Integrand size = 43, antiderivative size = 624 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{1920 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (15 A b^4-3560 a^3 b B-1330 a b^3 B-256 a^4 (4 A+5 C)-4 a^2 b^2 (809 A+1180 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{1920 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^5+96 a^5 B+240 a^3 b^2 B-10 a b^4 B+40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{128 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (45 A b^4-2840 a^3 b B-150 a b^3 B-256 a^4 (4 A+5 C)-12 a^2 b^2 (141 A+220 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{1920 a^2 d}+\frac {\left (15 A b^3+360 a^3 B+590 a b^2 B+4 a^2 b (193 A+260 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{960 a d}+\frac {\left (15 A b^2+110 a b B+16 a^2 (4 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{240 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \tan (c+d x)}{5 d} \] Output:
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*(14 1*A+220*C))*(a+b*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b /(a+b))^(1/2))/a^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-1/1920*(15*A*b^4-3560* B*a^3*b-1330*B*a*b^3-256*a^4*(4*A+5*C)-4*a^2*b^2*(809*A+1180*C))*((a+b*cos (d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2 ))/a/d/(a+b*cos(d*x+c))^(1/2)+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))*((a+b*cos(d*x+c))/(a+b))^(1/2) *EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(b/(a+b))^(1/2))/a^2/d/(a+b*cos(d *x+c))^(1/2)-1/1920*(45*A*b^4-2840*B*a^3*b-150*B*a*b^3-256*a^4*(4*A+5*C)-1 2*a^2*b^2*(141*A+220*C))*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a^2/d+1/960*(15 *A*b^3+360*B*a^3+590*B*a*b^2+4*a^2*b*(193*A+260*C))*(a+b*cos(d*x+c))^(1/2) *sec(d*x+c)*tan(d*x+c)/a/d+1/240*(15*A*b^2+110*B*a*b+16*a^2*(4*A+5*C))*(a+ b*cos(d*x+c))^(1/2)*sec(d*x+c)^2*tan(d*x+c)/d+1/8*(A*b+2*B*a)*(a+b*cos(d*x +c))^(3/2)*sec(d*x+c)^3*tan(d*x+c)/d+1/5*A*(a+b*cos(d*x+c))^(5/2)*sec(d*x+ c)^4*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 8.05 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.49 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sec[c + d*x]^6,x]
Output:
((2*(3088*a^3*A*b^2 + 60*a*A*b^4 + 1440*a^4*b*B + 2360*a^2*b^3*B + 4160*a^ 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/( a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(6176*a^4*A*b - 492*a^2*A*b^3 + 135 *A*b^5 + 2880*a^5*B + 4360*a^3*b^2*B - 450*a*b^4*B + 8320*a^4*b*C - 240*a^ 2*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2* b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-1024*a^4*A*b - 1692*a^2*A *b^3 + 45*A*b^5 - 2840*a^3*b^2*B - 150*a*b^4*B - 1280*a^4*b*C - 2640*a^2*b ^3*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)] *Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[ Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*Ellipt icPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], ( a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d* x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^ 2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^ 2)))/(7680*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^4*(21*a*A*b*S in[c + d*x] + 10*a^2*B*Sin[c + d*x]))/40 + (Sec[c + d*x]^3*(64*a^2*A*Sin[c + d*x] + 93*A*b^2*Sin[c + d*x] + 170*a*b*B*Sin[c + d*x] + 80*a^2*C*Sin[c + d*x]))/240 + (Sec[c + d*x]^2*(772*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 360*a^3*B*Sin[c + d*x] + 590*a*b^2*B*Sin[c + d*x] + 1040*a^2*b...
Time = 5.65 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.03, number of steps used = 30, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.698, Rules used = {3042, 3526, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^6(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{5} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (b (3 A+10 C) \cos ^2(c+d x)+2 (4 a A+5 b B+5 a C) \cos (c+d x)+5 (A b+2 a B)\right ) \sec ^5(c+d x)dx+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int (a+b \cos (c+d x))^{3/2} \left (b (3 A+10 C) \cos ^2(c+d x)+2 (4 a A+5 b B+5 a C) \cos (c+d x)+5 (A b+2 a B)\right ) \sec ^5(c+d x)dx+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (3 A+10 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 (4 a A+5 b B+5 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 (A b+2 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{4} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (16 (4 A+5 C) a^2+110 b B a+15 A b^2+b (39 A b+80 C b+30 a B) \cos ^2(c+d x)+2 \left (30 B a^2+b (59 A+80 C) a+40 b^2 B\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \int \sqrt {a+b \cos (c+d x)} \left (16 (4 A+5 C) a^2+110 b B a+15 A b^2+b (39 A b+80 C b+30 a B) \cos ^2(c+d x)+2 \left (30 B a^2+b (59 A+80 C) a+40 b^2 B\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (16 (4 A+5 C) a^2+110 b B a+15 A b^2+b (39 A b+80 C b+30 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (30 B a^2+b (59 A+80 C) a+40 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{3} \int \frac {\left (360 B a^3+2 (386 A b+520 C b) a^2+590 b^2 B a+15 A b^3+3 b \left (16 (4 A+5 C) a^2+170 b B a+b^2 (93 A+160 C)\right ) \cos ^2(c+d x)+2 \left (32 (4 A+5 C) a^3+490 b B a^2+3 b^2 (167 A+240 C) a+240 b^3 B\right ) \cos (c+d x)\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {\left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3+3 b \left (16 (4 A+5 C) a^2+170 b B a+b^2 (93 A+160 C)\right ) \cos ^2(c+d x)+2 \left (32 (4 A+5 C) a^3+490 b B a^2+3 b^2 (167 A+240 C) a+240 b^3 B\right ) \cos (c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3+3 b \left (16 (4 A+5 C) a^2+170 b B a+b^2 (93 A+160 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (32 (4 A+5 C) a^3+490 b B a^2+3 b^2 (167 A+240 C) a+240 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\int -\frac {\left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a-2 \left (360 B a^3+4 b (289 A+380 C) a^2+1610 b^2 B a+3 b^3 (191 A+320 C)\right ) \cos (c+d x) a+45 A b^4-b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\int \frac {\left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a-2 \left (360 B a^3+4 b (289 A+380 C) a^2+1610 b^2 B a+3 b^3 (191 A+320 C)\right ) \cos (c+d x) a+45 A b^4-b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\int \frac {-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a-2 \left (360 B a^3+4 b (289 A+380 C) a^2+1610 b^2 B a+3 b^3 (191 A+320 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+45 A b^4-b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\int -\frac {\left (b \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a+45 A b^4\right ) \cos ^2(c+d x)+2 a b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \cos (c+d x)+15 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\int \frac {\left (b \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a+45 A b^4\right ) \cos ^2(c+d x)+2 a b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \cos (c+d x)+15 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\int \frac {b \left (-256 (4 A+5 C) a^4-2840 b B a^3-12 b^2 (141 A+220 C) a^2-150 b^3 B a+45 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (360 B a^3+4 b (193 A+260 C) a^2+590 b^2 B a+15 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+15 \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \sqrt {a+b \cos (c+d x)}dx-\frac {\int -\frac {\left (15 b \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )-a b \left (-256 (4 A+5 C) a^4-3560 b B a^3-4 b^2 (809 A+1180 C) a^2-1330 b^3 B a+15 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \sqrt {a+b \cos (c+d x)}dx+\frac {\int \frac {\left (15 b \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )-a b \left (-256 (4 A+5 C) a^4-3560 b B a^3-4 b^2 (809 A+1180 C) a^2-1330 b^3 B a+15 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {15 b \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )-a b \left (-256 (4 A+5 C) a^4-3560 b B a^3-4 b^2 (809 A+1180 C) a^2-1330 b^3 B a+15 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {15 b \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )-a b \left (-256 (4 A+5 C) a^4-3560 b B a^3-4 b^2 (809 A+1180 C) a^2-1330 b^3 B a+15 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {15 b \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )-a b \left (-256 (4 A+5 C) a^4-3560 b B a^3-4 b^2 (809 A+1180 C) a^2-1330 b^3 B a+15 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\int \frac {15 b \left (96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right )-a b \left (-256 (4 A+5 C) a^4-3560 b B a^3-4 b^2 (809 A+1180 C) a^2-1330 b^3 B a+15 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {15 b \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-a b \left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {15 b \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {15 b \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b \left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {15 b \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a b \left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {15 b \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a b \left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\frac {15 b \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {\frac {15 b \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{8} \left (\frac {\tan (c+d x) \sec ^2(c+d x) \left (16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\tan (c+d x) \sec (c+d x) \left (360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{2 a d}-\frac {\frac {\tan (c+d x) \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)}}{a d}-\frac {\frac {2 \left (-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\frac {30 b \left (96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 a b \left (-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}}{2 a}}{4 a}\right )\right )+\frac {5 (2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec [c + d*x]^6,x]
Output:
(A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*(A* b + 2*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (((15*A*b^2 + 110*a*b*B + 16*a^2*(4*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Se c[c + d*x]^2*Tan[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))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d* x])/(2*a*d) - (-1/2*((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))*Sqrt[a + b*Cos[c + d*x]]*Elliptic E[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (( -2*a*b*(15*A*b^4 - 3560*a^3*b*B - 1330*a*b^3*B - 256*a^4*(4*A + 5*C) - 4*a ^2*b^2*(809*A + 1180*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (30*b*(3*A*b^5 + 9 6*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))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2 *b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]))/b)/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*Cos[c + d*x]]*Tan[c + d*x])/(a*d))/(4*a))/6)/8)/10
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[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
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/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/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[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(5170\) vs. \(2(600)=1200\).
Time = 1958.42 (sec) , antiderivative size = 5171, normalized size of antiderivative = 8.29
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(5171\) |
| parts | \(\text {Expression too large to display}\) | \(7471\) |
Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^6,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x +c)**6,x)
Output:
Timed out
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^6,x, algorithm="maxima")
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{6} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^6,x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*sec(d*x + c)^6, x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^6} \,d x \] Input:
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)^6,x)
Output:
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)^6, x)
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{6}d x \right ) a^{2} b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{6}d x \right ) b^{2} c +2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{6}d x \right ) a b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{6}d x \right ) b^{3}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{6}d x \right ) a^{2} c +3 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{6}d x \right ) a \,b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{6}d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x)
Output:
3*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**6,x)*a**2*b + in t(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**4*sec(c + d*x)**6,x)*b**2*c + 2*i nt(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**6,x)*a*b*c + int (sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**6,x)*b**3 + int(sq rt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**6,x)*a**2*c + 3*int(s qrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**6,x)*a*b**2 + int(sq rt(cos(c + d*x)*b + a)*sec(c + d*x)**6,x)*a**3