Integrand size = 43, antiderivative size = 628 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^5 d}-\frac {2 (a-b) \sqrt {a+b} \left (4 a^3 b (22 B-9 C)-48 a^4 C-6 a^2 b^2 (33 A-11 B+24 C)+3 b^4 (275 A-539 B+225 C)-3 a b^3 (627 A-143 B+471 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^4 d}-\frac {2 \left (44 a^3 b B-968 a b^3 B-24 a^4 C-75 b^4 (11 A+9 C)-3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^3 d}+\frac {2 \left (33 a^2 b B+539 b^3 B-18 a^3 C+6 a b^2 (132 A+101 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^2 d}+\frac {2 \left (99 A b^2+110 a b B+3 a^2 C+81 b^2 C\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{693 b d}+\frac {2 (11 b B+3 a C) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{99 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d} \]
-2/3465*(a-b)*(88*B*a^4*b+363*B*a^2*b^3+1617*B*b^5-48*a^5*C-18*a^3*b^2*(11 *A+6*C)+6*a*b^4*(451*A+348*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2) /(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/ 2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^5/d-2/3465*(a-b)*(4*a^3*b*(22*B-9*C)- 48*a^4*C-6*a^2*b^2*(33*A-11*B+24*C)+3*b^4*(275*A-539*B+225*C)-3*a*b^3*(627 *A-143*B+471*C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),( (a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec( d*x+c))/(a-b))^(1/2)/b^4/d+2/11*C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*tan( d*x+c)/d-2/3465*(44*B*a^3*b-968*B*a*b^3-24*a^4*C-75*b^4*(11*A+9*C)-3*a^2*b ^2*(33*A+19*C))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3/d+2/3465*(33*B*a^2*b +539*B*b^3-18*a^3*C+6*a*b^2*(132*A+101*C))*sec(d*x+c)*(a+b*sec(d*x+c))^(1/ 2)*tan(d*x+c)/b^2/d+2/693*(99*A*b^2+110*B*a*b+3*C*a^2+81*C*b^2)*sec(d*x+c) ^2*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b/d+2/99*(11*B*b+3*C*a)*sec(d*x+c)^3* (a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d
Time = 23.43 (sec) , antiderivative size = 1100, normalized size of antiderivative = 1.75 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left ((a+b) \left (-88 a^4 b B-363 a^2 b^3 B-1617 b^5 B+48 a^5 C+18 a^3 b^2 (11 A+6 C)-6 a b^4 (451 A+348 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+b (a+b) \left (-48 a^4 C+4 a^3 b (22 B+9 C)-6 a^2 b^2 (33 A+11 B+24 C)+3 b^4 (275 A+539 B+225 C)+3 a b^3 (627 A+143 B+471 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+\left (-88 a^4 b B-363 a^2 b^3 B-1617 b^5 B+48 a^5 C+18 a^3 b^2 (11 A+6 C)-6 a b^4 (451 A+348 C)\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (-1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (a \left (-1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-b \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{3465 b^4 d (b+a \cos (c+d x))^{3/2} (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {7}{2}}(c+d x) \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}}}+\frac {\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {4 \left (198 a^3 A b^2-2706 a A b^4-88 a^4 b B-363 a^2 b^3 B-1617 b^5 B+48 a^5 C+108 a^3 b^2 C-2088 a b^4 C\right ) \sin (c+d x)}{3465 b^4}+\frac {4}{99} \sec ^4(c+d x) (11 b B \sin (c+d x)+12 a C \sin (c+d x))+\frac {4 \sec ^3(c+d x) \left (99 A b^2 \sin (c+d x)+110 a b B \sin (c+d x)+3 a^2 C \sin (c+d x)+81 b^2 C \sin (c+d x)\right )}{693 b}+\frac {4 \sec ^2(c+d x) \left (792 a A b^2 \sin (c+d x)+33 a^2 b B \sin (c+d x)+539 b^3 B \sin (c+d x)-18 a^3 C \sin (c+d x)+606 a b^2 C \sin (c+d x)\right )}{3465 b^2}+\frac {4 \sec (c+d x) \left (99 a^2 A b^2 \sin (c+d x)+825 A b^4 \sin (c+d x)-44 a^3 b B \sin (c+d x)+968 a b^3 B \sin (c+d x)+24 a^4 C \sin (c+d x)+57 a^2 b^2 C \sin (c+d x)+675 b^4 C \sin (c+d x)\right )}{3465 b^3}+\frac {4}{11} b C \sec ^4(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))} \]
Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(4*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt [(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 16 17*b^5*B + 48*a^5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))*E llipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x) /2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan [(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C) - 6*a^2*b^2*(33*A + 11*B + 24*C) + 3*b^4*(275*A + 539*B + 225*C) + 3*a*b^3*( 627*A + 143*B + 471*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b )]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*T an[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-88*a^4*b*B - 363*a^ 2*b^3*B - 1617*b^5*B + 48*a^5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a*(-1 + Tan[(c + d* x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))))/(3465*b^4*d*(b + a*Cos[c + d*x])^ (3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2) *(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan [(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(198*a^3*A*b^2 - 2706*a*A*b^4 - 88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108 *a^3*b^2*C - 2088*a*b^4*C)*Sin[c + d*x])/(3465*b^4) + (4*Sec[c + d*x]^4*(1 1*b*B*Sin[c + d*x] + 12*a*C*Sin[c + d*x]))/99 + (4*Sec[c + d*x]^3*(99*A...
Time = 3.43 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.04, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4584, 27, 3042, 4584, 27, 3042, 4590, 27, 3042, 4580, 27, 3042, 4570, 27, 3042, 4493, 3042, 4319, 4492}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left ((11 b B+3 a C) \sec ^2(c+d x)+(11 A b+9 C b+11 a B) \sec (c+d x)+a (11 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left ((11 b B+3 a C) \sec ^2(c+d x)+(11 A b+9 C b+11 a B) \sec (c+d x)+a (11 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left ((11 b B+3 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(11 A b+9 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+a (11 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {\sec ^3(c+d x) \left (\left (3 C a^2+110 b B a+99 A b^2+81 b^2 C\right ) \sec ^2(c+d x)+\left (99 B a^2+198 A b a+156 b C a+77 b^2 B\right ) \sec (c+d x)+3 a (33 a A+22 b B+24 a C)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\sec ^3(c+d x) \left (\left (3 C a^2+110 b B a+99 A b^2+81 b^2 C\right ) \sec ^2(c+d x)+\left (99 B a^2+198 A b a+156 b C a+77 b^2 B\right ) \sec (c+d x)+3 a (33 a A+22 b B+24 a C)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\left (3 C a^2+110 b B a+99 A b^2+81 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (99 B a^2+198 A b a+156 b C a+77 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (33 a A+22 b B+24 a C)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2 \int \frac {\sec ^2(c+d x) \left (\left (-18 C a^3+33 b B a^2+6 b^2 (132 A+101 C) a+539 b^3 B\right ) \sec ^2(c+d x)+b \left ((693 A+519 C) a^2+1012 b B a+45 b^2 (11 A+9 C)\right ) \sec (c+d x)+4 a \left (3 C a^2+110 b B a+99 A b^2+81 b^2 C\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\int \frac {\sec ^2(c+d x) \left (\left (-18 C a^3+33 b B a^2+6 b^2 (132 A+101 C) a+539 b^3 B\right ) \sec ^2(c+d x)+b \left ((693 A+519 C) a^2+1012 b B a+45 b^2 (11 A+9 C)\right ) \sec (c+d x)+4 a \left (3 C a^2+110 b B a+99 A b^2+81 b^2 C\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (-18 C a^3+33 b B a^2+6 b^2 (132 A+101 C) a+539 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left ((693 A+519 C) a^2+1012 b B a+45 b^2 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+4 a \left (3 C a^2+110 b B a+99 A b^2+81 b^2 C\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4580 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {2 \int \frac {\sec (c+d x) \left (-3 \left (-24 C a^4+44 b B a^3-3 b^2 (33 A+19 C) a^2-968 b^3 B a-75 b^4 (11 A+9 C)\right ) \sec ^2(c+d x)+b \left (6 C a^3+2299 b B a^2+18 b^2 (242 A+191 C) a+1617 b^3 B\right ) \sec (c+d x)+2 a \left (-18 C a^3+33 b B a^2+6 b^2 (132 A+101 C) a+539 b^3 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\sec (c+d x) \left (-3 \left (-24 C a^4+44 b B a^3-3 b^2 (33 A+19 C) a^2-968 b^3 B a-75 b^4 (11 A+9 C)\right ) \sec ^2(c+d x)+b \left (6 C a^3+2299 b B a^2+18 b^2 (242 A+191 C) a+1617 b^3 B\right ) \sec (c+d x)+2 a \left (-18 C a^3+33 b B a^2+6 b^2 (132 A+101 C) a+539 b^3 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-3 \left (-24 C a^4+44 b B a^3-3 b^2 (33 A+19 C) a^2-968 b^3 B a-75 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (6 C a^3+2299 b B a^2+18 b^2 (242 A+191 C) a+1617 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a \left (-18 C a^3+33 b B a^2+6 b^2 (132 A+101 C) a+539 b^3 B\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {2 \int \frac {3 \sec (c+d x) \left (b \left (-12 C a^4+22 b B a^3+9 b^2 (187 A+141 C) a^2+2046 b^3 B a+75 b^4 (11 A+9 C)\right )+\left (-48 C a^5+88 b B a^4-18 b^2 (11 A+6 C) a^3+363 b^3 B a^2+6 b^4 (451 A+348 C) a+1617 b^5 B\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{3 b}-\frac {2 \tan (c+d x) \left (-24 a^4 C+44 a^3 b B-3 a^2 b^2 (33 A+19 C)-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\sec (c+d x) \left (b \left (-12 C a^4+22 b B a^3+9 b^2 (187 A+141 C) a^2+2046 b^3 B a+75 b^4 (11 A+9 C)\right )+\left (-48 C a^5+88 b B a^4-18 b^2 (11 A+6 C) a^3+363 b^3 B a^2+6 b^4 (451 A+348 C) a+1617 b^5 B\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 \tan (c+d x) \left (-24 a^4 C+44 a^3 b B-3 a^2 b^2 (33 A+19 C)-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (-12 C a^4+22 b B a^3+9 b^2 (187 A+141 C) a^2+2046 b^3 B a+75 b^4 (11 A+9 C)\right )+\left (-48 C a^5+88 b B a^4-18 b^2 (11 A+6 C) a^3+363 b^3 B a^2+6 b^4 (451 A+348 C) a+1617 b^5 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \tan (c+d x) \left (-24 a^4 C+44 a^3 b B-3 a^2 b^2 (33 A+19 C)-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4493 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx-(a-b) \left (-48 a^4 C+a^3 b (88 B-36 C)-6 a^2 b^2 (33 A-11 B+24 C)-3 a b^3 (627 A-143 B+471 C)+3 b^4 (275 A-539 B+225 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 \tan (c+d x) \left (-24 a^4 C+44 a^3 b B-3 a^2 b^2 (33 A+19 C)-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-(a-b) \left (-48 a^4 C+a^3 b (88 B-36 C)-6 a^2 b^2 (33 A-11 B+24 C)-3 a b^3 (627 A-143 B+471 C)+3 b^4 (275 A-539 B+225 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \tan (c+d x) \left (-24 a^4 C+44 a^3 b B-3 a^2 b^2 (33 A+19 C)-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-48 a^4 C+a^3 b (88 B-36 C)-6 a^2 b^2 (33 A-11 B+24 C)-3 a b^3 (627 A-143 B+471 C)+3 b^4 (275 A-539 B+225 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{b}-\frac {2 \tan (c+d x) \left (-24 a^4 C+44 a^3 b B-3 a^2 b^2 (33 A+19 C)-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2 \tan (c+d x) \sec ^2(c+d x) \left (3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{7 b d}+\frac {\frac {2 \tan (c+d x) \sec (c+d x) \left (-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}+\frac {\frac {-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-48 a^4 C+a^3 b (88 B-36 C)-6 a^2 b^2 (33 A-11 B+24 C)-3 a b^3 (627 A-143 B+471 C)+3 b^4 (275 A-539 B+225 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}}{b}-\frac {2 \tan (c+d x) \left (-24 a^4 C+44 a^3 b B-3 a^2 b^2 (33 A+19 C)-968 a b^3 B-75 b^4 (11 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}}{7 b}\right )+\frac {2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
(2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d) + ((2* (11*b*B + 3*a*C)*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9* d) + ((2*(99*A*b^2 + 110*a*b*B + 3*a^2*C + 81*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d) + ((2*(33*a^2*b*B + 539*b^3*B - 1 8*a^3*C + 6*a*b^2*(132*A + 101*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*T an[c + d*x])/(5*b*d) + (((-2*(a - b)*Sqrt[a + b]*(88*a^4*b*B + 363*a^2*b^3 *B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^4*(451*A + 34 8*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[ c + d*x]))/(a - b))])/(b^2*d) - (2*(a - b)*Sqrt[a + b]*(a^3*b*(88*B - 36*C ) - 48*a^4*C - 6*a^2*b^2*(33*A - 11*B + 24*C) + 3*b^4*(275*A - 539*B + 225 *C) - 3*a*b^3*(627*A - 143*B + 471*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[ a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d* x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d))/b - (2*(44*a ^3*b*B - 968*a*b^3*B - 24*a^4*C - 75*b^4*(11*A + 9*C) - 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(b*d))/(5*b))/(7*b))/9)/11
3.10.42.3.1 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[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(A - B) Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[B Int[Csc[e + f*x]*((1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B} , x] && NeQ[a^2 - b^2, 0] && NeQ[A^2 - B^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) )), x] + Simp[1/(b*(m + 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ (e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x _Symbol] :> Simp[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B* (m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] & & NeQ[a^2 - b^2, 0] && !LtQ[m, -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_. ))*(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]
Leaf count of result is larger than twice the leaf count of optimal. \(9222\) vs. \(2(586)=1172\).
Time = 61.38 (sec) , antiderivative size = 9223, normalized size of antiderivative = 14.69
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(9223\) |
| default | \(\text {Expression too large to display}\) | \(9339\) |
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \]
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
integral((C*b*sec(d*x + c)^6 + (C*a + B*b)*sec(d*x + c)^5 + A*a*sec(d*x + c)^3 + (B*a + A*b)*sec(d*x + c)^4)*sqrt(b*sec(d*x + c) + a), x)
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
Integral((a + b*sec(c + d*x))**(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)* *2)*sec(c + d*x)**3, x)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \]
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/ 2)*sec(d*x + c)^3, x)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^3} \,d x \]