Integrand size = 33, antiderivative size = 345 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {2 \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b \left (27 a A b+22 a^2 B+7 b^2 B\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b^2 (9 A b+13 a B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b B \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{9 d} \]
2/21*(21*A*a^2*b+5*A*b^3+7*B*a^3+15*B*a*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)/d +2/45*b*(27*A*a*b+22*B*a^2+7*B*b^2)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/63*b^2 *(9*A*b+13*B*a)*sec(d*x+c)^(7/2)*sin(d*x+c)/d+2/9*b*B*sec(d*x+c)^(5/2)*(a+ b*sec(d*x+c))^2*sin(d*x+c)/d+2/15*(15*A*a^3+27*A*a*b^2+27*B*a^2*b+7*B*b^3) *sin(d*x+c)*sec(d*x+c)^(1/2)/d-2/15*(15*A*a^3+27*A*a*b^2+27*B*a^2*b+7*B*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/21*(21*A*a^2*b+5*A*b^ 3+7*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)*Elli pticF(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 = 9.22 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.31 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\cos ^4(c+d x) \left (\frac {2 \left (-105 a^3 A-189 a A b^2-189 a^2 b B-49 b^3 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (105 a^2 A b+25 A b^3+35 a^3 B+75 a b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{105 d (b+a \cos (c+d x))^3 (B+A \cos (c+d x))}+\frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \left (\frac {2}{15} \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) \sin (c+d x)+\frac {2}{7} \sec ^3(c+d x) \left (A b^3 \sin (c+d x)+3 a b^2 B \sin (c+d x)\right )+\frac {2}{21} \sec (c+d x) \left (21 a^2 A b \sin (c+d x)+5 A b^3 \sin (c+d x)+7 a^3 B \sin (c+d x)+15 a b^2 B \sin (c+d x)\right )+\frac {2}{45} \sec ^2(c+d x) \left (27 a A b^2 \sin (c+d x)+27 a^2 b B \sin (c+d x)+7 b^3 B \sin (c+d x)\right )+\frac {2}{9} b^3 B \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^3 (B+A \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)} \]
(Cos[c + d*x]^4*((2*(-105*a^3*A - 189*a*A*b^2 - 189*a^2*b*B - 49*b^3*B)*El lipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(105* a^2*A*b + 25*A*b^3 + 35*a^3*B + 75*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[( c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d *x]))/(105*d*(b + a*Cos[c + d*x])^3*(B + A*Cos[c + d*x])) + ((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*((2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7 *b^3*B)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^3*(A*b^3*Sin[c + d*x] + 3*a*b^2 *B*Sin[c + d*x]))/7 + (2*Sec[c + d*x]*(21*a^2*A*b*Sin[c + d*x] + 5*A*b^3*S in[c + d*x] + 7*a^3*B*Sin[c + d*x] + 15*a*b^2*B*Sin[c + d*x]))/21 + (2*Sec [c + d*x]^2*(27*a*A*b^2*Sin[c + d*x] + 27*a^2*b*B*Sin[c + d*x] + 7*b^3*B*S in[c + d*x]))/45 + (2*b^3*B*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos [c + d*x])^3*(B + A*Cos[c + d*x])*Sec[c + d*x]^(7/2))
Time = 2.00 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.88, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 4514, 27, 3042, 4564, 27, 3042, 4535, 3042, 4255, 3042, 4258, 3042, 3120, 4534, 3042, 4255, 3042, 4258, 3042, 3119}
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.
| \(\displaystyle \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4514 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (b (9 A b+13 a B) \sec ^2(c+d x)+\left (7 B b^2+9 a (2 A b+a B)\right ) \sec (c+d x)+3 a (3 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (b (9 A b+13 a B) \sec ^2(c+d x)+\left (7 B b^2+9 a (2 A b+a B)\right ) \sec (c+d x)+3 a (3 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b (9 A b+13 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (7 B b^2+9 a (2 A b+a B)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (3 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sec ^{\frac {3}{2}}(c+d x) \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \sec ^2(c+d x)+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \sec (c+d x)\right )dx+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \sec ^2(c+d x)+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \sec (c+d x)\right )dx+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \sec ^{\frac {3}{2}}(c+d x) \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \sec ^2(c+d x)\right )dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \int \sec ^{\frac {5}{2}}(c+d x)dx\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \int \sec ^{\frac {3}{2}}(c+d x)dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}\) |
(2*b*B*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d) + ((2 *b^2*(9*A*b + 13*a*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d) + ((14*b*(27* a*A*b + 22*a^2*B + 7*b^2*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (21*( 15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*((-2*Sqrt[Cos[c + d*x]]*Elli pticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d))/5 + 9*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*((2*Sqrt[C os[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*Sec[ c + d*x]^(3/2)*Sin[c + d*x])/(3*d)))/7)/9
3.5.7.3.1 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_)]*(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. \(1165\) vs. \(2(365)=730\).
Time = 66.81 (sec) , antiderivative size = 1166, normalized size of antiderivative = 3.38
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1166\) |
| parts | \(\text {Expression too large to display}\) | \(1425\) |
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*a^3*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*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-(sin(1/2*d *x+1/2*c)^2)^(1/2)*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))+2*B*b^3*(-1/144*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)^5-7/180*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)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos (1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(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))-7/15*(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))-El lipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+2*b^2*(A*b+3*B*a)*(-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*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^2*(3*A*b+B *a)*(-1/6*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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.16 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {15 \, \sqrt {2} {\left (7 i \, B a^{3} + 21 i \, A a^{2} b + 15 i \, B a b^{2} + 5 i \, A b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-7 i \, B a^{3} - 21 i \, A a^{2} b - 15 i \, B a b^{2} - 5 i \, A b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, A a^{3} + 27 i \, B a^{2} b + 27 i \, A a b^{2} + 7 i \, B b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, A a^{3} - 27 i \, B a^{2} b - 27 i \, A a b^{2} - 7 i \, B b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left (15 \, A a^{3} + 27 \, B a^{2} b + 27 \, A a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )^{4} + 35 \, B b^{3} + 15 \, {\left (7 \, B a^{3} + 21 \, A a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, B a^{2} b + 27 \, A a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}} \]
-1/315*(15*sqrt(2)*(7*I*B*a^3 + 21*I*A*a^2*b + 15*I*B*a*b^2 + 5*I*A*b^3)*c os(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-7*I*B*a^3 - 21*I*A*a^2*b - 15*I*B*a*b^2 - 5*I*A*b^3)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt (2)*(15*I*A*a^3 + 27*I*B*a^2*b + 27*I*A*a*b^2 + 7*I*B*b^3)*cos(d*x + c)^4* weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-15*I*A*a^3 - 27*I*B*a^2*b - 27*I*A*a*b^2 - 7*I*B*b^ 3)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d* x + c) - I*sin(d*x + c))) - 2*(21*(15*A*a^3 + 27*B*a^2*b + 27*A*a*b^2 + 7* B*b^3)*cos(d*x + c)^4 + 35*B*b^3 + 15*(7*B*a^3 + 21*A*a^2*b + 15*B*a*b^2 + 5*A*b^3)*cos(d*x + c)^3 + 7*(27*B*a^2*b + 27*A*a*b^2 + 7*B*b^3)*cos(d*x + c)^2 + 45*(3*B*a*b^2 + A*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c )))/(d*cos(d*x + c)^4)
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]