Integrand size = 45, antiderivative size = 663 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\frac {2 \left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \]
2/3*(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))^(3/2)-2/3*(8*A*b^4+9*B*a^3*b-5*B*a*b^3-2*a^2*b^2*(6*A-C)-6*a^4*C) *sin(d*x+c)/a^2/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^(1/2)+2/15 *(128*A*b^5+5*a^5*B+80*a^3*b^2*B-80*a*b^4*B-4*a^2*b^3*(29*A-10*C)-a^4*b*(1 7*A+45*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^5/(a^2-b^2)/d/(a+b*sec(d*x+c))^(1/2)+2/15*(48*A*b^4+50*B*a^ 3*b-30*B*a*b^3+a^4*(3*A-35*C)-a^2*b^2*(71*A-15*C))*sin(d*x+c)*(a+b*sec(d*x +c))^(1/2)/a^3/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)-2/15*(64*A*b^5-5*a^5*B+65*a^ 3*b^2*B-40*a*b^4*B+2*a^4*b*(7*A-20*C)-2*a^2*b^3*(49*A-10*C))*sin(d*x+c)*(a +b*sec(d*x+c))^(1/2)/a^4/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)+2/15*(128*A*b^6-40 *a^5*b*B+140*a^3*b^3*B-80*a*b^5*B+5*a^4*b^2*(11*A-15*C)-4*a^2*b^4*(53*A-10 *C)+3*a^6*(3*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellip ticE(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^ 5/(a^2-b^2)^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 = 14.82 (sec) , antiderivative size = 9192, normalized size of antiderivative = 13.86 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Result too large to show} \]
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])^(5/2)),x]
Time = 5.39 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.01, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4588, 27, 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.
| \(\displaystyle \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \int \frac {-\left ((3 A-5 C) a^2\right )-5 b B a+3 (A b+C b-a B) \sec (c+d x) a+8 A b^2-6 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {-\left ((3 A-5 C) a^2\right )-5 b B a+3 (A b+C b-a B) \sec (c+d x) a+8 A b^2-6 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {-\left ((3 A-5 C) a^2\right )-5 b B a+3 (A b+C b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a+8 A b^2-6 \left (A b^2-a (b B-a C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {2 \int \frac {(3 A-35 C) a^4+50 b B a^3-b^2 (71 A-15 C) a^2-30 b^3 B a+\left (3 B a^3-2 b (3 A+2 C) a^2+b^2 B a+2 A b^3\right ) \sec (c+d x) a+48 A b^4-4 \left (-6 C a^4+9 b B a^3-2 b^2 (6 A-C) a^2-5 b^3 B a+8 A b^4\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {(3 A-35 C) a^4+50 b B a^3-b^2 (71 A-15 C) a^2-30 b^3 B a+\left (3 B a^3-2 b (3 A+2 C) a^2+b^2 B a+2 A b^3\right ) \sec (c+d x) a+48 A b^4-4 \left (-6 C a^4+9 b B a^3-2 b^2 (6 A-C) a^2-5 b^3 B a+8 A b^4\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {(3 A-35 C) a^4+50 b B a^3-b^2 (71 A-15 C) a^2-30 b^3 B a+\left (3 B a^3-2 b (3 A+2 C) a^2+b^2 B a+2 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^4-4 \left (-6 C a^4+9 b B a^3-2 b^2 (6 A-C) a^2-5 b^3 B a+8 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {-2 b \left ((3 A-35 C) a^4+50 b B a^3-b^2 (71 A-15 C) a^2-30 b^3 B a+48 A b^4\right ) \sec ^2(c+d x)+a \left (-3 (3 A+5 C) a^4+30 b B a^3-b^2 (27 A+5 C) a^2-10 b^3 B a+16 A b^4\right ) \sec (c+d x)+3 \left (-5 B a^5+2 b (7 A-20 C) a^4+65 b^2 B a^3-2 b^3 (49 A-10 C) a^2-40 b^4 B a+64 A b^5\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left ((3 A-35 C) a^4+50 b B a^3-b^2 (71 A-15 C) a^2-30 b^3 B a+48 A b^4\right ) \sec ^2(c+d x)+a \left (-3 (3 A+5 C) a^4+30 b B a^3-b^2 (27 A+5 C) a^2-10 b^3 B a+16 A b^4\right ) \sec (c+d x)+3 \left (-5 B a^5+2 b (7 A-20 C) a^4+65 b^2 B a^3-2 b^3 (49 A-10 C) a^2-40 b^4 B a+64 A b^5\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left ((3 A-35 C) a^4+50 b B a^3-b^2 (71 A-15 C) a^2-30 b^3 B a+48 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (-3 (3 A+5 C) a^4+30 b B a^3-b^2 (27 A+5 C) a^2-10 b^3 B a+16 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-5 B a^5+2 b (7 A-20 C) a^4+65 b^2 B a^3-2 b^3 (49 A-10 C) a^2-40 b^4 B a+64 A b^5\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {3 \left (3 (3 A+5 C) a^6-40 b B a^5+5 b^2 (11 A-15 C) a^4+140 b^3 B a^3-4 b^4 (53 A-10 C) a^2-80 b^5 B a+\left (5 B a^5-2 b (4 A+15 C) a^4+35 b^2 B a^3-2 b^3 (22 A-5 C) a^2-20 b^4 B a+32 A b^5\right ) \sec (c+d x) a+128 A b^6\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {3 (3 A+5 C) a^6-40 b B a^5+5 b^2 (11 A-15 C) a^4+140 b^3 B a^3-4 b^4 (53 A-10 C) a^2-80 b^5 B a+\left (5 B a^5-2 b (4 A+15 C) a^4+35 b^2 B a^3-2 b^3 (22 A-5 C) a^2-20 b^4 B a+32 A b^5\right ) \sec (c+d x) a+128 A b^6}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {3 (3 A+5 C) a^6-40 b B a^5+5 b^2 (11 A-15 C) a^4+140 b^3 B a^3-4 b^4 (53 A-10 C) a^2-80 b^5 B a+\left (5 B a^5-2 b (4 A+15 C) a^4+35 b^2 B a^3-2 b^3 (22 A-5 C) a^2-20 b^4 B a+32 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+128 A b^6}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {\left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
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])^(5/2)),x]
(2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^( 3/2)*(a + b*Sec[c + d*x])^(3/2)) - ((2*(8*A*b^4 + 9*a^3*b*B - 5*a*b^3*B - 2*a^2*b^2*(6*A - C) - 6*a^4*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*(48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B + a^4*(3*A - 35*C) - a^2*b^2*(71*A - 15*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[ c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (-(((2*(a^2 - b^2)*(128*A*b^5 + 5*a ^5*B + 80*a^3*b^2*B - 80*a*b^4*B - 4*a^2*b^3*(29*A - 10*C) - a^4*b*(17*A + 45*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) - 4*a^ 2*b^4*(53*A - 10*C) + 3*a^6*(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)]*Sqr t[Sec[c + d*x]]))/a) + (2*(64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B + 2*a^4*b*(7*A - 20*C) - 2*a^2*b^3*(49*A - 10*C))*Sqrt[a + b*Sec[c + d*x]] *Sin[c + d*x])/(a*d*Sqrt[Sec[c + d*x]]))/(5*a))/(a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
3.11.69.3.1 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. \(14012\) vs. \(2(681)=1362\).
Time = 26.05 (sec) , antiderivative size = 14013, normalized size of antiderivative = 21.14
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(14013\) |
| parts | \(\text {Expression too large to display}\) | \(14224\) |
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))^(5/2 ),x,method=_RETURNVERBOSE)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.34 (sec) , antiderivative size = 1771, normalized size of antiderivative = 2.67 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
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) )^(5/2),x, algorithm="fricas")
1/45*(sqrt(2)*(-15*I*B*a^7*b^2 + 6*I*(7*A + 20*C)*a^6*b^3 - 185*I*B*a^5*b^ 4 + 2*I*(121*A - 90*C)*a^4*b^5 + 340*I*B*a^3*b^6 - 40*I*(13*A - 2*C)*a^2*b ^7 - 160*I*B*a*b^8 + 256*I*A*b^9 + (-15*I*B*a^9 + 6*I*(7*A + 20*C)*a^8*b - 185*I*B*a^7*b^2 + 2*I*(121*A - 90*C)*a^6*b^3 + 340*I*B*a^5*b^4 - 40*I*(13 *A - 2*C)*a^4*b^5 - 160*I*B*a^3*b^6 + 256*I*A*a^2*b^7)*cos(d*x + c)^2 - 2* (15*I*B*a^8*b - 6*I*(7*A + 20*C)*a^7*b^2 + 185*I*B*a^6*b^3 - 2*I*(121*A - 90*C)*a^5*b^4 - 340*I*B*a^4*b^5 + 40*I*(13*A - 2*C)*a^3*b^6 + 160*I*B*a^2* b^7 - 256*I*A*a*b^8)*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*s in(d*x + c) + 2*b)/a) + sqrt(2)*(15*I*B*a^7*b^2 - 6*I*(7*A + 20*C)*a^6*b^3 + 185*I*B*a^5*b^4 - 2*I*(121*A - 90*C)*a^4*b^5 - 340*I*B*a^3*b^6 + 40*I*( 13*A - 2*C)*a^2*b^7 + 160*I*B*a*b^8 - 256*I*A*b^9 + (15*I*B*a^9 - 6*I*(7*A + 20*C)*a^8*b + 185*I*B*a^7*b^2 - 2*I*(121*A - 90*C)*a^6*b^3 - 340*I*B*a^ 5*b^4 + 40*I*(13*A - 2*C)*a^4*b^5 + 160*I*B*a^3*b^6 - 256*I*A*a^2*b^7)*cos (d*x + c)^2 - 2*(-15*I*B*a^8*b + 6*I*(7*A + 20*C)*a^7*b^2 - 185*I*B*a^6*b^ 3 + 2*I*(121*A - 90*C)*a^5*b^4 + 340*I*B*a^4*b^5 - 40*I*(13*A - 2*C)*a^3*b ^6 - 160*I*B*a^2*b^7 + 256*I*A*a*b^8)*cos(d*x + c))*sqrt(a)*weierstrassPIn verse(-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^7*b ^2 + 40*I*B*a^6*b^3 - 5*I*(11*A - 15*C)*a^5*b^4 - 140*I*B*a^4*b^5 + 4*I...
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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) )^(5/2),x, algorithm="maxima")
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
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) )^(5/2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((b*sec(d*x + c) + a)^(5 /2)*sec(d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + b/cos(c + d*x))^(5/2)*(1 /cos(c + d*x))^(5/2)),x)