Integrand size = 43, antiderivative size = 289 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 b B+4 a C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d} \]
2/21*(7*A*b^2+14*B*a*b+4*C*a^2+5*C*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/35 *b*(7*B*b+4*C*a)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/7*C*sec(d*x+c)^(3/2)*(a+b *sec(d*x+c))^2*sin(d*x+c)/d+2/5*(10*A*a*b+5*B*a^2+3*B*b^2+6*C*a*b)*sin(d*x +c)*sec(d*x+c)^(1/2)/d-2/5*(10*A*a*b+5*B*a^2+3*B*b^2+6*C*a*b)*(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*(14*B*a*b+7*a^2*(3*A+C)+b^2*(7*A+ 5*C))*(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 = 4.77 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.15 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-21 \left (5 a^2 B+3 b^2 B+2 a b (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+210 a A b \sin (c+d x)+105 a^2 B \sin (c+d x)+63 b^2 B \sin (c+d x)+126 a b C \sin (c+d x)+35 A b^2 \tan (c+d x)+70 a b B \tan (c+d x)+35 a^2 C \tan (c+d x)+25 b^2 C \tan (c+d x)+21 b^2 B \sec (c+d x) \tan (c+d x)+42 a b C \sec (c+d x) \tan (c+d x)+15 b^2 C \sec ^2(c+d x) \tan (c+d x)\right )}{105 d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {7}{2}}(c+d x)} \]
Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(4*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-21*(5* a^2*B + 3*b^2*B + 2*a*b*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x )/2, 2] + 5*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d* x]]*EllipticF[(c + d*x)/2, 2] + 210*a*A*b*Sin[c + d*x] + 105*a^2*B*Sin[c + d*x] + 63*b^2*B*Sin[c + d*x] + 126*a*b*C*Sin[c + d*x] + 35*A*b^2*Tan[c + d*x] + 70*a*b*B*Tan[c + d*x] + 35*a^2*C*Tan[c + d*x] + 25*b^2*C*Tan[c + d* x] + 21*b^2*B*Sec[c + d*x]*Tan[c + d*x] + 42*a*b*C*Sec[c + d*x]*Tan[c + d* x] + 15*b^2*C*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d*(b + a*Cos[c + d*x])^2* (A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2))
Time = 1.75 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.95, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 4584, 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.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left ((7 b B+4 a C) \sec ^2(c+d x)+(7 A b+5 C b+7 a B) \sec (c+d x)+a (7 A+C)\right )dx+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left ((7 b B+4 a C) \sec ^2(c+d x)+(7 A b+5 C b+7 a B) \sec (c+d x)+a (7 A+C)\right )dx+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left ((7 b B+4 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(7 A b+5 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+a (7 A+C)\right )dx+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\sec (c+d x)} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \sec ^2(c+d x)+7 \left (5 B a^2+10 A b a+6 b C a+3 b^2 B\right ) \sec (c+d x)\right )dx+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\sec (c+d x)} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \sec ^2(c+d x)+7 \left (5 B a^2+10 A b a+6 b C a+3 b^2 B\right ) \sec (c+d x)\right )dx+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (5 B a^2+10 A b a+6 b C a+3 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \int \sec ^{\frac {3}{2}}(c+d x)dx+\int \sqrt {\sec (c+d x)} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \sec ^2(c+d x)\right )dx\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 A+C) a^2+5 \left (4 C a^2+14 b B a+7 A b^2+5 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right )}{3 d}+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right )}{3 d}+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right )}{3 d}+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right )}{3 d}+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right )}{3 d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right )}{3 d}+7 \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
(2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2*b *(7*b*B + 4*a*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + ((10*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x) /2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (10*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b ^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + 7*(10*a*A*b + 5*a^2*B + 3*b ^2*B + 6*a*b*C)*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec [c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d))/5)/7
3.10.91.3.1 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[(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_)]*(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]
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[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(m + n + 1) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a *B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(919\) vs. \(2(313)=626\).
Time = 6.86 (sec) , antiderivative size = 920, normalized size of antiderivative = 3.18
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(920\) |
| parts | \(\text {Expression too large to display}\) | \(1172\) |
int(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A*a^2*(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*C*b^2*(-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*a*(2*A*b+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)-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d *x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2))+2/5*b*(B*b+2*C*a)/(8*si n(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*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*EllipticE(c os(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1 /2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2* c)^4+12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^( 1/2)*(sin(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*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2* d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2))*(-2*sin(1/2*d*x+1/2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.24 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{2} + 14 i \, B a b + i \, {\left (7 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{2} - 14 i \, B a b - i \, {\left (7 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, B a^{2} + 2 i \, {\left (5 \, A + 3 \, C\right )} a b + 3 i \, B b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-5 i \, B a^{2} - 2 i \, {\left (5 \, A + 3 \, C\right )} a b - 3 i \, B b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left (5 \, B a^{2} + 2 \, {\left (5 \, A + 3 \, C\right )} a b + 3 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, C b^{2} + 5 \, {\left (7 \, C a^{2} + 14 \, B a b + {\left (7 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
-1/105*(5*sqrt(2)*(7*I*(3*A + C)*a^2 + 14*I*B*a*b + I*(7*A + 5*C)*b^2)*cos (d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5* sqrt(2)*(-7*I*(3*A + C)*a^2 - 14*I*B*a*b - I*(7*A + 5*C)*b^2)*cos(d*x + c) ^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)* (5*I*B*a^2 + 2*I*(5*A + 3*C)*a*b + 3*I*B*b^2)*cos(d*x + c)^3*weierstrassZe ta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21* sqrt(2)*(-5*I*B*a^2 - 2*I*(5*A + 3*C)*a*b - 3*I*B*b^2)*cos(d*x + c)^3*weie rstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c ))) - 2*(21*(5*B*a^2 + 2*(5*A + 3*C)*a*b + 3*B*b^2)*cos(d*x + c)^3 + 15*C* b^2 + 5*(7*C*a^2 + 14*B*a*b + (7*A + 5*C)*b^2)*cos(d*x + c)^2 + 21*(2*C*a* b + B*b^2)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^ 3)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {\sec \left (d x + c\right )} \,d x } \]
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^2*s qrt(sec(d*x + c)), x)
\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {\sec \left (d x + c\right )} \,d x } \]
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^2*s qrt(sec(d*x + c)), x)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]