Integrand size = 23, antiderivative size = 330 \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {a \left (32 a^2-29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{60 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (32 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{60 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{60 \left (a^2-b^2\right )^2 d} \]
-1/30*sec(d*x+c)^3*(b-8*a*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/d+1/5*sec(d*x +c)^5*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/d-1/60*sec(d*x+c)*(b*(8*a^4- 13*a^2*b^2+5*b^4)-a*(32*a^4-61*a^2*b^2+29*b^4)*sin(d*x+c))*(a+b*sin(d*x+c) )^(1/2)/(a^2-b^2)^2/d+1/60*a*(32*a^2-29*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)/(a^2-b^2)/d/((a+b*sin(d*x+c))/( a+b))^(1/2)-1/60*(32*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)/d/(a+b*sin(d*x+c))^(1/2)
Time = 6.30 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.10 \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {a+b \sin (c+d x)} \left (\frac {1}{5} \sec ^5(c+d x) (b+a \sin (c+d x))+\frac {1}{30} \sec ^3(c+d x) (-b+8 a \sin (c+d x))+\frac {\sec (c+d x) \left (-8 a^2 b+5 b^3+32 a^3 \sin (c+d x)-29 a b^2 \sin (c+d x)\right )}{60 \left (a^2-b^2\right )}\right )}{d}-\frac {b \left (-\frac {2 \left (8 a^2 b-5 b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {\left (32 a^3-29 a b^2\right ) \left (\frac {2 (a+b) E\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}\right )}{b}\right )}{120 (a-b) (a+b) d} \]
(Sqrt[a + b*Sin[c + d*x]]*((Sec[c + d*x]^5*(b + a*Sin[c + d*x]))/5 + (Sec[ c + d*x]^3*(-b + 8*a*Sin[c + d*x]))/30 + (Sec[c + d*x]*(-8*a^2*b + 5*b^3 + 32*a^3*Sin[c + d*x] - 29*a*b^2*Sin[c + d*x]))/(60*(a^2 - b^2))))/d - (b*( (-2*(8*a^2*b - 5*b^3)*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^3 - 29*a*b ^2)*((2*(a + b)*EllipticE[(-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]] - (2*a*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]]))/b))/(120*(a - b)*(a + b)*d)
Time = 2.00 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.21, 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, 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.
| \(\displaystyle \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^{3/2}}{\cos (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}-\frac {1}{5} \int -\frac {\sec ^4(c+d x) \left (8 a^2+7 b \sin (c+d x) a-b^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int \frac {\sec ^4(c+d x) \left (8 a^2+7 b \sin (c+d x) a-b^2\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \int \frac {8 a^2+7 b \sin (c+d x) a-b^2}{\cos (c+d x)^4 \sqrt {a+b \sin (c+d x)}}dx+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {1}{10} \left (-\frac {\int -\frac {\sec ^2(c+d x) \left (32 a^4-37 b^2 a^2+24 b \left (a^2-b^2\right ) \sin (c+d x) a+5 b^4\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {\int \frac {\sec ^2(c+d x) \left (32 a^4-37 b^2 a^2+24 b \left (a^2-b^2\right ) \sin (c+d x) a+5 b^4\right )}{\sqrt {a+b \sin (c+d x)}}dx}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {\int \frac {32 a^4-37 b^2 a^2+24 b \left (a^2-b^2\right ) \sin (c+d x) a+5 b^4}{\cos (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\int \frac {\left (8 a^4-13 b^2 a^2+5 b^4\right ) b^2+a \left (32 a^4-61 b^2 a^2+29 b^4\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\int \frac {\left (8 a^4-13 b^2 a^2+5 b^4\right ) b^2+a \left (32 a^4-61 b^2 a^2+29 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\int \frac {\left (8 a^4-13 b^2 a^2+5 b^4\right ) b^2+a \left (32 a^4-61 b^2 a^2+29 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (32 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (32 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\frac {a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (32 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\frac {a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (32 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\frac {2 a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (32 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\frac {2 a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\frac {2 a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (8 a^4-13 a^2 b^2+5 b^4\right )-a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\frac {2 a \left (32 a^4-61 a^2 b^2+29 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (32 a^2-5 b^2\right ) \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-8 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{5 d}\) |
(Sec[c + d*x]^5*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(5*d) + (-1 /3*(Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(b*(a^2 - b^2) - 8*a*(a^2 - b^ 2)*Sin[c + d*x]))/((a^2 - b^2)*d) + (-((Sec[c + d*x]*Sqrt[a + b*Sin[c + d* x]]*(b*(8*a^4 - 13*a^2*b^2 + 5*b^4) - a*(32*a^4 - 61*a^2*b^2 + 29*b^4)*Sin [c + d*x]))/((a^2 - b^2)*d)) - ((2*a*(32*a^4 - 61*a^2*b^2 + 29*b^4)*Ellipt icE[(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*(32*a^2 - 5*b^2)*(a^2 - b^2)^2*Elliptic F[(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)))/10
3.5.93.3.1 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*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. \(1518\) vs. \(2(372)=744\).
Time = 4.84 (sec) , antiderivative size = 1519, normalized size of antiderivative = 4.60
1/120*(-2*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*a*b^2*(32*a^2-2 9*b^2)*cos(d*x+c)^6+2*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*b*( 32*a^4-37*a^2*b^2+5*b^4)*cos(d*x+c)^4*sin(d*x+c)+2*(b*cos(d*x+c)^2*sin(d*x +c)+a*cos(d*x+c)^2)^(1/2)*(32*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/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^5-61*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/(a+b)*sin(d*x+c)+b/ (a+b))^(1/2)*a^3*b^2+29*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/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a*b^4-32*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/(a+b)*sin(d*x+c)+b/(a+b ))^(1/2)*a^4*b+24*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/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^3*b^2+37*EllipticF((b/(a-b)*s in(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/(a+b)*sin(d*x+c)+b/(a+b))^( 1/2)*a^2*b^3-24*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)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.85 \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {2} {\left (64 \, a^{4} - 82 \, a^{2} b^{2} + 15 \, b^{4}\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 ) + \sqrt {2} {\left (64 \, a^{4} - 82 \, a^{2} b^{2} + 15 \, b^{4}\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^{3} b - 29 i \, a 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^{3} b + 29 i \, a 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 ({\left (8 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )^{4} - 12 \, a^{2} b^{2} + 12 \, b^{4} + 2 \, {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (32 \, a^{3} b - 29 \, a b^{3}\right )} \cos \left (d x + c\right )^{4} + 12 \, a^{3} b - 12 \, a b^{3} + 16 \, {\left (a^{3} b - a 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 \, {\left (a^{2} b - b^{3}\right )} d \cos \left (d x + c\right )^{5}} \]
1/360*(sqrt(2)*(64*a^4 - 82*a^2*b^2 + 15*b^4)*sqrt(I*b)*cos(d*x + c)^5*wei erstrassPInverse(-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)*(64*a^4 - 82*a^2*b^2 + 15*b^4)*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^3*b - 29*I*a*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*sqrt(2)*(-32*I*a^3*b + 29*I*a*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)) - 6*( (8*a^2*b^2 - 5*b^4)*cos(d*x + c)^4 - 12*a^2*b^2 + 12*b^4 + 2*(a^2*b^2 - b^ 4)*cos(d*x + c)^2 - ((32*a^3*b - 29*a*b^3)*cos(d*x + c)^4 + 12*a^3*b - 12* a*b^3 + 16*(a^3*b - a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/((a^2*b - b^3)*d*cos(d*x + c)^5)
Timed out. \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{6} \,d x } \]
Timed out. \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Hanged} \]