Integrand size = 45, antiderivative size = 461 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=-\frac {2 \left (48 A b^3-5 a^3 B-40 a b^2 B+6 a^2 b (2 A+5 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 a^4 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (48 A b^4+25 a^3 b B-40 a b^3 B-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^4 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (24 A b^3+5 a^3 B-20 a b^2 B-a^2 (9 A b-15 b C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}} \]
2*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/(a^2-b^2)/d/sec(d*x+c)^(3/2)/(a+b*sec(d *x+c))^(1/2)-2/15*(48*A*b^3-5*B*a^3-40*B*a*b^2+6*a^2*b*(2*A+5*C))*(cos(1/2 *d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/ 2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/a^4/d/ (a+b*sec(d*x+c))^(1/2)-2/5*(6*A*b^2-5*B*a*b-a^2*(A-5*C))*sin(d*x+c)*(a+b*s ec(d*x+c))^(1/2)/a^2/(a^2-b^2)/d/sec(d*x+c)^(3/2)+2/15*(24*A*b^3+5*B*a^3-2 0*B*a*b^2-a^2*(9*A*b-15*C*b))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^3/(a^2-b ^2)/d/sec(d*x+c)^(1/2)-2/15*(48*A*b^4+25*B*a^3*b-40*B*a*b^3-6*a^2*b^2*(4*A -5*C)-3*a^4*(3*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell ipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/ a^4/(a^2-b^2)/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.35 (sec) , antiderivative size = 6134, normalized size of antiderivative = 13.31 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Result too large to show} \]
Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]
Time = 3.54 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.01, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.489, Rules used = {3042, 4588, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {2 \int \frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \sec (c+d x) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \sec (c+d x) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {-\left ((A-5 C) a^2\right )-5 b B a+(A b+C b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a+6 A b^2-4 \left (A b^2-a (b B-a C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {5 B a^3-(9 A b-15 b C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \sec (c+d x) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {5 B a^3-(9 A b-15 b C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \sec (c+d x) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {5 B a^3-(9 A b-15 b C) a^2-20 b^2 B a+\left ((3 A+5 C) a^2-5 b B a+2 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+24 A b^3-2 b \left (-\left ((A-5 C) a^2\right )-5 b B a+6 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \sec (c+d x) a+48 A b^4}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \sec (c+d x) a+48 A b^4}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-3 (3 A+5 C) a^4+25 b B a^3-6 b^2 (4 A-5 C) a^2-40 b^3 B a+\left (-5 B a^3+3 b (A+5 C) a^2-10 b^2 B a+12 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^4}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^2 (A-5 C)\right )-5 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{3 a}}{5 a}}{a \left (a^2-b^2\right )}\) |
Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec [c + d*x])^(3/2)),x]
(2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/ 2)*Sqrt[a + b*Sec[c + d*x]]) - ((2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqr t[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (-1/3*((2 *(a^2 - b^2)*(48*A*b^3 - 5*a^3*B - 40*a*b^2*B + 6*a^2*b*(2*A + 5*C))*Sqrt[ (b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[S ec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*EllipticE[(c + d* x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d* x])/(a + b)]*Sqrt[Sec[c + d*x]]))/a + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt [Sec[c + d*x]]))/(5*a))/(a*(a^2 - b^2))
3.11.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[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_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc [e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f *x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*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] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*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] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(4990\) vs. \(2(487)=974\).
Time = 18.34 (sec) , antiderivative size = 4991, normalized size of antiderivative = 10.83
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(4991\) |
| default | \(\text {Expression too large to display}\) | \(5355\) |
int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c))^(3/2 ),x,method=_RETURNVERBOSE)
2/5*A/d/a^4/(a+b)/((a-b)/(a+b))^(1/2)*(a+b*sec(d*x+c))^(1/2)/(b+a*cos(d*x+ c))/sec(d*x+c)^(5/2)/(1+cos(d*x+c))*(-6*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d *x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot (d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*sec(d*x+c)-32*(1/(a+b)*(b+a* cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b )/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^4*sec(d*x+ c)+8*((a-b)/(a+b))^(1/2)*a*b^3*tan(d*x+c)*sec(d*x+c)+3*EllipticE(((a-b)/(a +b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*co s(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^4*sec(d*x+c)^2+ 3*((a-b)/(a+b))^(1/2)*a^3*b*tan(d*x+c)*sec(d*x+c)-3*(1/(a+b)*(b+a*cos(d*x+ c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b)) ^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*sec(d*x+c)^2-16* (1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*E llipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2) )*b^4*sec(d*x+c)^2+6*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c) ),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1 /(1+cos(d*x+c)))^(1/2)*a^4*sec(d*x+c)-((a-b)/(a+b))^(1/2)*a^3*b*sin(d*x+c) -2*((a-b)/(a+b))^(1/2)*a^2*b^2*sin(d*x+c)+((a-b)/(a+b))^(1/2)*a^4*cos(d*x+ c)*sin(d*x+c)-3*(1/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-c ot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 1018, normalized size of antiderivative = 2.21 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c) )^(3/2),x, algorithm="fricas")
-1/45*(sqrt(2)*(15*I*B*a^5*b - 3*I*(9*A + 25*C)*a^4*b^2 + 80*I*B*a^3*b^3 - 12*I*(7*A - 5*C)*a^2*b^4 - 80*I*B*a*b^5 + 96*I*A*b^6 + (15*I*B*a^6 - 3*I* (9*A + 25*C)*a^5*b + 80*I*B*a^4*b^2 - 12*I*(7*A - 5*C)*a^3*b^3 - 80*I*B*a^ 2*b^4 + 96*I*A*a*b^5)*cos(d*x + c))*sqrt(a)*weierstrassPInverse(-4/3*(3*a^ 2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a* sin(d*x + c) + 2*b)/a) + sqrt(2)*(-15*I*B*a^5*b + 3*I*(9*A + 25*C)*a^4*b^2 - 80*I*B*a^3*b^3 + 12*I*(7*A - 5*C)*a^2*b^4 + 80*I*B*a*b^5 - 96*I*A*b^6 + (-15*I*B*a^6 + 3*I*(9*A + 25*C)*a^5*b - 80*I*B*a^4*b^2 + 12*I*(7*A - 5*C) *a^3*b^3 + 80*I*B*a^2*b^4 - 96*I*A*a*b^5)*cos(d*x + c))*sqrt(a)*weierstras sPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*c os(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(3*I*(3*A + 5*C)*a^ 5*b - 25*I*B*a^4*b^2 + 6*I*(4*A - 5*C)*a^3*b^3 + 40*I*B*a^2*b^4 - 48*I*A*a *b^5 + (3*I*(3*A + 5*C)*a^6 - 25*I*B*a^5*b + 6*I*(4*A - 5*C)*a^4*b^2 + 40* I*B*a^3*b^3 - 48*I*A*a^2*b^4)*cos(d*x + c))*sqrt(a)*weierstrassZeta(-4/3*( 3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*( 3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3* I*a*sin(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(-3*I*(3*A + 5*C)*a^5*b + 25*I*B*a ^4*b^2 - 6*I*(4*A - 5*C)*a^3*b^3 - 40*I*B*a^2*b^4 + 48*I*A*a*b^5 + (-3*I*( 3*A + 5*C)*a^6 + 25*I*B*a^5*b - 6*I*(4*A - 5*C)*a^4*b^2 - 40*I*B*a^3*b^3 + 48*I*A*a^2*b^4)*cos(d*x + c))*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*...
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c) )^(3/2),x, algorithm="maxima")
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c) )^(3/2),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)^(5/2)), x)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + b/cos(c + d*x))^(3/2)*(1 /cos(c + d*x))^(5/2)),x)