b^2\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\left (27 a^2+5 b^2\right ) b^2+a \left (3 a^2+29 b^2\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\left (27 a^2+5 b^2\right ) b^2+a \left (3 a^2+29 b^2\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {-\frac {a \left (3 a^2+29 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {a \left (3 a^2+29 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (3 a^4+2 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}}{3 \left (a^2-b^2\right )}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}+\frac {\frac {16 a b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\frac {2 a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (3 a^4+2 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}}{a^2-b^2}}{3 \left (a^2-b^2\right )}\) |
(2*b*Sec[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + ((16*a*b *Sec[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (-((Sec[c + d*x] *Sqrt[a + b*Sin[c + d*x]]*(b*(27*a^2 + 5*b^2) - a*(3*a^2 + 29*b^2)*Sin[c + d*x]))/((a^2 - b^2)*d)) - ((2*a*(3*a^2 + 29*b^2)*EllipticE[(c - Pi/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*(3*a^4 + 2*a^2*b^2 - 5*b^4)*EllipticF[(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)))/(a^2 - b^2))/(3*(a^2 - b^2))
3.6.37.3.1 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. \(1652\) vs. \(2(367)=734\).
Time = 3.37 (sec) , antiderivative size = 1653, normalized size of antiderivative = 5.09
(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)*(1/2/(a+b)^3/b/cos(d*x+c)^2/(a+b*s in(d*x+c))*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*((-b/(a+b)*sin (d*x+c)+b/(a+b))^(1/2)*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a-b)*sin(d* x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a +b))^(1/2))*a^2-(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(b/(a-b)*sin(d*x+c)+a/ (a-b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d* x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^2-cos(d*x+c)^2*b^2+sin(d*x+c)*a *b+sin(d*x+c)*b^2+a*b+b^2)+1/2/(a-b)^2*(-(-sin(d*x+c)^2*b-a*sin(d*x+c)+b*s in(d*x+c)+a)/(a-b)/((1+sin(d*x+c))*(sin(d*x+c)-1)*(-b*sin(d*x+c)-a))^(1/2) -2*b/(2*a-2*b)*(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))-b/( a-b)*(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))+Ell ipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))))-b^2/(a-b)/(a+ b)*(2/3/b/(a^2-b^2)*(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)/(sin(d*x+c)+a/ b)^2+8/3*b*cos(d*x+c)^2/(a^2-b^2)^2*a/(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1 /2)+2*(3*a^2+b^2)/(3*a^4-6*a^2*b^2+3*b^4)*(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)/(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.06 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
1/18*((sqrt(2)*(6*a^4*b^2 - 23*a^2*b^4 - 15*b^6)*cos(d*x + c)^3 - 2*sqrt(2 )*(6*a^5*b - 23*a^3*b^3 - 15*a*b^5)*cos(d*x + c)*sin(d*x + c) - sqrt(2)*(6 *a^6 - 17*a^4*b^2 - 38*a^2*b^4 - 15*b^6)*cos(d*x + c))*sqrt(I*b)*weierstra ssPInverse(-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) + (sqrt(2)*(6*a^4*b^2 - 23*a^2*b^4 - 15*b^6)*cos(d*x + c)^3 - 2*sqrt(2)*(6*a^5*b - 23*a^3*b^3 - 1 5*a*b^5)*cos(d*x + c)*sin(d*x + c) - sqrt(2)*(6*a^6 - 17*a^4*b^2 - 38*a^2* b^4 - 15*b^6)*cos(d*x + c))*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 *sin(d*x + c) + 2*I*a)/b) - 3*(sqrt(2)*(-3*I*a^3*b^3 - 29*I*a*b^5)*cos(d*x + c)^3 + 2*sqrt(2)*(3*I*a^4*b^2 + 29*I*a^2*b^4)*cos(d*x + c)*sin(d*x + c) + sqrt(2)*(3*I*a^5*b + 32*I*a^3*b^3 + 29*I*a*b^5)*cos(d*x + c))*sqrt(I*b) *weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3 , 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*sin(d*x + c) - 2*I*a)/b)) - 3*(sqrt(2 )*(3*I*a^3*b^3 + 29*I*a*b^5)*cos(d*x + c)^3 + 2*sqrt(2)*(-3*I*a^4*b^2 - 29 *I*a^2*b^4)*cos(d*x + c)*sin(d*x + c) + sqrt(2)*(-3*I*a^5*b - 32*I*a^3*b^3 - 29*I*a*b^5)*cos(d*x + c))*sqrt(-I*b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^ 2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 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) +...
\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]