Integrand size = 43, antiderivative size = 774 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1920 a^2 b d}-\frac {\sqrt {a+b} \left (45 A b^4-30 a b^3 (A+5 B)-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1920 a^2 d}-\frac {\sqrt {a+b} \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{128 a^3 d}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {(A b+2 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d} \]
1/8*(A*b+2*B*a)*cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/5*A*cos (d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d-1/1920*(a-b)*(45*A*b^4-2840* B*a^3*b-150*B*a*b^3-256*a^4*(4*A+5*C)-12*a^2*b^2*(141*A+220*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^2/b/ d-1/1920*(45*A*b^4-30*a*b^3*(A+5*B)-16*a^4*(64*A+45*B+80*C)-8*a^3*b*(193*A +355*B+260*C)-4*a^2*b^2*(423*A+295*B+660*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^2/d-1/128*(3*A*b^5+96*a ^5*B+240*a^3*b^2*B-10*a*b^4*B+40*a^2*b^3*(A+2*C)+80*a^4*b*(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^3/d-1/1920*(45*A*b^4-2840*B*a^3*b-150*B*a*b^3-256*a^4*(4*A+5*C)- 12*a^2*b^2*(141*A+220*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^2/d+1/960*(1 5*A*b^3+360*B*a^3+590*B*a*b^2+4*a^2*b*(193*A+260*C))*cos(d*x+c)*sin(d*x+c) *(a+b*sec(d*x+c))^(1/2)/a/d+1/240*(15*A*b^2+110*B*a*b+16*a^2*(4*A+5*C))*co s(d*x+c)^2*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d
Time = 28.29 (sec) , antiderivative size = 711, normalized size of antiderivative = 0.92 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{480} \left (88 a^2 A+93 A b^2+170 a b B+80 a^2 C\right ) \sin (c+d x)+\frac {\left (1024 a^2 A b+15 A b^3+480 a^3 B+590 a b^2 B+1040 a^2 b C\right ) \sin (2 (c+d x))}{960 a}+\frac {1}{480} \left (100 a^2 A+93 A b^2+170 a b B+80 a^2 C\right ) \sin (3 (c+d x))+\frac {1}{160} a (21 A b+10 a B) \sin (4 (c+d x))+\frac {1}{40} a^2 A \sin (5 (c+d x))\right )}{d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}-\frac {\cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {\left ((a+b) \left (-45 A b^4+2840 a^3 b B+150 a b^3 B+256 a^4 (4 A+5 C)+12 a^2 b^2 (141 A+220 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )-2 a \left (-15 A b^4+720 a^4 B-4 a^2 b^2 (193 A-805 B+260 C)+8 a^3 b (289 A-45 B+380 C)+2 a b^3 (573 A-295 B+960 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )+30 \left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}}{(b+a \cos (c+d x)) \sqrt {\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )}}-\left (-45 A b^4+2840 a^3 b B+150 a b^3 B+256 a^4 (4 A+5 C)+12 a^2 b^2 (141 A+220 C)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{960 a^2 d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))} \]
Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(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)*(((88*a^2*A + 93*A*b^2 + 170*a*b*B + 80*a^2*C)*Sin[c + d*x])/480 + ((1024*a^2*A*b + 15*A*b^3 + 480*a^3*B + 590*a*b^2*B + 1040*a^2*b*C)*Sin[ 2*(c + d*x)])/(960*a) + ((100*a^2*A + 93*A*b^2 + 170*a*b*B + 80*a^2*C)*Sin [3*(c + d*x)])/480 + (a*(21*A*b + 10*a*B)*Sin[4*(c + d*x)])/160 + (a^2*A*S in[5*(c + d*x)])/40))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x ] + A*Cos[2*c + 2*d*x])) - (Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-((((a + b)*(-45*A*b^4 + 2840*a^3*b*B + 150*a*b^3*B + 256*a^4*(4*A + 5*C) + 12*a^2*b^2*(141*A + 220*C))*Ellipti cE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*(-15*A*b^4 + 720*a^4*B - 4*a^2*b^2*(193*A - 805*B + 260*C) + 8*a^3*b*(289*A - 45*B + 380*C) + 2* a*b^3*(573*A - 295*B + 960*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b) /(a + b)] + 30*(3*A*b^5 + 96*a^5*B + 240*a^3*b^2*B - 10*a*b^4*B + 40*a^2*b ^3*(A + 2*C) + 80*a^4*b*(3*A + 4*C))*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2 ]], (a - b)/(a + b)])*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/((b + a*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2])) - (-45* A*b^4 + 2840*a^3*b*B + 150*a*b^3*B + 256*a^4*(4*A + 5*C) + 12*a^2*b^2*(141 *A + 220*C))*Tan[(c + d*x)/2]))/(960*a^2*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))
Time = 4.58 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.02, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.535, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 27, 3042, 4592, 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.
| \(\displaystyle \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{5} \int \frac {1}{2} \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (b (3 A+10 C) \sec ^2(c+d x)+2 (4 a A+5 b B+5 a C) \sec (c+d x)+5 (A b+2 a B)\right )dx+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (b (3 A+10 C) \sec ^2(c+d x)+2 (4 a A+5 b B+5 a C) \sec (c+d x)+5 (A b+2 a B)\right )dx+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (3 A+10 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 (4 a A+5 b B+5 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+5 (A b+2 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{4} \int \frac {1}{2} \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (16 (4 A+5 C) a^2+110 b B a+15 A b^2+b (39 A b+80 C b+30 a B) \sec ^2(c+d x)+2 \left (30 B a^2+b (59 A+80 C) a+40 b^2 B\right ) \sec (c+d x)\right )dx+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (16 (4 A+5 C) a^2+110 b B a+15 A b^2+b (39 A b+80 C b+30 a B) \sec ^2(c+d x)+2 \left (30 B a^2+b (59 A+80 C) a+40 b^2 B\right ) \sec (c+d x)\right )dx+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (16 (4 A+5 C) a^2+110 b B a+15 A b^2+b (39 A b+80 C b+30 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (30 B a^2+b (59 A+80 C) a+40 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{3} \int \frac {\cos ^2(c+d x) \left (360 B a^3+2 (386 A b+520 C b) a^2+590 b^2 B a+15 A b^3+3 b \left (16 (4 A+5 C) a^2+170 b B a+b^2 (93 A+160 C)\right ) \sec ^2(c+d x)+2 \left (32 (4 A+5 C) a^3+490 b B a^2+3 b^2 (167 A+240 C) a+240 b^3 B\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {\cos ^2(c+d x) \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3+3 b \left (16 (4 A+5 C) a^2+170 b B a+b^2 (93 A+160 C)\right ) \sec ^2(c+d x)+2 \left (32 (4 A+5 C) a^3+490 b B a^2+3 b^2 (167 A+240 C) a+240 b^3 B\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3+3 b \left (16 (4 A+5 C) a^2+170 b B a+b^2 (93 A+160 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (32 (4 A+5 C) a^3+490 b B a^2+3 b^2 (167 A+240 C) a+240 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a-2 \left (360 B a^3+4 b (289 A+380 C) a^2+1610 b^2 B a+3 b^3 (191 A+320 C)\right ) \sec (c+d x) a+45 A b^4-b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a-2 \left (360 B a^3+4 b (289 A+380 C) a^2+1610 b^2 B a+3 b^3 (191 A+320 C)\right ) \sec (c+d x) a+45 A b^4-b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a-2 \left (360 B a^3+4 b (289 A+380 C) a^2+1610 b^2 B a+3 b^3 (191 A+320 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+45 A b^4-b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a+45 A b^4\right ) \sec ^2(c+d x)+2 a b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \sec (c+d x)+15 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a+45 A b^4\right ) \sec ^2(c+d x)+2 a b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \sec (c+d x)+15 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a+45 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+15 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {15 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )+\left (2 a b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right )-b \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a+45 A b^4\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {15 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )+\left (2 a b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right )-b \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a+45 A b^4\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)-30 a b^3 (A+5 B)+45 A b^4\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+15 \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)-30 a b^3 (A+5 B)+45 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)-30 a b^3 (A+5 B)+45 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 \sqrt {a+b} \cot (c+d x) \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \cot (c+d x) \left (-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)-30 a b^3 (A+5 B)+45 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {30 \sqrt {a+b} \cot (c+d x) \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-\frac {2 \sqrt {a+b} \cot (c+d x) \left (-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)-30 a b^3 (A+5 B)+45 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {30 \sqrt {a+b} \cot (c+d x) \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}\right )\right )+\frac {5 (2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}\) |
(A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d) + ((5*(A* b + 2*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + (((15*A*b^2 + 110*a*b*B + 16*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sqrt[a + b*S ec[c + d*x]]*Sin[c + d*x])/(3*d) + (((15*A*b^3 + 360*a^3*B + 590*a*b^2*B + 4*a^2*b*(193*A + 260*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d* x])/(2*a*d) - (-1/2*((-2*(a - b)*Sqrt[a + b]*(45*A*b^4 - 2840*a^3*b*B - 15 0*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*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]*(45*A*b^4 - 30*a*b^3*(A + 5*B) - 16*a^4*(64*A + 45*B + 80*C) - 8*a^3*b*(193*A + 355*B + 260*C) - 4*a^2*b^2*(423*A + 295*B + 660*C))*Cot[c + d*x]*EllipticF[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))])/d - (30*Sqrt[a + b]*(3*A*b^5 + 96*a^5*B + 240*a ^3*b^2*B - 10*a*b^4*B + 40*a^2*b^3*(A + 2*C) + 80*a^4*b*(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 + ((45*A*b^4 - 2840*a^3*b*B - 150*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Sqrt[a + b*Sec[c + d*x ]]*Sin[c + d*x])/(a*d))/(4*a))/6)/8)/10
3.10.57.3.1 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. \(9714\) vs. \(2(721)=1442\).
Time = 1264.89 (sec) , antiderivative size = 9715, normalized size of antiderivative = 12.55
int(cos(d*x+c)^5*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
integrate(cos(d*x+c)^5*(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)^5*sec(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^5*sec(d*x + c)^3 + A*a^2*cos(d*x + c)^5 + (C*a^2 + 2*B*a*b + A*b^2)*c os(d*x + c)^5*sec(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c)^5*sec(d*x + c))*sqrt(b*sec(d*x + c) + a), x)
Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
integrate(cos(d*x+c)^5*(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)^5, x)
\[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{5} \,d x } \]
integrate(cos(d*x+c)^5*(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)^5, x)
Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^5\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]