Integrand size = 27, antiderivative size = 626 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=-\frac {2 \left (41 a^2 A b^4-15 A b^6-3 a^6 C-29 a^4 b^2 (2 A+C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 b^2 \sqrt {a+b} \left (a^2-b^2\right )^2 d}+\frac {2 \left (36 a^2 A b^3-5 a A b^4-15 A b^5+3 a^5 C+a^3 b^2 (13 A+5 C)-3 a^4 b (15 A+8 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{15 a^3 b \sqrt {a+b} \left (a^2-b^2\right )^2 d}-\frac {2 A \sqrt {a+b} \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^4 d}+\frac {2 \left (A b^2+a^2 C\right ) \tan (c+d x)}{5 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{5/2}}-\frac {2 \left (5 A b^4-3 a^4 C-a^2 b^2 (13 A+5 C)\right ) \tan (c+d x)}{15 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (41 a^2 A b^4-15 A b^6-3 a^6 C-29 a^4 b^2 (2 A+C)\right ) \tan (c+d x)}{15 a^3 \left (a^2-b^2\right )^3 d \sqrt {a+b \sec (c+d x)}} \]
-2/15*(41*a^2*A*b^4-15*A*b^6-3*a^6*C-29*a^4*b^2*(2*A+C))*cot(d*x+c)*Ellipt icE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+ c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/b^2/(a^2-b^2)^2/d/(a+ b)^(1/2)+2/15*(36*a^2*A*b^3-5*a*A*b^4-15*A*b^5+3*a^5*C+a^3*b^2*(13*A+5*C)- 3*a^4*b*(15*A+8*C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2 ),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/( a-b))^(1/2)/a^3/b/(a^2-b^2)^2/d/(a+b)^(1/2)-2*A*cot(d*x+c)*EllipticPi((a+b *sec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b *(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/d+2/5*(A* b^2+C*a^2)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^(5/2)-2/15*(5*A*b^4-3 *a^4*C-a^2*b^2*(13*A+5*C))*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^( 3/2)-2/15*(41*a^2*A*b^4-15*A*b^6-3*a^6*C-29*a^4*b^2*(2*A+C))*tan(d*x+c)/a^ 3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(2188\) vs. \(2(626)=1252\).
Time = 24.28 (sec) , antiderivative size = 2188, normalized size of antiderivative = 3.50 \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\text {Result too large to show} \]
((b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*((4*(58*a^4* A*b^2 - 41*a^2*A*b^4 + 15*A*b^6 + 3*a^6*C + 29*a^4*b^2*C)*Sin[c + d*x])/(1 5*a^3*b*(-a^2 + b^2)^3) + (4*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]) )/(5*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^3) + (4*(-19*a^2*A*b^3*Sin[c + d *x] + 11*A*b^5*Sin[c + d*x] - 9*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Sin[c + d *x]))/(15*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (4*(74*a^4*A*b^2*Sin [c + d*x] - 65*a^2*A*b^4*Sin[c + d*x] + 23*A*b^6*Sin[c + d*x] + 9*a^6*C*Si n[c + d*x] + 25*a^4*b^2*C*Sin[c + d*x] - 2*a^2*b^4*C*Sin[c + d*x]))/(15*a^ 3*(a^2 - b^2)^3*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])* (a + b*Sec[c + d*x])^(7/2)) - (4*(b + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^( 3/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)] *(58*a^5*A*b^2*Tan[(c + d*x)/2] + 58*a^4*A*b^3*Tan[(c + d*x)/2] - 41*a^3*A *b^4*Tan[(c + d*x)/2] - 41*a^2*A*b^5*Tan[(c + d*x)/2] + 15*a*A*b^6*Tan[(c + d*x)/2] + 15*A*b^7*Tan[(c + d*x)/2] + 3*a^7*C*Tan[(c + d*x)/2] + 3*a^6*b *C*Tan[(c + d*x)/2] + 29*a^5*b^2*C*Tan[(c + d*x)/2] + 29*a^4*b^3*C*Tan[(c + d*x)/2] - 116*a^5*A*b^2*Tan[(c + d*x)/2]^3 + 82*a^3*A*b^4*Tan[(c + d*x)/ 2]^3 - 30*a*A*b^6*Tan[(c + d*x)/2]^3 - 6*a^7*C*Tan[(c + d*x)/2]^3 - 58*a^5 *b^2*C*Tan[(c + d*x)/2]^3 + 58*a^5*A*b^2*Tan[(c + d*x)/2]^5 - 58*a^4*A*b^3 *Tan[(c + d*x)/2]^5 - 41*a^3*A*b^4*Tan[(c + d*x)/2]^5 + 41*a^2*A*b^5*Ta...
Time = 2.77 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3042, 4549, 27, 3042, 4548, 27, 3042, 4548, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4549 |
| \(\displaystyle \frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}-\frac {2 \int -\frac {3 \left (C a^2+A b^2\right ) \sec ^2(c+d x)-5 a b (A+C) \sec (c+d x)+5 A \left (a^2-b^2\right )}{2 (a+b \sec (c+d x))^{5/2}}dx}{5 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 \left (C a^2+A b^2\right ) \sec ^2(c+d x)-5 a b (A+C) \sec (c+d x)+5 A \left (a^2-b^2\right )}{(a+b \sec (c+d x))^{5/2}}dx}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 \left (C a^2+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-5 a b (A+C) \csc \left (c+d x+\frac {\pi }{2}\right )+5 A \left (a^2-b^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {15 A \left (a^2-b^2\right )^2-\left (-3 C a^4-b^2 (13 A+5 C) a^2+5 A b^4\right ) \sec ^2(c+d x)+6 a b \left (A b^2-a^2 (5 A+4 C)\right ) \sec (c+d x)}{2 (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {15 A \left (a^2-b^2\right )^2-\left (-3 C a^4-b^2 (13 A+5 C) a^2+5 A b^4\right ) \sec ^2(c+d x)+6 a b \left (A b^2-a^2 (5 A+4 C)\right ) \sec (c+d x)}{(a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {15 A \left (a^2-b^2\right )^2+\left (3 C a^4+b^2 (13 A+5 C) a^2-5 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+6 a b \left (A b^2-a^2 (5 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {15 A \left (a^2-b^2\right )^3+\left (-3 C a^6-29 b^2 (2 A+C) a^4+41 A b^4 a^2-15 A b^6\right ) \sec ^2(c+d x)-a b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) \sec (c+d x)}{2 \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {15 A \left (a^2-b^2\right )^3+\left (-3 C a^6-29 b^2 (2 A+C) a^4+41 A b^4 a^2-15 A b^6\right ) \sec ^2(c+d x)-a b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {15 A \left (a^2-b^2\right )^3+\left (-3 C a^6-29 b^2 (2 A+C) a^4+41 A b^4 a^2-15 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {\frac {\frac {\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {15 A \left (a^2-b^2\right )^3+\left (3 C a^6+29 b^2 (2 A+C) a^4-41 A b^4 a^2-b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) a+15 A b^6\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {15 A \left (a^2-b^2\right )^3+\left (3 C a^6+29 b^2 (2 A+C) a^4-41 A b^4 a^2-b \left (9 (5 A+3 C) a^4-b^2 (23 A-5 C) a^2+10 A b^4\right ) a+15 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {\frac {\frac {15 A \left (a^2-b^2\right )^3 \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {15 A \left (a^2-b^2\right )^3 \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {\frac {\frac {\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+(a-b) \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 A \sqrt {a+b} \left (a^2-b^2\right )^3 \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {\frac {\frac {\left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 A \sqrt {a+b} \left (a^2-b^2\right )^3 \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 (a-b) \sqrt {a+b} \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}+\frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {2 \left (a^2 C+A b^2\right ) \tan (c+d x)}{5 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{5/2}}+\frac {\frac {\frac {-\frac {30 A \sqrt {a+b} \left (a^2-b^2\right )^3 \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}-\frac {2 (a-b) \sqrt {a+b} \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}-\frac {2 \left (-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{5 a \left (a^2-b^2\right )}\) |
(2*(A*b^2 + a^2*C)*Tan[c + d*x])/(5*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^( 5/2)) + ((-2*(5*A*b^4 - 3*a^4*C - a^2*b^2*(13*A + 5*C))*Tan[c + d*x])/(3*a *(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (((-2*(a - b)*Sqrt[a + b]*(41 *a^2*A*b^4 - 15*A*b^6 - 3*a^6*C - 29*a^4*b^2*(2*A + C))*Cot[c + d*x]*Ellip ticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[( b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b ^2*d) + (2*(a - b)*Sqrt[a + b]*(36*a^2*A*b^3 - 5*a*A*b^4 - 15*A*b^5 + 3*a^ 5*C + a^3*b^2*(13*A + 5*C) - 3*a^4*b*(15*A + 8*C))*Cot[c + d*x]*EllipticF[ ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (30*A*Sqrt[a + b]*(a^2 - b^2)^3*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin [Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[ c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d))/(a*(a^ 2 - b^2)) - (2*(41*a^2*A*b^4 - 15*A*b^6 - 3*a^6*C - 29*a^4*b^2*(2*A + C))* Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]))/(3*a*(a^2 - b^2) ))/(5*a*(a^2 - b^2))
3.8.54.3.1 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[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) *((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[d Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C )*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C Int[Csc[e + f*x]*(( 1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A , B, C}, x] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(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)/(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)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x ] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 + a^2*C)*Cot[e + f*x]*((a + b*Csc[ e + f*x])^(m + 1)/(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)*Simp[A*(a^2 - b^2)*(m + 1) - a*b* (A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m ] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(14041\) vs. \(2(583)=1166\).
Time = 15.93 (sec) , antiderivative size = 14042, normalized size of antiderivative = 22.43
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(14042\) |
| default | \(\text {Expression too large to display}\) | \(14230\) |
\[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
integral((C*sec(d*x + c)^2 + A)*sqrt(b*sec(d*x + c) + a)/(b^4*sec(d*x + c) ^4 + 4*a*b^3*sec(d*x + c)^3 + 6*a^2*b^2*sec(d*x + c)^2 + 4*a^3*b*sec(d*x + c) + a^4), x)
\[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\int \frac {A + C \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]