Integrand size = 23, antiderivative size = 322 \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{5 d}-\frac {\left (32 a^2-9 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{60 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {a \left (32 a^2-17 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{60 d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (5 a b+\left (8 a^2-3 b^2\right ) \sin (c+d x)\right )}{30 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{60 \left (a^2-b^2\right ) d} \]
1/5*sec(d*x+c)^5*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^(3/2)/d+1/30*sec(d*x+c) ^3*(5*a*b+(8*a^2-3*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/d-1/60*sec(d*x+ c)*(8*a*b*(a^2-b^2)-(32*a^4-41*a^2*b^2+9*b^4)*sin(d*x+c))*(a+b*sin(d*x+c)) ^(1/2)/(a^2-b^2)/d+1/60*(32*a^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)*EllipticE(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)/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/60 *a*(32*a^2-17*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)/d/(a+b*sin(d*x+c))^(1/2)
Time = 6.33 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.09 \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\sqrt {a+b \sin (c+d x)} \left (\frac {1}{60} \sec (c+d x) \left (-8 a b+32 a^2 \sin (c+d x)-9 b^2 \sin (c+d x)\right )+\frac {1}{30} \sec ^3(c+d x) \left (-a b+8 a^2 \sin (c+d x)-3 b^2 \sin (c+d x)\right )+\frac {1}{5} \sec ^5(c+d x) \left (2 a b+a^2 \sin (c+d x)+b^2 \sin (c+d x)\right )\right )}{d}-\frac {b \left (-\frac {16 a b \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {\left (32 a^2-9 b^2\right ) \left (\frac {2 (a+b) E\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )}{b}\right )}{120 d} \]
(Sqrt[a + b*Sin[c + d*x]]*((Sec[c + d*x]*(-8*a*b + 32*a^2*Sin[c + d*x] - 9 *b^2*Sin[c + d*x]))/60 + (Sec[c + d*x]^3*(-(a*b) + 8*a^2*Sin[c + d*x] - 3* b^2*Sin[c + d*x]))/30 + (Sec[c + d*x]^5*(2*a*b + a^2*Sin[c + d*x] + b^2*Si n[c + d*x]))/5))/d - (b*((-16*a*b*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((32* a^2 - 9*b^2)*((2*(a + b)*EllipticE[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqr t[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*a*EllipticF [(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/S qrt[a + b*Sin[c + d*x]]))/b))/(120*d)
Time = 1.84 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.11, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 3170, 27, 3042, 3340, 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.
| \(\displaystyle \int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^{5/2}}{\cos (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}-\frac {1}{5} \int -\frac {1}{2} \sec ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2+5 b \sin (c+d x) a-3 b^2\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int \sec ^4(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2+5 b \sin (c+d x) a-3 b^2\right )dx+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \int \frac {\sqrt {a+b \sin (c+d x)} \left (8 a^2+5 b \sin (c+d x) a-3 b^2\right )}{\cos (c+d x)^4}dx+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3340 |
| \(\displaystyle \frac {1}{10} \left (\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}-\frac {1}{3} \int -\frac {\sec ^2(c+d x) \left (a \left (32 a^2-17 b^2\right )+3 b \left (8 a^2-3 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \int \frac {\sec ^2(c+d x) \left (a \left (32 a^2-17 b^2\right )+3 b \left (8 a^2-3 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \int \frac {a \left (32 a^2-17 b^2\right )+3 b \left (8 a^2-3 b^2\right ) \sin (c+d x)}{\cos (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\int \frac {8 a \left (a^2-b^2\right ) b^2+\left (32 a^4-41 b^2 a^2+9 b^4\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\int \frac {8 a \left (a^2-b^2\right ) b^2+\left (32 a^4-41 b^2 a^2+9 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\int \frac {8 a \left (a^2-b^2\right ) b^2+\left (32 a^4-41 b^2 a^2+9 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\left (32 a^4-41 a^2 b^2+9 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-a \left (32 a^4-49 a^2 b^2+17 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\left (32 a^4-41 a^2 b^2+9 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-a \left (32 a^4-49 a^2 b^2+17 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\frac {\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-a \left (32 a^4-49 a^2 b^2+17 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\frac {\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-a \left (32 a^4-49 a^2 b^2+17 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\frac {2 \left (32 a^4-41 a^2 b^2+9 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-a \left (32 a^4-49 a^2 b^2+17 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\frac {2 \left (32 a^4-41 a^2 b^2+9 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^4-49 a^2 b^2+17 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{6} \left (-\frac {\frac {2 \left (32 a^4-41 a^2 b^2+9 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (32 a^4-49 a^2 b^2+17 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{10} \left (\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (8 a^2-3 b^2\right ) \sin (c+d x)+5 a b\right )}{3 d}+\frac {1}{6} \left (-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b \left (a^2-b^2\right )-\left (32 a^4-41 a^2 b^2+9 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\frac {2 \left (32 a^4-41 a^2 b^2+9 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a \left (32 a^4-49 a^2 b^2+17 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}\right )\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}\) |
(Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(5*d) + ( (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(5*a*b + (8*a^2 - 3*b^2)*Sin[c + d*x]))/(3*d) + (-((Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(8*a*b*(a^2 - b^2 ) - (32*a^4 - 41*a^2*b^2 + 9*b^4)*Sin[c + d*x]))/((a^2 - b^2)*d)) - ((2*(3 2*a^4 - 41*a^2*b^2 + 9*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*S qrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*a*(32 *a^4 - 49*a^2*b^2 + 17*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*S qrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]))/(2*(a^2 - b^2)))/6)/10
3.6.4.3.1 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[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x ])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g }, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2* p] || IntegerQ[m])
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[(-(g* Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Si n[e + f*x])^(m - 1)*Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ [m, 0] && LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])
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. \(1258\) vs. \(2(364)=728\).
Time = 4.22 (sec) , antiderivative size = 1259, normalized size of antiderivative = 3.91
\[\text {Expression too large to display}\]
-1/60*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*(32*(-b/(a+b)*sin(d *x+c)+b/(a+b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/( a+b))^(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)*a^3*b*cos(d*x+c)^4-24*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*Ellipt icF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(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)*a^2*b^2*cos(d*x+c )^4-17*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a /(a-b))^(1/2),((a-b)/(a+b))^(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)*a*b^3*cos(d*x+c)^4+9*(-b/(a+b)*sin(d*x+c)+ b/(a+b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^ (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^4*cos(d*x+c)^4-32*(-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/(a-b)*sin(d*x+c)-b /(a-b))^(1/2)*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*a^4*cos(d*x+c)^4+41*(-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/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(b/(a-b)*sin(d* x+c)+a/(a-b))^(1/2)*a^2*b^2*cos(d*x+c)^4-9*(-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 /(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*b^4*co s(d*x+c)^4+32*cos(d*x+c)^6*a^2*b^2-9*cos(d*x+c)^6*b^4-32*cos(d*x+c)^4*s...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.74 \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {2 \, \sqrt {2} {\left (32 \, a^{3} - 21 \, a b^{2}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, \sqrt {2} {\left (32 \, a^{3} - 21 \, a b^{2}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-32 i \, a^{2} b + 9 i \, b^{3}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (32 i \, a^{2} b - 9 i \, b^{3}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 6 \, {\left (8 \, a b^{2} \cos \left (d x + c\right )^{4} + 2 \, a b^{2} \cos \left (d x + c\right )^{2} - 24 \, a b^{2} - {\left ({\left (32 \, a^{2} b - 9 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 12 \, a^{2} b + 12 \, b^{3} + 2 \, {\left (8 \, a^{2} b - 3 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{360 \, b d \cos \left (d x + c\right )^{5}} \]
1/360*(2*sqrt(2)*(32*a^3 - 21*a*b^2)*sqrt(I*b)*cos(d*x + c)^5*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) + 2*sqrt(2)*(32*a^3 - 21*a *b^2)*sqrt(-I*b)*cos(d*x + c)^5*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)*(-32*I*a^2*b + 9*I*b^3)*sqrt(I*b)*cos(d*x + c)^5*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*sq rt(2)*(32*I*a^2*b - 9*I*b^3)*sqrt(-I*b)*cos(d*x + c)^5*weierstrassZeta(-4/ 3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 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)) - 6*(8*a*b^2*cos(d*x + c)^4 + 2*a*b^2*cos(d*x + c)^2 - 24*a*b^2 - ((32*a^2*b - 9*b^3)*cos(d*x + c)^4 + 12*a^2*b + 12*b^3 + 2*(8*a^2*b - 3*b^3)*cos(d*x + c)^2)*sin(d*x + c))*sqrt (b*sin(d*x + c) + a))/(b*d*cos(d*x + c)^5)
Timed out. \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{6} \,d x } \]
Timed out. \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Hanged} \]