Integrand size = 43, antiderivative size = 652 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a b d}+\frac {\sqrt {a+b} \left (15 A b^3+8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}+\frac {\sqrt {a+b} \left (5 A b^4-160 a^3 b B-40 a b^3 B-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d} \]
1/24*(5*A*b+8*B*a)*cos(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/4*A* cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d+1/192*(a-b)*(15*A*b^3+128 *B*a^3+264*B*a*b^2+4*a^2*b*(71*A+108*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x +c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/ (a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a/b/d+1/192*(15*A*b^3+8*a^3*( 9*A+16*B+12*C)+4*a^2*b*(71*A+52*B+108*C)+2*a*b^2*(59*A+132*B+192*C))*cot(d *x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a +b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a /d+1/64*(5*A*b^4-160*B*a^3*b-40*B*a*b^3-120*a^2*b^2*(A+2*C)-16*a^4*(3*A+4* C))*cot(d*x+c)*EllipticPi((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b )/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+ c))/(a-b))^(1/2)/a^2/d+1/192*(15*A*b^3+128*B*a^3+264*B*a*b^2+4*a^2*b*(71*A +108*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d+1/32*(5*A*b^2+24*B*a*b+4*a^ 2*(3*A+4*C))*cos(d*x+c)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(5667\) vs. \(2(652)=1304\).
Time = 37.57 (sec) , antiderivative size = 5667, normalized size of antiderivative = 8.69 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]
Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Time = 3.50 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}
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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (b (A+8 C) \sec ^2(c+d x)+2 (3 a A+4 b B+4 a C) \sec (c+d x)+5 A b+8 a B\right )dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (b (A+8 C) \sec ^2(c+d x)+2 (3 a A+4 b B+4 a C) \sec (c+d x)+5 A b+8 a B\right )dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (A+8 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 (3 a A+4 b B+4 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+5 A b+8 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {1}{2} \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (11 A b+48 C b+8 a B) \sec ^2(c+d x)+2 \left (16 B a^2+b (31 A+48 C) a+24 b^2 B\right ) \sec (c+d x)+3 \left (4 (3 A+4 C) a^2+24 b B a+5 A b^2\right )\right )dx+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (11 A b+48 C b+8 a B) \sec ^2(c+d x)+2 \left (16 B a^2+b (31 A+48 C) a+24 b^2 B\right ) \sec (c+d x)+3 \left (4 (3 A+4 C) a^2+24 b B a+5 A b^2\right )\right )dx+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (11 A b+48 C b+8 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (16 B a^2+b (31 A+48 C) a+24 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (4 (3 A+4 C) a^2+24 b B a+5 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {\cos (c+d x) \left (128 B a^3+4 b (71 A+108 C) a^2+264 b^2 B a+15 A b^3+b \left (12 (3 A+4 C) a^2+104 b B a+b^2 (59 A+192 C)\right ) \sec ^2(c+d x)+2 \left (12 (3 A+4 C) a^3+152 b B a^2+b^2 (161 A+288 C) a+96 b^3 B\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\cos (c+d x) \left (128 B a^3+4 b (71 A+108 C) a^2+264 b^2 B a+15 A b^3+b \left (12 (3 A+4 C) a^2+104 b B a+b^2 (59 A+192 C)\right ) \sec ^2(c+d x)+2 \left (12 (3 A+4 C) a^3+152 b B a^2+b^2 (161 A+288 C) a+96 b^3 B\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {128 B a^3+4 b (71 A+108 C) a^2+264 b^2 B a+15 A b^3+b \left (12 (3 A+4 C) a^2+104 b B a+b^2 (59 A+192 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (12 (3 A+4 C) a^3+152 b B a^2+b^2 (161 A+288 C) a+96 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (128 B a^3+4 b (71 A+108 C) a^2+264 b^2 B a+15 A b^3\right ) \sec ^2(c+d x)-2 a b \left (12 (3 A+4 C) a^2+104 b B a+b^2 (59 A+192 C)\right ) \sec (c+d x)+3 \left (-16 (3 A+4 C) a^4-160 b B a^3-120 b^2 (A+2 C) a^2-40 b^3 B a+5 A b^4\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (128 B a^3+4 b (71 A+108 C) a^2+264 b^2 B a+15 A b^3\right ) \sec ^2(c+d x)-2 a b \left (12 (3 A+4 C) a^2+104 b B a+b^2 (59 A+192 C)\right ) \sec (c+d x)+3 \left (-16 (3 A+4 C) a^4-160 b B a^3-120 b^2 (A+2 C) a^2-40 b^3 B a+5 A b^4\right )}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (128 B a^3+4 b (71 A+108 C) a^2+264 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (12 (3 A+4 C) a^2+104 b B a+b^2 (59 A+192 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-16 (3 A+4 C) a^4-160 b B a^3-120 b^2 (A+2 C) a^2-40 b^3 B a+5 A b^4\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {3 \left (-16 (3 A+4 C) a^4-160 b B a^3-120 b^2 (A+2 C) a^2-40 b^3 B a+5 A b^4\right )+\left (-b \left (128 B a^3+4 b (71 A+108 C) a^2+264 b^2 B a+15 A b^3\right )-2 a b \left (12 (3 A+4 C) a^2+104 b B a+b^2 (59 A+192 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {3 \left (-16 (3 A+4 C) a^4-160 b B a^3-120 b^2 (A+2 C) a^2-40 b^3 B a+5 A b^4\right )+\left (-b \left (128 B a^3+4 b (71 A+108 C) a^2+264 b^2 B a+15 A b^3\right )-2 a b \left (12 (3 A+4 C) a^2+104 b B a+b^2 (59 A+192 C)\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+3 \left (-16 a^4 (3 A+4 C)-160 a^3 b B-120 a^2 b^2 (A+2 C)-40 a b^3 B+5 A b^4\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (-16 a^4 (3 A+4 C)-160 a^3 b B-120 a^2 b^2 (A+2 C)-40 a b^3 B+5 A b^4\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-16 a^4 (3 A+4 C)-160 a^3 b B-120 a^2 b^2 (A+2 C)-40 a b^3 B+5 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \cot (c+d x) \left (8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-16 a^4 (3 A+4 C)-160 a^3 b B-120 a^2 b^2 (A+2 C)-40 a b^3 B+5 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}\right )+\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {3 \sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 d}+\frac {1}{4} \left (\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-\frac {2 \sqrt {a+b} \cot (c+d x) \left (8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-16 a^4 (3 A+4 C)-160 a^3 b B-120 a^2 b^2 (A+2 C)-40 a b^3 B+5 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}\right )\right )+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
(A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d) + (((5*A* b + 8*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*(5*A*b^2 + 24*a*b*B + 4*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[ c + d*x]]*Sin[c + d*x])/(2*d) + (-1/2*((-2*(a - b)*Sqrt[a + b]*(15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Cot[c + d*x]*EllipticE[ ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*Sqrt[a + b]*(15*A*b^3 + 8*a^3*(9*A + 16*B + 12*C) + 4*a^2*b*(71*A + 52 *B + 108*C) + 2*a*b^2*(59*A + 132*B + 192*C))*Cot[c + d*x]*EllipticF[ArcSi n[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec [c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (6*Sqrt[ a + b]*(5*A*b^4 - 160*a^3*b*B - 40*a*b^3*B - 120*a^2*b^2*(A + 2*C) - 16*a^ 4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b) ]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d))/a + ((15*A*b^3 + 128*a^3 *B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d))/4)/6)/8
3.10.56.3.1 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[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) *((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[d Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C )*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C Int[Csc[e + f*x]*(( 1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A , B, C}, x] && 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. \(8027\) vs. \(2(603)=1206\).
Time = 519.44 (sec) , antiderivative size = 8028, normalized size of antiderivative = 12.31
int(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
integral((C*b^2*cos(d*x + c)^4*sec(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^4*sec(d*x + c)^3 + A*a^2*cos(d*x + c)^4 + (C*a^2 + 2*B*a*b + A*b^2)*c os(d*x + c)^4*sec(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c)^4*sec(d*x + c))*sqrt(b*sec(d*x + c) + a), x)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(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)^(5/ 2)*cos(d*x + c)^4, x)
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(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)^(5/ 2)*cos(d*x + c)^4, x)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]