Integrand size = 45, antiderivative size = 550 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{192 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{64 b d \sqrt {a+b \sec (c+d x)}}-\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d} \] Output:
1/192*(472*B*a^2*b+128*B*b^3+4*a*b^2*(132*A+89*C)+a^3*(384*A+133*C))*((b+a *cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^ (1/2))*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+1/64*(40*B*a^3*b+160*B*a* b^3-5*a^4*C+120*a^2*b^2*(2*A+C)+16*b^4*(4*A+3*C))*((b+a*cos(d*x+c))/(a+b)) ^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(a/(a+b))^(1/2))*sec(d*x+c) ^(1/2)/b/d/(a+b*sec(d*x+c))^(1/2)-1/192*(264*B*a^2*b+128*B*b^3+15*a^3*C+4* a*b^2*(108*A+71*C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))* (a+b*sec(d*x+c))^(1/2)/b/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2) +1/192*(264*B*a^2*b+128*B*b^3+15*a^3*C+4*a*b^2*(108*A+71*C))*sec(d*x+c)^(1 /2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/b/d+1/32*(16*A*b^2+24*B*a*b+5*C*a^2+ 12*C*b^2)*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d+1/24*(8*B*b +5*C*a)*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/4*C*sec(d*x +c)^(3/2)*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d
Result contains complex when optimal does not.
Time = 12.99 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.68 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x ] + C*Sec[c + d*x]^2),x]
Output:
-1/384*((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2) *((2*(-768*a^3*A*b - 192*a*A*b^3 - 416*a^2*b^2*B - 236*a^3*b*C - 144*a*b^3 *C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b )])/Sqrt[b + a*Cos[c + d*x]] + (2*(-1008*a^2*A*b^2 - 384*A*b^4 + 24*a^3*b* B - 832*a*b^3*B + 45*a^4*C - 436*a^2*b^2*C - 288*b^4*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)*(432*a^2*A*b^2 + 264*a^3*b*B + 128*a*b^3*B + 15*a^4*C + 284*a^2*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)*EllipticE[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))))/(b*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)*((S ec[c + d*x]^3*(8*b^2*B*Sin[c + d*x] + 17*a*b*C*Sin[c + d*x]))/12 + (Sec[c + d*x]^2*(48*A*b^2*Sin[c + d*x] + 104*a*b*B*Sin[c + d*x] + 59*a^2*C*Sin[c + d*x] + 36*b^2*C*Sin[c + d*x]))/48 + (Sec[c + d*x]*(432*a*A*b^2*Sin[c ...
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.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left ((8 b B+5 a C) \sec ^2(c+d x)+2 (4 A b+3 C b+4 a B) \sec (c+d x)+a (8 A+C)\right )dx+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left ((8 b B+5 a C) \sec ^2(c+d x)+2 (4 A b+3 C b+4 a B) \sec (c+d x)+a (8 A+C)\right )dx+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((8 b B+5 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 (4 A b+3 C b+4 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+a (8 A+C)\right )dx+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {1}{2} \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \left (3 \left (5 C a^2+24 b B a+16 A b^2+12 b^2 C\right ) \sec ^2(c+d x)+2 \left (24 B a^2+b (48 A+31 C) a+16 b^2 B\right ) \sec (c+d x)+a (48 a A+8 b B+11 a C)\right )dx+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \left (3 \left (5 C a^2+24 b B a+16 A b^2+12 b^2 C\right ) \sec ^2(c+d x)+2 \left (24 B a^2+b (48 A+31 C) a+16 b^2 B\right ) \sec (c+d x)+a (48 a A+8 b B+11 a C)\right )dx+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (3 \left (5 C a^2+24 b B a+16 A b^2+12 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (24 B a^2+b (48 A+31 C) a+16 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (48 a A+8 b B+11 a C)\right )dx+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {\sqrt {\sec (c+d x)} \left (\left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right ) \sec ^2(c+d x)+2 \left (96 B a^3+b (288 A+161 C) a^2+152 b^2 B a+12 b^3 (4 A+3 C)\right ) \sec (c+d x)+a \left (2 \left (96 A+\frac {59 C}{2}\right ) a^2+104 b B a+12 b^2 (4 A+3 C)\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\sqrt {\sec (c+d x)} \left (\left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right ) \sec ^2(c+d x)+2 \left (96 B a^3+b (288 A+161 C) a^2+152 b^2 B a+12 b^3 (4 A+3 C)\right ) \sec (c+d x)+a \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (96 B a^3+b (288 A+161 C) a^2+152 b^2 B a+12 b^3 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\int -\frac {-3 \left (-5 C a^4+40 b B a^3+120 b^2 (2 A+C) a^2+160 b^3 B a+16 b^4 (4 A+3 C)\right ) \sec ^2(c+d x)-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \sec (c+d x)+a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{b}+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {-3 \left (-5 C a^4+40 b B a^3+120 b^2 (2 A+C) a^2+160 b^3 B a+16 b^4 (4 A+3 C)\right ) \sec ^2(c+d x)-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \sec (c+d x)+a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {-3 \left (-5 C a^4+40 b B a^3+120 b^2 (2 A+C) a^2+160 b^3 B a+16 b^4 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4596 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx-3 \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-3 \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {a \left (15 C a^3+264 b B a^2+4 b^2 (108 A+71 C) a+128 b^3 B\right )-2 a b \left ((192 A+59 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-b \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-\frac {b \sqrt {\sec (c+d x)} \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-\frac {b \sqrt {\sec (c+d x)} \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-\frac {b \sqrt {\sec (c+d x)} \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 \sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}\right )+\frac {3 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
Input:
Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C* Sec[c + d*x]^2),x]
Output:
$Aborted
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[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[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(m + n + 1) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a *B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LeQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 )*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[C/d^2 Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*C sc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[ a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 83.72 (sec) , antiderivative size = 4126, normalized size of antiderivative = 7.50
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(4126\) |
| parts | \(\text {Expression too large to display}\) | \(4255\) |
Input:
int(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x,method=_RETURNVERBOSE)
Output:
1/192/d/b/((a-b)/(a+b))^(1/2)*(a+b*sec(d*x+c))^(1/2)*sec(d*x+c)^(1/2)/(cos (d*x+c)^2*a+a*cos(d*x+c)+b*cos(d*x+c)+b)*((72*cos(d*x+c)^3+72*cos(d*x+c)^2 +48*cos(d*x+c)+48)*C*((a-b)/(a+b))^(1/2)*b^4*tan(d*x+c)*sec(d*x+c)^2+A*(1/ (a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*b^4* EllipticF(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2) )*(-192*cos(d*x+c)^3-384*cos(d*x+c)^2-192*cos(d*x+c))+C*(1/(a+b)*(b+a*cos( d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^4*EllipticF(((a-b )/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(30*cos(d*x+c )^3+60*cos(d*x+c)^2+30*cos(d*x+c))+C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c) +1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*b^4*EllipticF(((a-b)/(a+b))^(1/2)*(csc (d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-144*cos(d*x+c)^3-288*cos(d*x+c )^2-144*cos(d*x+c))+A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/( cos(d*x+c)+1))^(1/2)*b^4*EllipticPi(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d* x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(384*cos(d*x+c)^3+768*cos(d*x+c)^ 2+384*cos(d*x+c))+C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(co s(d*x+c)+1))^(1/2)*a^4*EllipticPi(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+ c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(-30*cos(d*x+c)^3-60*cos(d*x+c)^2-3 0*cos(d*x+c))+C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d* x+c)+1))^(1/2)*b^4*EllipticPi(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)), (a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(288*cos(d*x+c)^3+576*cos(d*x+c)^2+2...
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**(1/2)*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec( d*x+c)**2),x)
Output:
Timed out
\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)*sqrt(sec(d*x + c)), x)
\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)*sqrt(sec(d*x + c)), x)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:
int((a + b/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
Output:
int((a + b/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2), x)
\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{2}d x \right ) a^{2} c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{2}d x \right ) a \,b^{2}+3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}d x \right ) a^{3} \] Input:
int(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x)
Output:
int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**4,x)*b**2*c + 2*int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**3,x)*a*b *c + int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**3,x)*b* *3 + int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**2,x)*a* *2*c + 3*int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**2,x )*a*b**2 + 3*int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x), x)*a**2*b + int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a),x)*a**3