Integrand size = 35, antiderivative size = 550 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 a (a-b) \sqrt {a+b} \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1155 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)+3 a b^3 (209 A+157 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{1155 b^4 d}+\frac {2 \left (8 a^4 C+25 b^4 (11 A+9 C)+a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^3 d}+\frac {4 a \left (132 A b^2-3 a^2 C+101 b^2 C\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{1155 b^2 d}+\frac {2 \left (a^2 C+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{231 b d}+\frac {2 a C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{33 d}+\frac {2 C \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{11 d} \] Output:
4/1155*a*(a-b)*(a+b)^(1/2)*(8*a^4*C+3*a^2*b^2*(11*A+6*C)-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))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^5/d+ 2/1155*(a-b)*(a+b)^(1/2)*(16*a^4*C+12*a^3*b*C+6*a^2*b^2*(11*A+8*C)-25*b^4* (11*A+9*C)+3*a*b^3*(209*A+157*C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1 /2)/(a+b)^(1/2),((a+b)/(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+2/1155*(8*a^4*C+25*b^4*(11*A+9*C)+a^2*b^2* (33*A+19*C))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3/d+4/1155*a*(132*A*b^2-3 *C*a^2+101*C*b^2)*sec(d*x+c)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^2/d+2/231 *(C*a^2+3*b^2*(11*A+9*C))*sec(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b /d+2/33*a*C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d+2/11*C*sec(d* x+c)^3*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(3988\) vs. \(2(550)=1100\).
Time = 24.25 (sec) , antiderivative size = 3988, normalized size of antiderivative = 7.25 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2) ,x]
Output:
(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((-8*a*( 33*a^2*A*b^2 - 451*A*b^4 + 8*a^4*C + 18*a^2*b^2*C - 348*b^4*C)*Sin[c + d*x ])/(1155*b^4) + (4*Sec[c + d*x]^3*(33*A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d *x] + 27*b^2*C*Sin[c + d*x]))/(231*b) + (8*Sec[c + d*x]^2*(132*a*A*b^2*Sin [c + d*x] - 3*a^3*C*Sin[c + d*x] + 101*a*b^2*C*Sin[c + d*x]))/(1155*b^2) + (4*Sec[c + d*x]*(33*a^2*A*b^2*Sin[c + d*x] + 275*A*b^4*Sin[c + d*x] + 8*a ^4*C*Sin[c + d*x] + 19*a^2*b^2*C*Sin[c + d*x] + 225*b^4*C*Sin[c + d*x]))/( 1155*b^3) + (16*a*C*Sec[c + d*x]^3*Tan[c + d*x])/33 + (4*b*C*Sec[c + d*x]^ 4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x] )) + (8*((4*a^3*A)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (1 64*a*A*b)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^5*C)/( 1155*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (24*a^3*C)/(385*b* Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (464*a*b*C)/(385*Sqrt[b + a *Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (62*a^2*A*Sqrt[Sec[c + d*x]])/(105*Sq rt[b + a*Cos[c + d*x]]) + (4*a^4*A*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a* Cos[c + d*x]]) + (10*A*b^2*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x] ]) - (26*a^2*C*Sqrt[Sec[c + d*x]])/(55*Sqrt[b + a*Cos[c + d*x]]) + (32*a^6 *C*Sqrt[Sec[c + d*x]])/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]) + (64*a^4*C*Sqr t[Sec[c + d*x]])/(1155*b^2*Sqrt[b + a*Cos[c + d*x]]) + (30*b^2*C*Sqrt[Sec[ c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) - (164*a^2*A*Cos[2*(c + d*x)]*...
Time = 3.03 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.05, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4585, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4585 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (3 a C \sec ^2(c+d x)+b (11 A+9 C) \sec (c+d x)+a (11 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (3 a C \sec ^2(c+d x)+b (11 A+9 C) \sec (c+d x)+a (11 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (3 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (11 A+9 C) \csc \left (c+d x+\frac {\pi }{2}\right )+a (11 A+6 C)\right )dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {3 \sec ^3(c+d x) \left (3 (11 A+8 C) a^2+2 b (33 A+26 C) \sec (c+d x) a+\left (C a^2+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \int \frac {\sec ^3(c+d x) \left (3 (11 A+8 C) a^2+2 b (33 A+26 C) \sec (c+d x) a+\left (C a^2+3 b^2 (11 A+9 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (3 (11 A+8 C) a^2+2 b (33 A+26 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a+\left (C a^2+3 b^2 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {2 \int \frac {\sec ^2(c+d x) \left (2 a \left (-3 C a^2+132 A b^2+101 b^2 C\right ) \sec ^2(c+d x)+b \left ((231 A+173 C) a^2+15 b^2 (11 A+9 C)\right ) \sec (c+d x)+4 a \left (C a^2+3 b^2 (11 A+9 C)\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\int \frac {\sec ^2(c+d x) \left (2 a \left (-3 C a^2+132 A b^2+101 b^2 C\right ) \sec ^2(c+d x)+b \left ((231 A+173 C) a^2+15 b^2 (11 A+9 C)\right ) \sec (c+d x)+4 a \left (C a^2+3 b^2 (11 A+9 C)\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (2 a \left (-3 C a^2+132 A b^2+101 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left ((231 A+173 C) a^2+15 b^2 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+4 a \left (C a^2+3 b^2 (11 A+9 C)\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4580 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {2 \int \frac {\sec (c+d x) \left (4 \left (-3 C a^2+132 A b^2+101 b^2 C\right ) a^2+2 b \left (726 A b^2+\left (a^2+573 b^2\right ) C\right ) \sec (c+d x) a+3 \left (8 C a^4+b^2 (33 A+19 C) a^2+25 b^4 (11 A+9 C)\right ) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {\int \frac {\sec (c+d x) \left (4 \left (-3 C a^2+132 A b^2+101 b^2 C\right ) a^2+2 b \left (726 A b^2+\left (a^2+573 b^2\right ) C\right ) \sec (c+d x) a+3 \left (8 C a^4+b^2 (33 A+19 C) a^2+25 b^4 (11 A+9 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (4 \left (-3 C a^2+132 A b^2+101 b^2 C\right ) a^2+2 b \left (726 A b^2+\left (a^2+573 b^2\right ) C\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+3 \left (8 C a^4+b^2 (33 A+19 C) a^2+25 b^4 (11 A+9 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \int -\frac {3 \sec (c+d x) \left (b \left (4 C a^4-3 b^2 (187 A+141 C) a^2-25 b^4 (11 A+9 C)\right )+2 a \left (8 C a^4+3 b^2 (11 A+6 C) a^2-b^4 (451 A+348 C)\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{3 b}+\frac {2 \left (8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {\sec (c+d x) \left (b \left (4 C a^4-3 b^2 (187 A+141 C) a^2-25 b^4 (11 A+9 C)\right )+2 a \left (8 C a^4+3 b^2 (11 A+6 C) a^2-b^4 (451 A+348 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{b}}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (4 C a^4-3 b^2 (187 A+141 C) a^2-25 b^4 (11 A+9 C)\right )+2 a \left (8 C a^4+3 b^2 (11 A+6 C) a^2-b^4 (451 A+348 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4493 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx-(a-b) \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)+3 a b^3 (209 A+157 C)-25 b^4 (11 A+9 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{b}}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-(a-b) \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)+3 a b^3 (209 A+157 C)-25 b^4 (11 A+9 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {\frac {\frac {2 \left (8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)+3 a b^3 (209 A+157 C)-25 b^4 (11 A+9 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{b}}{5 b}+\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (\frac {2 \left (C \left (a^2+27 b^2\right )+33 A b^2\right ) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}+\frac {\frac {4 a \left (-3 a^2 C+132 A b^2+101 b^2 C\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}+\frac {\frac {2 \left (8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {-\frac {4 a (a-b) \sqrt {a+b} \left (8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}-\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)+3 a b^3 (209 A+157 C)-25 b^4 (11 A+9 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{b}}{5 b}}{7 b}\right )+\frac {2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}\) |
Input:
Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]
Output:
(2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d) + ((2* a*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d) + ((2*(33* A*b^2 + (a^2 + 27*b^2)*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d) + ((4*a*(132*A*b^2 - 3*a^2*C + 101*b^2*C)*Sec[c + d*x]*Sqrt[ a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d) + (-(((-4*a*(a - b)*Sqrt[a + b]* (8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cot[c + d*x]*Elli pticE[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]*(16*a^4*C + 12*a^3*b*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Cot[c + d*x]*Ellipt icF[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*(8*a^4*C + 25*b^4*(11*A + 9*C) + 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))/3)/11
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_)]^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*Csc[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 + b*(A*(m + n + 1) + C*(m + n))*Csc[e + f*x ] + a*C*m*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, 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. \(2616\) vs. \(2(508)=1016\).
Time = 109.79 (sec) , antiderivative size = 2617, normalized size of antiderivative = 4.76
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2617\) |
| parts | \(\text {Expression too large to display}\) | \(2625\) |
Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x,method=_RETUR NVERBOSE)
Output:
2/1155/d/b^4*(a+b*sec(d*x+c))^(1/2)/(cos(d*x+c)^2*a+a*cos(d*x+c)+b*cos(d*x +c)+b)*((696*cos(d*x+c)^4+221*cos(d*x+c)^3+221*cos(d*x+c)^2+145*cos(d*x+c) +145)*C*a^2*b^4*tan(d*x+c)*sec(d*x+c)^2+(225*cos(d*x+c)^5+921*cos(d*x+c)^4 +337*cos(d*x+c)^3+337*cos(d*x+c)^2+245*cos(d*x+c)+245)*C*a*b^5*tan(d*x+c)* sec(d*x+c)^3+(19*cos(d*x+c)^3-17*cos(d*x+c)^2-cos(d*x+c)-1)*C*a^3*b^3*tan( d*x+c)*sec(d*x+c)+11*(25*cos(d*x+c)^3+107*cos(d*x+c)^2+39*cos(d*x+c)+39)*A *b^5*a*tan(d*x+c)*sec(d*x+c)+16*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*C*(1/(a+b)* (b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a ^6*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+275*(-cos(d*x+c)^ 2-2*cos(d*x+c)-1)*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d *x+c)/(cos(d*x+c)+1))^(1/2)*b^6*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a +b))^(1/2))+225*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*C*(1/(a+b)*(b+a*cos(d*x+c)) /(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^6*EllipticF(-cs c(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))-66*A*a^4*b^2*cos(d*x+c)*sin(d*x+c )+11*(27+82*cos(d*x+c)^2+27*cos(d*x+c))*a^2*A*b^4*tan(d*x+c)+15*(15*cos(d* x+c)^5+15*cos(d*x+c)^4+9*cos(d*x+c)^3+9*cos(d*x+c)^2+7*cos(d*x+c)+7)*C*b^6 *tan(d*x+c)*sec(d*x+c)^4+55*(5*cos(d*x+c)^3+5*cos(d*x+c)^2+3*cos(d*x+c)+3) *A*b^6*tan(d*x+c)*sec(d*x+c)^2+2*(-18*cos(d*x+c)^2+cos(d*x+c)+1)*a^4*b^2*C *tan(d*x+c)+33*sin(d*x+c)*(cos(d*x+c)-1)*A*a^3*b^3+8*sin(d*x+c)*(cos(d*x+c )-1)*a^5*C*b+36*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*C*(1/(a+b)*(b+a*cos(d*x+...
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algori thm="fricas")
Output:
integral((C*b*sec(d*x + c)^6 + C*a*sec(d*x + c)^5 + A*b*sec(d*x + c)^4 + A *a*sec(d*x + c)^3)*sqrt(b*sec(d*x + c) + a), x)
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**(3/2)*(A+C*sec(d*x+c)**2),x)
Output:
Integral((A + C*sec(c + d*x)**2)*(a + b*sec(c + d*x))**(3/2)*sec(c + d*x)* *3, x)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algori thm="maxima")
Output:
Timed out
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algori thm="giac")
Output:
integrate((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c) + a)^(3/2)*sec(d*x + c)^3 , x)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:
int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2))/cos(c + d*x)^3,x)
Output:
int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2))/cos(c + d*x)^3, x)
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{6}d x \right ) b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) a b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) a^{2} \] Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**6,x)*b*c + int(sqrt(sec(c + d*x )*b + a)*sec(c + d*x)**5,x)*a*c + int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x )**4,x)*a*b + int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**3,x)*a**2