Integrand size = 35, antiderivative size = 425 \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a (4 A b+3 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
2/9*a*A*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/315*(a^2-b^ 2)*(114*A*a^2*b-10*A*b^3+75*B*a^3+45*B*a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2) /cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))* ((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/a^2/d/(a+b*sec(d*x+c))^(1/ 2)+2/21*a*(4*A*b+3*B*a)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(5/ 2)+2/315*(49*A*a^2+75*A*b^2+135*B*a*b)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d /sec(d*x+c)^(3/2)+2/315*(163*A*a^2*b+5*A*b^3+75*B*a^3+135*B*a*b^2)*sin(d*x +c)*(a+b*sec(d*x+c))^(1/2)/a/d/sec(d*x+c)^(1/2)+2/315*(147*A*a^4+279*A*a^2 *b^2-10*A*b^4+435*B*a^3*b+45*B*a*b^3)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2 *d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec (d*x+c))^(1/2)/a^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)
Time = 7.59 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^{5/2} \left (8 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \left (a^2 \left (261 a^2 A b+155 A b^3+75 a^3 B+405 a b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )+\left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )\right )\right )+a (b+a \cos (c+d x)) \left (2 \left (747 a^2 A b+20 A b^3+345 a^3 B+540 a b^2 B\right ) \sin (c+d x)+a \left (\left (266 a^2 A+300 A b^2+540 a b B\right ) \sin (2 (c+d x))+5 a (2 (19 A b+9 a B) \sin (3 (c+d x))+7 a A \sin (4 (c+d x)))\right )\right )\right )}{1260 a^2 d (b+a \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \]
((a + b*Sec[c + d*x])^(5/2)*(8*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(26 1*a^2*A*b + 155*A*b^3 + 75*a^3*B + 405*a*b^2*B)*EllipticF[(c + d*x)/2, (2* a)/(a + b)] + (147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b ^3*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d* x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[c + d*x])*(2*(747*a^2*A*b + 20*A*b^3 + 345*a^3*B + 540*a*b^2*B)*Sin[c + d*x] + a*((266*a^2*A + 300*A*b^2 + 540 *a*b*B)*Sin[2*(c + d*x)] + 5*a*(2*(19*A*b + 9*a*B)*Sin[3*(c + d*x)] + 7*a* A*Sin[4*(c + d*x)])))))/(1260*a^2*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5 /2))
Time = 3.49 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.02, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4513, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
\(\Big \downarrow \) 4513 |
\(\displaystyle \frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int -\frac {\sqrt {a+b \sec (c+d x)} \left (b (4 a A+9 b B) \sec ^2(c+d x)+\left (7 A a^2+18 b B a+9 A b^2\right ) \sec (c+d x)+3 a (4 A b+3 a B)\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int \frac {\sqrt {a+b \sec (c+d x)} \left (b (4 a A+9 b B) \sec ^2(c+d x)+\left (7 A a^2+18 b B a+9 A b^2\right ) \sec (c+d x)+3 a (4 A b+3 a B)\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (4 a A+9 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (7 A a^2+18 b B a+9 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (4 A b+3 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4582 |
\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {b \left (36 B a^2+76 A b a+63 b^2 B\right ) \sec ^2(c+d x)+\left (45 B a^3+137 A b a^2+189 b^2 B a+63 A b^3\right ) \sec (c+d x)+a \left (49 A a^2+135 b B a+75 A b^2\right )}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {b \left (36 B a^2+76 A b a+63 b^2 B\right ) \sec ^2(c+d x)+\left (45 B a^3+137 A b a^2+189 b^2 B a+63 A b^3\right ) \sec (c+d x)+a \left (49 A a^2+135 b B a+75 A b^2\right )}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {b \left (36 B a^2+76 A b a+63 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^3+137 A b a^2+189 b^2 B a+63 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (49 A a^2+135 b B a+75 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int -\frac {2 a b \left (49 A a^2+135 b B a+75 A b^2\right ) \sec ^2(c+d x)+a \left (147 A a^3+585 b B a^2+605 A b^2 a+315 b^3 B\right ) \sec (c+d x)+3 a \left (75 B a^3+163 A b a^2+135 b^2 B a+5 A b^3\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\int \frac {2 a b \left (49 A a^2+135 b B a+75 A b^2\right ) \sec ^2(c+d x)+a \left (147 A a^3+585 b B a^2+605 A b^2 a+315 b^3 B\right ) \sec (c+d x)+3 a \left (75 B a^3+163 A b a^2+135 b^2 B a+5 A b^3\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\int \frac {2 a b \left (49 A a^2+135 b B a+75 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (147 A a^3+585 b B a^2+605 A b^2 a+315 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (75 B a^3+163 A b a^2+135 b^2 B a+5 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}-\frac {2 \int -\frac {3 \left (\left (75 B a^3+261 A b a^2+405 b^2 B a+155 A b^3\right ) \sec (c+d x) a^2+\left (147 A a^4+435 b B a^3+279 A b^2 a^2+45 b^3 B a-10 A b^4\right ) a\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\int \frac {\left (75 B a^3+261 A b a^2+405 b^2 B a+155 A b^3\right ) \sec (c+d x) a^2+\left (147 A a^4+435 b B a^3+279 A b^2 a^2+45 b^3 B a-10 A b^4\right ) a}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\int \frac {\left (75 B a^3+261 A b a^2+405 b^2 B a+155 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (147 A a^4+435 b B a^3+279 A b^2 a^2+45 b^3 B a-10 A b^4\right ) a}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4523 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4343 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4345 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}+\frac {\frac {2 \left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 a (3 a B+4 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
(2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((6*a*(4*A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Sqrt[a + b*Sec[c + d* x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (((2*(a^2 - b^2)*(114*a^2*A*b - 10*A*b^3 + 75*a^3*B + 45*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*El lipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[ c + d*x]]) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a *b^3*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d *Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]))/a + (2*(163*a^2*A *b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d* x])/(d*Sqrt[Sec[c + d*x]]))/(5*a))/7)/9
3.5.54.3.1 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[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] + Sim p[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[ a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] & & LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(6589\) vs. \(2(443)=886\).
Time = 20.30 (sec) , antiderivative size = 6590, normalized size of antiderivative = 15.51
method | result | size |
parts | \(\text {Expression too large to display}\) | \(6590\) |
default | \(\text {Expression too large to display}\) | \(6634\) |
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left (-225 i \, B a^{5} - 489 i \, A a^{4} b - 345 i \, B a^{3} b^{2} + 93 i \, A a^{2} b^{3} + 90 i \, B a b^{4} - 20 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (225 i \, B a^{5} + 489 i \, A a^{4} b + 345 i \, B a^{3} b^{2} - 93 i \, A a^{2} b^{3} - 90 i \, B a b^{4} + 20 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-147 i \, A a^{5} - 435 i \, B a^{4} b - 279 i \, A a^{3} b^{2} - 45 i \, B a^{2} b^{3} + 10 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (147 i \, A a^{5} + 435 i \, B a^{4} b + 279 i \, A a^{3} b^{2} + 45 i \, B a^{2} b^{3} - 10 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (35 \, A a^{5} \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B a^{5} + 19 \, A a^{4} b\right )} \cos \left (d x + c\right )^{3} + {\left (49 \, A a^{5} + 135 \, B a^{4} b + 75 \, A a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (75 \, B a^{5} + 163 \, A a^{4} b + 135 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{945 \, a^{3} d} \]
1/945*(sqrt(2)*(-225*I*B*a^5 - 489*I*A*a^4*b - 345*I*B*a^3*b^2 + 93*I*A*a^ 2*b^3 + 90*I*B*a*b^4 - 20*I*A*b^5)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*s in(d*x + c) + 2*b)/a) + sqrt(2)*(225*I*B*a^5 + 489*I*A*a^4*b + 345*I*B*a^3 *b^2 - 93*I*A*a^2*b^3 - 90*I*B*a*b^4 + 20*I*A*b^5)*sqrt(a)*weierstrassPInv erse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d* x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(-147*I*A*a^5 - 435*I*B* a^4*b - 279*I*A*a^3*b^2 - 45*I*B*a^2*b^3 + 10*I*A*a*b^4)*sqrt(a)*weierstra ssZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPI nverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos( d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(147*I*A*a^5 + 435*I* B*a^4*b + 279*I*A*a^3*b^2 + 45*I*B*a^2*b^3 - 10*I*A*a*b^4)*sqrt(a)*weierst rassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrass PInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*co s(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a)) + 6*(35*A*a^5*cos(d*x + c)^4 + 5*(9*B*a^5 + 19*A*a^4*b)*cos(d*x + c)^3 + (49*A*a^5 + 135*B*a^4*b + 75*A*a ^3*b^2)*cos(d*x + c)^2 + (75*B*a^5 + 163*A*a^4*b + 135*B*a^3*b^2 + 5*A*a^2 *b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*sin(d*x + c)/s qrt(cos(d*x + c)))/(a^3*d)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]