Integrand size = 45, antiderivative size = 419 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\left (10 a^3 B+20 a b^2 B-b^3 (16 A-15 C)+4 a^2 b (4 A+15 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 d \sqrt {a+b \sec (c+d x)}}+\frac {b^2 (2 b B+5 a C) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{d \sqrt {a+b \sec (c+d x)}}+\frac {\left (70 a b B+b^2 (46 A-15 C)+6 a^2 (3 A+5 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}-\frac {b (16 A b+10 a B-15 b C) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (A b+a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)} \]
2/5*A*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/3*(A*b+B*a)*( a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+1/15*(10*B*a^3+20*B*a* b^2-b^3*(16*A-15*C)+4*a^2*b*(4*A+15*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)*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+b^2*(2* B*b+5*C*a)*(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*cos(d*x+c))/(a+b))^(1/2)*s ec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+1/15*(70*B*a*b+b^2*(46*A-15*C)+6* a^2*(3*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(s in(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/d/((b+a* cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)-1/15*b*(16*A*b+10*B*a-15*C*b)*si n(d*x+c)*sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/d
Result contains complex when optimal does not.
Time = 14.06 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (68 a^2 A b+60 A b^3+20 a^3 B+180 a b^2 B+180 a^2 b C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 \left (18 a^3 A+46 a A b^2+70 a^2 b B+60 b^3 B+30 a^3 C+135 a b^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 i \left (18 a^3 A+46 a A b^2+70 a^2 b B+30 a^3 C-15 a b^2 C\right ) \sqrt {\frac {a-a \cos (c+d x)}{a+b}} \sqrt {\frac {a+a \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (-2 b (a+b) E\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right )|\frac {-a+b}{a+b}\right )+a \left (2 b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )+a \operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )\right )\right ) \sin (c+d x)}{\sqrt {\frac {1}{a-b}} b \sqrt {1-\cos ^2(c+d x)} \sqrt {\frac {a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2-4 b (b+a \cos (c+d x))+2 (b+a \cos (c+d x))^2\right )}\right )}{30 d (b+a \cos (c+d x))^{5/2} (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4}{15} a (11 A b+5 a B) \sin (c+d x)+\frac {2}{5} a^2 A \sin (2 (c+d x))+2 b^2 C \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]
Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x] ^2))/Sec[c + d*x]^(5/2),x]
((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(6 8*a^2*A*b + 60*A*b^3 + 20*a^3*B + 180*a*b^2*B + 180*a^2*b*C)*Sqrt[(b + a*C os[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Co s[c + d*x]] + (2*(18*a^3*A + 46*a*A*b^2 + 70*a^2*b*B + 60*b^3*B + 30*a^3*C + 135*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x) /2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(18*a^3*A + 46*a*A*b ^2 + 70*a^2*b*B + 30*a^3*C - 15*a*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b) ]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*Ellipt icE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x ]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1) ]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*( -a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(30 *d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2* d*x])*Sec[c + d*x]^(9/2)) + ((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x ] + C*Sec[c + d*x]^2)*((4*a*(11*A*b + 5*a*B)*Sin[c + d*x])/15 + (2*a^2*A*S in[2*(c + d*x)])/5 + 2*b^2*C*Tan[c + d*x]))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))
Time = 4.41 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.02, number of steps used = 29, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.644, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4584, 27, 3042, 4596, 3042, 4346, 3042, 3286, 3042, 3284, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {2}{5} \int \frac {(a+b \sec (c+d x))^{3/2} \left (-b (2 A-5 C) \sec ^2(c+d x)+(3 a A+5 b B+5 a C) \sec (c+d x)+5 (A b+a B)\right )}{2 \sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {(a+b \sec (c+d x))^{3/2} \left (-b (2 A-5 C) \sec ^2(c+d x)+(3 a A+5 b B+5 a C) \sec (c+d x)+5 (A b+a B)\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-b (2 A-5 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(3 a A+5 b B+5 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+5 (A b+a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {\sqrt {a+b \sec (c+d x)} \left (-b (16 A b-15 C b+10 a B) \sec ^2(c+d x)+\left (5 B a^2+8 A b a+30 b C a+15 b^2 B\right ) \sec (c+d x)+3 \left ((3 A+5 C) a^2+10 b B a+5 A b^2\right )\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {\sqrt {a+b \sec (c+d x)} \left (-b (16 A b-15 C b+10 a B) \sec ^2(c+d x)+\left (5 B a^2+8 A b a+30 b C a+15 b^2 B\right ) \sec (c+d x)+3 \left ((3 A+5 C) a^2+10 b B a+5 A b^2\right )\right )}{\sqrt {\sec (c+d x)}}dx+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (-b (16 A b-15 C b+10 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 B a^2+8 A b a+30 b C a+15 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left ((3 A+5 C) a^2+10 b B a+5 A b^2\right )\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\int \frac {15 b^2 (2 b B+5 a C) \sec ^2(c+d x)+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \sec (c+d x)+a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {15 b^2 (2 b B+5 a C) \sec ^2(c+d x)+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \sec (c+d x)+a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {15 b^2 (2 b B+5 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4596 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx+15 b^2 (5 a C+2 b B) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 b^2 (5 a C+2 b B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {15 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {15 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {15 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {15 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {a \left (6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right )+2 \left (5 B a^3+b (17 A+45 C) a^2+45 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx+\left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {2 \left (6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 \sqrt {\sec (c+d x)} \left (10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {30 b^2 (5 a C+2 b B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {10 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\) |
Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/S ec[c + d*x]^(5/2),x]
(2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + ( (10*(A*b + a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (((2*(10*a^3*B + 20*a*b^2*B - b^3*(16*A - 15*C) + 4*a^2*b*(4*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (30*b^2*(2*b*B + 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a )/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt [Sec[c + d*x]]))/2 - (b*(16*A*b + 10*a*B - 15*b*C)*Sqrt[Sec[c + d*x]]*Sqrt [a + b*Sec[c + d*x]]*Sin[c + d*x])/d)/3)/5
3.11.48.3.1 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[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[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[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/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[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[(-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_. ))/(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]
Result contains complex when optimal does not.
Time = 18.78 (sec) , antiderivative size = 7222, normalized size of antiderivative = 17.24
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(7222\) |
| default | \(\text {Expression too large to display}\) | \(7227\) |
int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2 ),x,method=_RETURNVERBOSE)
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(5/2),x, algorithm="fricas")
integral((C*b^2*sec(d*x + c)^4 + (2*C*a*b + B*b^2)*sec(d*x + c)^3 + A*a^2 + (C*a^2 + 2*B*a*b + A*b^2)*sec(d*x + c)^2 + (B*a^2 + 2*A*a*b)*sec(d*x + c ))*sqrt(b*sec(d*x + c) + a)/sec(d*x + c)^(5/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(5/2),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)/sec(d*x + c)^(5/2), x)
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(5/2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)/sec(d*x + c)^(5/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
int(((a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/( 1/cos(c + d*x))^(5/2),x)