Integrand size = 35, antiderivative size = 588 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=-\frac {2 \left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \] Output:
-2/15*(17*A*a^4*b+116*A*a^2*b^3-128*A*b^5-5*B*a^5-80*B*a^3*b^2+80*B*a*b^4) *((b+a*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/( a+b))^(1/2))*sec(d*x+c)^(1/2)/a^5/(a^2-b^2)/d/(a+b*sec(d*x+c))^(1/2)+2/15* (9*A*a^6+55*A*a^4*b^2-212*A*a^2*b^4+128*A*b^6-40*B*a^5*b+140*B*a^3*b^3-80* B*a*b^5)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d* x+c))^(1/2)/a^5/(a^2-b^2)^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1 /2)+2/3*b*(A*b-B*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*b*(12*A*a^2*b-8*A*b^3-9*B*a^3+5*B*a*b^2)*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*(3*A*a^4-71*A*a^2* b^2+48*A*b^4+50*B*a^3*b-30*B*a*b^3)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a^3/ (a^2-b^2)^2/d/sec(d*x+c)^(3/2)-2/15*(14*A*a^4*b-98*A*a^2*b^3+64*A*b^5-5*B* a^5+65*B*a^3*b^2-40*B*a*b^4)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a^4/(a^2-b^ 2)^2/d/sec(d*x+c)^(1/2)
Time = 3.55 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.67 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\frac {(b+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x) \left (-\frac {2 \left (\frac {b+a \cos (c+d x)}{a+b}\right )^{3/2} \left (a^2 \left (8 a^4 A b+44 a^2 A b^3-32 A b^5-5 a^5 B-35 a^3 b^2 B+20 a b^4 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )-\left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}+a \left (\frac {10 b^4 (A b-a B) \sin (c+d x)}{-a^2+b^2}-\frac {10 b^3 \left (-15 a^2 A b+11 A b^3+12 a^3 B-8 a b^2 B\right ) (b+a \cos (c+d x)) \sin (c+d x)}{\left (a^2-b^2\right )^2}-2 (14 A b-5 a B) (b+a \cos (c+d x))^2 \sin (c+d x)+3 a A (b+a \cos (c+d x))^2 \sin (2 (c+d x))\right )\right )}{15 a^5 d (a+b \sec (c+d x))^{5/2}} \] Input:
Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5 /2)),x]
Output:
((b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((-2*((b + a*Cos[c + d*x])/(a + b ))^(3/2)*(a^2*(8*a^4*A*b + 44*a^2*A*b^3 - 32*A*b^5 - 5*a^5*B - 35*a^3*b^2* B + 20*a*b^4*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] - (9*a^6*A + 55*a^4* A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5* B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/ 2, (2*a)/(a + b)])))/((a - b)^2*(a + b)) + a*((10*b^4*(A*b - a*B)*Sin[c + d*x])/(-a^2 + b^2) - (10*b^3*(-15*a^2*A*b + 11*A*b^3 + 12*a^3*B - 8*a*b^2* B)*(b + a*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 - 2*(14*A*b - 5*a*B)*( b + a*Cos[c + d*x])^2*Sin[c + d*x] + 3*a*A*(b + a*Cos[c + d*x])^2*Sin[2*(c + d*x)])))/(15*a^5*d*(a + b*Sec[c + d*x])^(5/2))
Time = 4.81 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.01, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4518, 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.
| \(\displaystyle \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4518 |
| \(\displaystyle \frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \int -\frac {3 A a^2+5 b B a-3 (A b-a B) \sec (c+d x) a-8 A b^2+6 b (A b-a B) \sec ^2(c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 A a^2+5 b B a-3 (A b-a B) \sec (c+d x) a-8 A b^2+6 b (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 A a^2+5 b B a-3 (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a-8 A b^2+6 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {2 \int -\frac {3 A a^4+50 b B a^3-71 A b^2 a^2-30 b^3 B a-\left (-3 B a^3+6 A b a^2-b^2 B a-2 A b^3\right ) \sec (c+d x) a+48 A b^4+4 b \left (-9 B a^3+12 A b a^2+5 b^2 B a-8 A b^3\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 A a^4+50 b B a^3-71 A b^2 a^2-30 b^3 B a-\left (-3 B a^3+6 A b a^2-b^2 B a-2 A b^3\right ) \sec (c+d x) a+48 A b^4+4 b \left (-9 B a^3+12 A b a^2+5 b^2 B a-8 A b^3\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 A a^4+50 b B a^3-71 A b^2 a^2-30 b^3 B a-\left (-3 B a^3+6 A b a^2-b^2 B a-2 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^4+4 b \left (-9 B a^3+12 A b a^2+5 b^2 B a-8 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {-2 b \left (3 A a^4+50 b B a^3-71 A b^2 a^2-30 b^3 B a+48 A b^4\right ) \sec ^2(c+d x)-a \left (9 A a^4-30 b B a^3+27 A b^2 a^2+10 b^3 B a-16 A b^4\right ) \sec (c+d x)+3 \left (-5 B a^5+14 A b a^4+65 b^2 B a^3-98 A b^3 a^2-40 b^4 B a+64 A b^5\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (3 A a^4+50 b B a^3-71 A b^2 a^2-30 b^3 B a+48 A b^4\right ) \sec ^2(c+d x)-a \left (9 A a^4-30 b B a^3+27 A b^2 a^2+10 b^3 B a-16 A b^4\right ) \sec (c+d x)+3 \left (-5 B a^5+14 A b a^4+65 b^2 B a^3-98 A b^3 a^2-40 b^4 B a+64 A b^5\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (3 A a^4+50 b B a^3-71 A b^2 a^2-30 b^3 B a+48 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (9 A a^4-30 b B a^3+27 A b^2 a^2+10 b^3 B a-16 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-5 B a^5+14 A b a^4+65 b^2 B a^3-98 A b^3 a^2-40 b^4 B a+64 A b^5\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {3 \left (9 A a^6-40 b B a^5+55 A b^2 a^4+140 b^3 B a^3-212 A b^4 a^2-80 b^5 B a-\left (-5 B a^5+8 A b a^4-35 b^2 B a^3+44 A b^3 a^2+20 b^4 B a-32 A b^5\right ) \sec (c+d x) a+128 A b^6\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {9 A a^6-40 b B a^5+55 A b^2 a^4+140 b^3 B a^3-212 A b^4 a^2-80 b^5 B a-\left (-5 B a^5+8 A b a^4-35 b^2 B a^3+44 A b^3 a^2+20 b^4 B a-32 A b^5\right ) \sec (c+d x) a+128 A b^6}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {9 A a^6-40 b B a^5+55 A b^2 a^4+140 b^3 B a^3-212 A b^4 a^2-80 b^5 B a-\left (-5 B a^5+8 A b a^4-35 b^2 B a^3+44 A b^3 a^2+20 b^4 B a-32 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+128 A b^6}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}}{a}}{5 a}}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
Input:
Int[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x ]
Output:
(2*b*(A*b - a*B)*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*b*(12*a^2*A*b - 8*A*b^3 - 9*a^3*B + 5*a*b^2*B )*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x ]]) + ((2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B)*Sq rt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (-(((-2* (a^2 - b^2)*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2 *B + 80*a*b^4*B)*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*(9 *a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b ^3*B - 80*a*b^5*B)*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*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a* b^4*B)*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))
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_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[b*(A*b - a*B)*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] + Simp[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^2*(m + 1) - b^2*(m + n + 1)) + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m + n + 2) *Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A* b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && IL tQ[n, 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. \(3924\) vs. \(2(557)=1114\).
Time = 46.53 (sec) , antiderivative size = 3925, normalized size of antiderivative = 6.68
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3925\) |
| parts | \(\text {Expression too large to display}\) | \(3959\) |
Input:
int((A+B*sec(d*x+c))/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c))^(5/2),x,method=_RET URNVERBOSE)
Output:
2/15/d/a^5/(a-b)/(a+b)^2/((a-b)/(a+b))^(1/2)*(a+b*sec(d*x+c))^(1/2)/(cos(d *x+c)^2*(1+cos(d*x+c))*a^2+cos(d*x+c)*(2+2*cos(d*x+c))*a*b+(1+cos(d*x+c))* b^2)/sec(d*x+c)^(5/2)*((-3*cos(d*x+c)^4-11*cos(d*x+c)^3+31*cos(d*x+c)^2-7* cos(d*x+c)+9)*A*((a-b)/(a+b))^(1/2)*a^5*b^2*tan(d*x+c)*sec(d*x+c)+(-3*cos( d*x+c)^4+5*cos(d*x+c)^3+47*cos(d*x+c)^2+89*cos(d*x+c)-5)*A*((a-b)/(a+b))^( 1/2)*a^4*b^3*tan(d*x+c)*sec(d*x+c)+(8*cos(d*x+c)^3-40*cos(d*x+c)^2+214*cos (d*x+c)+50)*A*((a-b)/(a+b))^(1/2)*a^3*b^4*tan(d*x+c)*sec(d*x+c)+(-5*cos(d* x+c)^3-35*cos(d*x+c)^2-65*cos(d*x+c)+5)*B*((a-b)/(a+b))^(1/2)*a^5*b^2*tan( d*x+c)*sec(d*x+c)+B*(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^3*b^4*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d *x+c)),(-(a+b)/(a-b))^(1/2))*(140+280*sec(d*x+c)+140*sec(d*x+c)^2)+B*(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*b^6* EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2) )*(-80-160*sec(d*x+c)-80*sec(d*x+c)^2)+A*(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^6*b*EllipticF(((a-b)/(a+b))^(1/2 )*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-17*cos(d*x+c)-43-35*sec( d*x+c)-9*sec(d*x+c)^2)+A*(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^5*b^2*EllipticF(((a-b)/(a+b))^(1/2)*(csc(d*x+c)- cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-72*cos(d*x+c)-161-106*sec(d*x+c)-17*se c(d*x+c)^2)+A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos...
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1531, normalized size of antiderivative = 2.60 \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sec(d*x+c))/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c))^(5/2),x, algo rithm="fricas")
Output:
1/45*(sqrt(2)*(-15*I*B*a^7*b^2 + 42*I*A*a^6*b^3 - 185*I*B*a^5*b^4 + 242*I* A*a^4*b^5 + 340*I*B*a^3*b^6 - 520*I*A*a^2*b^7 - 160*I*B*a*b^8 + 256*I*A*b^ 9 + (-15*I*B*a^9 + 42*I*A*a^8*b - 185*I*B*a^7*b^2 + 242*I*A*a^6*b^3 + 340* I*B*a^5*b^4 - 520*I*A*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 - 42*I*A*a^7*b^2 + 185*I*B*a^6*b^3 - 242*I*A*a^5 *b^4 - 340*I*B*a^4*b^5 + 520*I*A*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*sin(d*x + c) + 2*b)/ a) + sqrt(2)*(15*I*B*a^7*b^2 - 42*I*A*a^6*b^3 + 185*I*B*a^5*b^4 - 242*I*A* a^4*b^5 - 340*I*B*a^3*b^6 + 520*I*A*a^2*b^7 + 160*I*B*a*b^8 - 256*I*A*b^9 + (15*I*B*a^9 - 42*I*A*a^8*b + 185*I*B*a^7*b^2 - 242*I*A*a^6*b^3 - 340*I*B *a^5*b^4 + 520*I*A*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 + 42*I*A*a^7*b^2 - 185*I*B*a^6*b^3 + 242*I*A*a^5*b ^4 + 340*I*B*a^4*b^5 - 520*I*A*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*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(-9*I*A*a^7*b^2 + 40*I*B*a^6*b^3 - 55*I*A*a^5*b^4 - 140*I*B*a ^4*b^5 + 212*I*A*a^3*b^6 + 80*I*B*a^2*b^7 - 128*I*A*a*b^8 + (-9*I*A*a^9 + 40*I*B*a^8*b - 55*I*A*a^7*b^2 - 140*I*B*a^6*b^3 + 212*I*A*a^5*b^4 + 80*I*B *a^4*b^5 - 128*I*A*a^3*b^6)*cos(d*x + c)^2 + 2*(-9*I*A*a^8*b + 40*I*B*a...
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*sec(d*x+c))/sec(d*x+c)**(5/2)/(a+b*sec(d*x+c))**(5/2),x)
Output:
Timed out
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((A+B*sec(d*x+c))/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c))^(5/2),x, algo rithm="maxima")
Output:
Timed out
\[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((A+B*sec(d*x+c))/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c))^(5/2),x, algo rithm="giac")
Output:
integrate((B*sec(d*x + c) + A)/((b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^(5 /2)), x)
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \] Input:
int((A + B/cos(c + d*x))/((a + b/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(5/2 )),x)
Output:
int((A + B/cos(c + d*x))/((a + b/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(5/2 )), x)
\[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{5} b^{2}+2 \sec \left (d x +c \right )^{4} a b +\sec \left (d x +c \right )^{3} a^{2}}d x \] Input:
int((A+B*sec(d*x+c))/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c))^(5/2),x)
Output:
int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a))/(sec(c + d*x)**5*b**2 + 2*sec(c + d*x)**4*a*b + sec(c + d*x)**3*a**2),x)