Integrand size = 43, antiderivative size = 667 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^4 (a+b)^3 d}+\frac {\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
-1/12*(15*B*a^3*b-33*B*a*b^3-a^2*b^2*(3*A-61*C)+b^4*(21*A-8*C)-35*a^4*C)*s ec(d*x+c)^(3/2)*sin(d*x+c)/b^3/(a^2-b^2)^2/d-1/2*(A*b^2-a*(B*b-C*a))*sec(d *x+c)^(7/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+1/4*(5*A*b^4+3*B*a ^3*b-9*B*a*b^3-7*a^4*C+a^2*b^2*(A+13*C))*sec(d*x+c)^(5/2)*sin(d*x+c)/b^2/( a^2-b^2)^2/d/(a+b*sec(d*x+c))+1/4*(15*B*a^4*b-29*B*a^2*b^3+8*B*b^5-a^3*b^2 *(3*A-65*C)+3*a*b^4*(3*A-8*C)-35*a^5*C)*sin(d*x+c)*sec(d*x+c)^(1/2)/b^4/(a ^2-b^2)^2/d-1/4*(15*B*a^4*b-29*B*a^2*b^3+8*B*b^5-a^3*b^2*(3*A-65*C)+3*a*b^ 4*(3*A-8*C)-35*a^5*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elli pticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/b^4/(a ^2-b^2)^2/d-1/12*(15*B*a^3*b-33*B*a*b^3-a^2*b^2*(3*A-61*C)+b^4*(21*A-8*C)- 35*a^4*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/b^3/(a^2-b^2)^2/d+ 1/4*(15*A*b^6-15*a^5*b*B+38*a^3*b^3*B-35*a*b^5*B+a^4*b^2*(3*A-86*C)-3*a^2* b^4*(2*A-21*C)+35*a^6*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*E llipticPi(sin(1/2*d*x+1/2*c),2*a/(a+b),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c )^(1/2)/(a-b)^2/b^4/(a+b)^3/d
Time = 16.40 (sec) , antiderivative size = 1156, normalized size of antiderivative = 1.73 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x))^3 \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (27 a^4 A b^2-57 a^2 A b^4+48 A b^6-135 a^5 b B+285 a^3 b^3 B-168 a b^5 B+315 a^6 C-641 a^4 b^2 C+328 a^2 b^4 C+16 b^6 C\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (24 a^3 A b^3-96 a A b^5-120 a^4 b^2 B+240 a^2 b^4 B-48 b^6 B+280 a^5 b C-512 a^3 b^3 C+160 a b^5 C\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (9 a^4 A b^2-27 a^2 A b^4-45 a^5 b B+87 a^3 b^3 B-24 a b^5 B+105 a^6 C-195 a^4 b^2 C+72 a^2 b^4 C\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-2 a (a-2 b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 a^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 b^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}\right )}{24 (a-b)^2 b^4 (a+b)^2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {(b+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {\left (3 a^3 A b^2-9 a A b^4-15 a^4 b B+29 a^2 b^3 B-8 b^5 B+35 a^5 C-65 a^3 b^2 C+24 a b^4 C\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2}+\frac {-a A b^2 \sin (c+d x)+a^2 b B \sin (c+d x)-a^3 C \sin (c+d x)}{b^2 \left (-a^2+b^2\right ) (b+a \cos (c+d x))^2}+\frac {a^3 A b^2 \sin (c+d x)-7 a A b^4 \sin (c+d x)-5 a^4 b B \sin (c+d x)+11 a^2 b^3 B \sin (c+d x)+9 a^5 C \sin (c+d x)-15 a^3 b^2 C \sin (c+d x)}{2 b^3 \left (-a^2+b^2\right )^2 (b+a \cos (c+d x))}+\frac {4 C \tan (c+d x)}{3 b^3}\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3} \]
Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]
((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^ 2)*((2*(27*a^4*A*b^2 - 57*a^2*A*b^4 + 48*A*b^6 - 135*a^5*b*B + 285*a^3*b^3 *B - 168*a*b^5*B + 315*a^6*C - 641*a^4*b^2*C + 328*a^2*b^4*C + 16*b^6*C)*C os[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/ a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2* (24*a^3*A*b^3 - 96*a*A*b^5 - 120*a^4*b^2*B + 240*a^2*b^4*B - 48*b^6*B + 28 0*a^5*b*C - 512*a^3*b^3*C + 160*a*b^5*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x ]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^4 *A*b^2 - 27*a^2*A*b^4 - 45*a^5*b*B + 87*a^3*b^3*B - 24*a*b^5*B + 105*a^6*C - 195*a^4*b^2*C + 72*a^2*b^4*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4 *a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2 ] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[ c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x]) /(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(24*(a - b)^2*b^4*(a + b)^2*d*(A + 2*C + 2*B*Cos[c +...
Time = 4.96 (sec) , antiderivative size = 660, normalized size of antiderivative = 0.99, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {3042, 4586, 27, 3042, 4586, 27, 3042, 4590, 27, 3042, 4590, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 3284}
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.
| \(\displaystyle \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4586 |
| \(\displaystyle -\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (-\left (\left (7 C a^2-3 b B a+3 A b^2-4 b^2 C\right ) \sec ^2(c+d x)\right )+4 b (b B-a (A+C)) \sec (c+d x)+5 \left (A b^2-a (b B-a C)\right )\right )}{2 (a+b \sec (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (-\left (\left (7 C a^2-3 b B a+3 A b^2-4 b^2 C\right ) \sec ^2(c+d x)\right )+4 b (b B-a (A+C)) \sec (c+d x)+5 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \sec (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (\left (-7 C a^2+3 b B a-3 A b^2+4 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 b (b B-a (A+C)) \csc \left (c+d x+\frac {\pi }{2}\right )+5 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4586 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right ) \sec ^2(c+d x)\right )+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \sec (c+d x)+3 \left (-7 C a^4+3 b B a^3+b^2 (A+13 C) a^2-9 b^3 B a+5 A b^4\right )\right )}{2 (a+b \sec (c+d x))}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right ) \sec ^2(c+d x)\right )+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \sec (c+d x)+3 \left (-7 C a^4+3 b B a^3+b^2 (A+13 C) a^2-9 b^3 B a+5 A b^4\right )\right )}{a+b \sec (c+d x)}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\left (35 C a^4-15 b B a^3+b^2 (3 A-61 C) a^2+33 b^3 B a-b^4 (21 A-8 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-7 C a^4+3 b B a^3+b^2 (A+13 C) a^2-9 b^3 B a+5 A b^4\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle -\frac {-\frac {\frac {2 \int -\frac {\sqrt {\sec (c+d x)} \left (-3 \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right ) \sec ^2(c+d x)-4 b \left (-7 C a^4+3 b B a^3+b^2 (3 A+14 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right ) \sec (c+d x)+a \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )\right )}{2 (a+b \sec (c+d x))}dx}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {\sqrt {\sec (c+d x)} \left (-3 \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right ) \sec ^2(c+d x)-4 b \left (-7 C a^4+3 b B a^3+b^2 (3 A+14 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right ) \sec (c+d x)+a \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )\right )}{a+b \sec (c+d x)}dx}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (-3 \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-4 b \left (-7 C a^4+3 b B a^3+b^2 (3 A+14 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {2 \int \frac {\left (-105 C a^6+45 b B a^5-b^2 (9 A-223 C) a^4-99 b^3 B a^3+b^4 (15 A-128 C) a^2+72 b^5 B a-8 b^6 (3 A+C)\right ) \sec ^2(c+d x)+4 b \left (-35 C a^5+15 b B a^4-b^2 (3 A-64 C) a^3-30 b^3 B a^2+4 b^4 (3 A-5 C) a+6 b^5 B\right ) \sec (c+d x)+3 a \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right )}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {\left (-105 C a^6+45 b B a^5-b^2 (9 A-223 C) a^4-99 b^3 B a^3+b^4 (15 A-128 C) a^2+72 b^5 B a-8 b^6 (3 A+C)\right ) \sec ^2(c+d x)+4 b \left (-35 C a^5+15 b B a^4-b^2 (3 A-64 C) a^3-30 b^3 B a^2+4 b^4 (3 A-5 C) a+6 b^5 B\right ) \sec (c+d x)+3 a \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right )}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {\left (-105 C a^6+45 b B a^5-b^2 (9 A-223 C) a^4-99 b^3 B a^3+b^4 (15 A-128 C) a^2+72 b^5 B a-8 b^6 (3 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (-35 C a^5+15 b B a^4-b^2 (3 A-64 C) a^3-30 b^3 B a^2+4 b^4 (3 A-5 C) a+6 b^5 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4594 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\int \frac {3 \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right ) a^2+b \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right ) \sec (c+d x) a^2}{\sqrt {\sec (c+d x)}}dx}{a^2}-3 \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\int \frac {3 \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right ) a^2+b \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}-3 \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a^2 b \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right ) \int \sqrt {\sec (c+d x)}dx+3 a^2 \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx}{a^2}-3 \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a^2 b \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+3 a^2 \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}-3 \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a^2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+3 a^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right ) \int \sqrt {\cos (c+d x)}dx}{a^2}-3 \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a^2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}-3 \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a^2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{d}}{a^2}-3 \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\frac {2 a^2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{d}+\frac {6 a^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{d}}{a^2}-3 \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4336 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\frac {2 a^2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{d}+\frac {6 a^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{d}}{a^2}-3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {\frac {2 a^2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{d}+\frac {6 a^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{d}}{a^2}-3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {\sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {-\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}-\frac {\frac {\frac {\frac {2 a^2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{d}+\frac {6 a^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{d}}{a^2}-\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{b}-\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
-1/2*((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(b*(a^2 - b ^2)*d*(a + b*Sec[c + d*x])^2) - (-(((5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a ^4*C + a^2*b^2*(A + 13*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2) *d*(a + b*Sec[c + d*x]))) - ((-2*(15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3* b*d) - ((((6*a^2*(15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65* C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x )/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^2*b*(15*a^3*b*B - 33*a*b^3*B - a^2*b^ 2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticF [(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d)/a^2 - (6*(15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*S qrt[Sec[c + d*x]])/((a + b)*d))/b - (6*(15*a^4*b*B - 29*a^2*b^3*B + 8*b^5* B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*Sqrt[Sec[c + d* x]]*Sin[c + d*x])/(b*d))/(3*b))/(2*b*(a^2 - b^2)))/(4*b*(a^2 - b^2))
3.11.25.3.1 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[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]] Int[ 1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && 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[(-d)*(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 - 1)/(b*f*(a^2 - b^2)*(m + 1)) ), x] + Simp[d/(b*(a^2 - b^2)*(m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*( d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C }, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[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[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 )*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2) Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2 Int[(a*A - (A*b - a *B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2157\) vs. \(2(711)=1422\).
Time = 359.78 (sec) , antiderivative size = 2158, normalized size of antiderivative = 3.24
int(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*C/b^3*(-1/6* cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(c os(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d* x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*E llipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(B*b-3*C*a)/b^4/sin(1/2*d*x+1/2*c) ^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^ 2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-EllipticE(cos(1/2*d*x+ 1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1 /2))-2*a*(B*b-2*C*a)/b^3*(a^2/b/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d *x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*a-a+b)-1/2 /(a+b)/b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(- 2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1 /2*c),2^(1/2))+1/2*a/b/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2) *EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2*a/b/(a^2-b^2)*(sin(1/2*d*x+1/2* c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin (1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/b/(a^2- b^2)/(a^2-a*b)*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1 )^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(co s(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))+3/2*b/(a^2-b^2)/(a^2-a*b)*a*(sin(1/...
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^3,x, algorithm="fricas")
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^3,x, algorithm="maxima")
\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^3,x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sec(d*x + c)^(7/2)/(b*se c(d*x + c) + a)^3, x)
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
int(((1/cos(c + d*x))^(7/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + b/cos(c + d*x))^3,x)