Integrand size = 33, antiderivative size = 295 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {2 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (5 a^3 A+21 a A b^2+21 a^2 b B+21 b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2 A+18 A b^2+21 a b B\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a^2 (11 A b+7 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))^2 \sin (c+d x)}{7 d} \]
2/21*a*(5*A*a^2+18*A*b^2+21*B*a*b)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/35*a^2* (11*A*b+7*B*a)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/7*a*A*sec(d*x+c)^(3/2)*(b+a *sec(d*x+c))^2*sin(d*x+c)/d+2/5*(9*A*a^2*b+5*A*b^3+3*B*a^3+15*B*a*b^2)*sin (d*x+c)*sec(d*x+c)^(1/2)/d-2/5*(9*A*a^2*b+5*A*b^3+3*B*a^3+15*B*a*b^2)*(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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/21*(5*A*a^3+21*A*a*b^2+21*B* a^2*b+21*B*b^3)*(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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d
Time = 3.82 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.76 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {2 \sqrt {\sec (c+d x)} \left (-21 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (5 a^3 A+21 a A b^2+21 a^2 b B+21 b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+21 \left (9 a^2 A b+5 A b^3+3 a^3 B+15 a b^2 B\right ) \sin (c+d x)+5 a \left (5 a^2 A+21 A b^2+21 a b B\right ) \tan (c+d x)+21 a^2 (3 A b+a B) \sec (c+d x) \tan (c+d x)+15 a^3 A \sec ^2(c+d x) \tan (c+d x)\right )}{105 d} \]
(2*Sqrt[Sec[c + d*x]]*(-21*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sq rt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(5*a^3*A + 21*a*A*b^2 + 21* a^2*b*B + 21*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 21*(9*a ^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sin[c + d*x] + 5*a*(5*a^2*A + 21* A*b^2 + 21*a*b*B)*Tan[c + d*x] + 21*a^2*(3*A*b + a*B)*Sec[c + d*x]*Tan[c + d*x] + 15*a^3*A*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d)
Time = 1.94 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.94, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3439, 3042, 4514, 27, 3042, 4564, 27, 3042, 4535, 3042, 4255, 3042, 4258, 3042, 3119, 4534, 3042, 4258, 3042, 3120}
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.
| \(\displaystyle \int \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{9/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3439 |
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a \sec (c+d x)+b)^3 (A \sec (c+d x)+B)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )^3 \left (A \csc \left (c+d x+\frac {\pi }{2}\right )+B\right )dx\) |
| \(\Big \downarrow \) 4514 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} \sqrt {\sec (c+d x)} (b+a \sec (c+d x)) \left (a (11 A b+7 a B) \sec ^2(c+d x)+\left (5 A a^2+7 b (A b+2 a B)\right ) \sec (c+d x)+b (a A+7 b B)\right )dx+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \sqrt {\sec (c+d x)} (b+a \sec (c+d x)) \left (a (11 A b+7 a B) \sec ^2(c+d x)+\left (5 A a^2+7 b (A b+2 a B)\right ) \sec (c+d x)+b (a A+7 b B)\right )dx+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a (11 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 A a^2+7 b (A b+2 a B)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+b (a A+7 b B)\right )dx+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\sec (c+d x)} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \sec ^2(c+d x)+7 \left (3 B a^3+9 A b a^2+15 b^2 B a+5 A b^3\right ) \sec (c+d x)\right )dx+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\sec (c+d x)} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \sec ^2(c+d x)+7 \left (3 B a^3+9 A b a^2+15 b^2 B a+5 A b^3\right ) \sec (c+d x)\right )dx+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (3 B a^3+9 A b a^2+15 b^2 B a+5 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\sec (c+d x)} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \sec ^2(c+d x)\right )dx+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \int \sec ^{\frac {3}{2}}(c+d x)dx\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (a A+7 b B) b^2+5 a \left (5 A a^2+21 b B a+18 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 a \left (5 a^2 A+21 a b B+18 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 a \left (5 a^2 A+21 a b B+18 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 a \left (5 a^2 A+21 a b B+18 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a \left (5 a^2 A+21 a b B+18 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {1}{5} \left (\frac {10 a \left (5 a^2 A+21 a b B+18 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {10 \left (5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+7 \left (3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}\) |
(2*a*A*Sec[c + d*x]^(3/2)*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2 *a^2*(11*A*b + 7*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + ((10*(5*a^3 *A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (10*a*(5*a^2*A + 18*A*b^2 + 21*a*b *B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + 7*(9*a^2*A*b + 5*A*b^3 + 3*a^ 3*B + 15*a*b^2*B)*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[S ec[c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d))/5)/7
3.6.61.3.1 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[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* (x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim p[g^(m + n) Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(d + c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(m + n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n* Simp[a^2*A*(m + n) + a*b*B*n + (a*(2*A*b + a*B)*(m + n) + b^2*B*(m + n - 1) )*Csc[e + f*x] + b*(A*b*(m + n) + a*B*(2*m + n - 1))*Csc[e + f*x]^2, x], x] , x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && !(IGtQ[n, 1] && !IntegerQ[m])
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
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_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(916\) vs. \(2(319)=638\).
Time = 1876.70 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.11
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(917\) |
| parts | \(\text {Expression too large to display}\) | \(1175\) |
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B*b^3*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1 /2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 2*A*a^3*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2 *c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin( 1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+ 5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*si n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c ),2^(1/2)))+2/5*a^2*(3*A*b+B*a)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2 *c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*cos(1/2*d*x+1/2*c )*sin(1/2*d*x+1/2*c)^6-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4-2 4*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1 /2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin (1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(sin(1/2*d*x +1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/ 2*c),2^(1/2)))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*b^2* (A*b+3*B*a)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d* x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x +1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*E...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.23 \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {5 \, \sqrt {2} {\left (5 i \, A a^{3} + 21 i \, B a^{2} b + 21 i \, A a b^{2} + 21 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-5 i \, A a^{3} - 21 i \, B a^{2} b - 21 i \, A a b^{2} - 21 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (3 i \, B a^{3} + 9 i \, A a^{2} b + 15 i \, B a b^{2} + 5 i \, A b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-3 i \, B a^{3} - 9 i \, A a^{2} b - 15 i \, B a b^{2} - 5 i \, A b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, A a^{3} + 21 \, {\left (3 \, B a^{3} + 9 \, A a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (5 \, A a^{3} + 21 \, B a^{2} b + 21 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
-1/105*(5*sqrt(2)*(5*I*A*a^3 + 21*I*B*a^2*b + 21*I*A*a*b^2 + 21*I*B*b^3)*c os(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-5*I*A*a^3 - 21*I*B*a^2*b - 21*I*A*a*b^2 - 21*I*B*b^3)*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt (2)*(3*I*B*a^3 + 9*I*A*a^2*b + 15*I*B*a*b^2 + 5*I*A*b^3)*cos(d*x + c)^3*we ierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-3*I*B*a^3 - 9*I*A*a^2*b - 15*I*B*a*b^2 - 5*I*A*b^3)*c os(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(15*A*a^3 + 21*(3*B*a^3 + 9*A*a^2*b + 15*B*a*b^2 + 5*A*b^3)*cos(d*x + c)^3 + 5*(5*A*a^3 + 21*B*a^2*b + 21*A*a*b^2)*cos(d*x + c)^2 + 21*(B*a^3 + 3*A*a^2*b)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^3)
Timed out. \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
\[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
Timed out. \[ \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]