Integrand size = 45, antiderivative size = 674 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\left (1330 a^3 b B+3560 a b^3 B-15 a^4 C+256 b^4 (5 A+4 C)+4 a^2 b^2 (1180 A+809 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{1920 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{128 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{1920 b^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {\left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {\left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{960 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt {\cos (c+d x)}}+\frac {(2 b B+a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)} \]
1/8*(2*B*b+C*a)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(5/2)+1/5*C *(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d/cos(d*x+c)^(5/2)+1/1920*(1330*B*a^3*b +3560*B*a*b^3-15*a^4*C+256*b^4*(5*A+4*C)+4*a^2*b^2*(1180*A+809*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)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)/b/d/cos(d*x+c)^(1/2)/( a+b*sec(d*x+c))^(1/2)-1/128*(10*B*a^4*b-240*B*a^2*b^3-96*B*b^5-3*a^5*C-40* a^3*b^2*(2*A+C)-80*a*b^4*(4*A+3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d *x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(a/(a+b))^(1/2))*((b+a*c os(d*x+c))/(a+b))^(1/2)/b^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+1/24 0*(80*A*b^2+110*B*a*b+15*C*a^2+64*C*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2) /d/cos(d*x+c)^(5/2)+1/960*(590*B*a^2*b+360*B*b^3+15*a^3*C+4*a*b^2*(260*A+1 93*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/b/d/cos(d*x+c)^(3/2)+1/1920*(150* B*a^3*b+2840*B*a*b^3-45*a^4*C+256*b^4*(5*A+4*C)+12*a^2*b^2*(220*A+141*C))* sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/b^2/d/cos(d*x+c)^(1/2)-1/1920*(150*B*a^3 *b+2840*B*a*b^3-45*a^4*C+256*b^4*(5*A+4*C)+12*a^2*b^2*(220*A+141*C))*(cos( 1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^ (1/2)*(a/(a+b))^(1/2))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/b^2/d/((b+a *cos(d*x+c))/(a+b))^(1/2)
Result contains complex when optimal does not.
Time = 43.30 (sec) , antiderivative size = 335495, normalized size of antiderivative = 497.77 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Result too large to show} \]
Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x] ^2))/Cos[c + d*x]^(3/2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )}{\cos (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4753 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+B \csc \left (c+d x+\frac {\pi }{2}\right )+A\right )dx\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \int \frac {1}{2} \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (5 (2 b B+a C) \sec ^2(c+d x)+2 (5 A b+4 C b+5 a B) \sec (c+d x)+a (10 A+3 C)\right )dx+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (5 (2 b B+a C) \sec ^2(c+d x)+2 (5 A b+4 C b+5 a B) \sec (c+d x)+a (10 A+3 C)\right )dx+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (5 (2 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 (5 A b+4 C b+5 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+a (10 A+3 C)\right )dx+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{4} \int \frac {1}{2} \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (\left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right ) \sec ^2(c+d x)+2 \left (40 B a^2+b (80 A+59 C) a+30 b^2 B\right ) \sec (c+d x)+a (80 a A+30 b B+39 a C)\right )dx+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \int \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (\left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right ) \sec ^2(c+d x)+2 \left (40 B a^2+b (80 A+59 C) a+30 b^2 B\right ) \sec (c+d x)+a (80 a A+30 b B+39 a C)\right )dx+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (15 C a^2+110 b B a+80 A b^2+64 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (40 B a^2+b (80 A+59 C) a+30 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (80 a A+30 b B+39 a C)\right )dx+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{3} \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (\left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sec ^2(c+d x)+2 \left (240 B a^3+(720 A b+501 C b) a^2+490 b^2 B a+32 b^3 (5 A+4 C)\right ) \sec (c+d x)+3 a \left ((160 A+93 C) a^2+170 b B a+16 b^2 (5 A+4 C)\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (\left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sec ^2(c+d x)+2 \left (240 B a^3+(720 A b+501 C b) a^2+490 b^2 B a+32 b^3 (5 A+4 C)\right ) \sec (c+d x)+3 a \left ((160 A+93 C) a^2+170 b B a+16 b^2 (5 A+4 C)\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (240 B a^3+(720 A b+501 C b) a^2+490 b^2 B a+32 b^3 (5 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left ((160 A+93 C) a^2+170 b B a+16 b^2 (5 A+4 C)\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sec ^2(c+d x)+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \sec (c+d x)+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sec ^2(c+d x)+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \sec (c+d x)+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left ((960 A+573 C) a^3+1610 b B a^2+4 b^2 (380 A+289 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\int -\frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \sec ^2(c+d x)-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sec (c+d x)+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{b}+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \sec ^2(c+d x)-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sec (c+d x)+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {15 \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 4596 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx+15 \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {15 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {15 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {15 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {15 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right )-2 a b \left (15 C a^3+590 b B a^2+4 b^2 (260 A+193 C) a+360 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {30 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (-15 a^4 C+1330 a^3 b B+4 a^2 b^2 (1180 A+809 C)+3560 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx+\frac {30 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (-15 a^4 C+1330 a^3 b B+4 a^2 b^2 (1180 A+809 C)+3560 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {30 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (-15 a^4 C+1330 a^3 b B+4 a^2 b^2 (1180 A+809 C)+3560 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\frac {30 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (-15 a^4 C+1330 a^3 b B+4 a^2 b^2 (1180 A+809 C)+3560 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\frac {30 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (-15 a^4 C+1330 a^3 b B+4 a^2 b^2 (1180 A+809 C)+3560 a b^3 B+256 b^4 (5 A+4 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {30 \sqrt {\sec (c+d x)} \left (-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{2 b d}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (a C+2 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\right )\) |
Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/C os[c + d*x]^(3/2),x]
3.14.55.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x ]]/Sqrt[a + b*Csc[e + f*x]]) Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]] ), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-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]
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)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 )*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*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] && GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[C/d^2 Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*C sc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[ a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x ]
Result contains complex when optimal does not.
Time = 31.78 (sec) , antiderivative size = 8402, normalized size of antiderivative = 12.47
int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2 ),x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c )^(3/2),x, algorithm="fricas")
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c )^(3/2),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)^(3/2), x)
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c )^(3/2),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)^(3/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^{3/2}} \,d x \]
int(((a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/c os(c + d*x)^(3/2),x)