Integrand size = 45, antiderivative size = 457 \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^4 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^4 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 (A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d}+\frac {2 A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d} \] Output:
-2/315*(a^2-b^2)*(16*A*b^3-75*B*a^3-24*B*a*b^2+6*a^2*b*(6*A+7*C))*((b+a*co s(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^(1/ 2))/a^4/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-2/315*(16*A*b^4-57*B*a^3 *b-24*B*a*b^3+6*a^2*b^2*(4*A+7*C)-21*a^4*(7*A+9*C))*cos(d*x+c)^(1/2)*Ellip ticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/a^ 4/d/((b+a*cos(d*x+c))/(a+b))^(1/2)+2/315*(8*A*b^3+75*B*a^3-12*B*a*b^2+a^2* b*(13*A+21*C))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a^3/d-2/ 315*(6*A*b^2-9*B*a*b-7*a^2*(7*A+9*C))*cos(d*x+c)^(3/2)*(a+b*sec(d*x+c))^(1 /2)*sin(d*x+c)/a^2/d+2/63*(A*b+9*B*a)*cos(d*x+c)^(5/2)*(a+b*sec(d*x+c))^(1 /2)*sin(d*x+c)/a/d+2/9*A*cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c )/d
Result contains complex when optimal does not.
Time = 26.08 (sec) , antiderivative size = 3595, normalized size of antiderivative = 7.87 \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Cos[c + d*x]^(9/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(((57*a^2*A*b + 32*A*b^3 + 34 5*a^3*B - 48*a*b^2*B + 84*a^2*b*C)*Sin[c + d*x])/(630*a^3) + ((133*a^2*A - 12*A*b^2 + 18*a*b*B + 126*a^2*C)*Sin[2*(c + d*x)])/(630*a^2) + ((A*b + 9* a*B)*Sin[3*(c + d*x)])/(126*a) + (A*Sin[4*(c + d*x)])/36))/d - (2*Cos[c + d*x]^(3/2)*((7*a*A*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[S ec[c + d*x]]) - (8*A*b^2*Sqrt[Cos[c + d*x]])/(105*a*Sqrt[b + a*Cos[c + d*x ]]*Sqrt[Sec[c + d*x]]) - (16*A*b^4*Sqrt[Cos[c + d*x]])/(315*a^3*Sqrt[b + a *Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (19*b*B*Sqrt[Cos[c + d*x]])/(105*Sqrt [b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*b^3*B*Sqrt[Cos[c + d*x]])/(1 05*a^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*a*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b^2*C*Sqrt[Cos [c + d*x]])/(15*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (37*A*b*S qrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (4* A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(315*a^2*Sqrt[b + a*Cos[c + d *x]]) + (5*a*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[b + a*Cos[c + d*x]]) + (7*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*Sqr t[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sqrt[a + b *Sec[c + d*x]]*((-I)*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2* b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)...
Time = 4.28 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.09, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4753, 3042, 4582, 27, 3042, 4592, 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.
| \(\displaystyle \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{9/2} \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4753 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sqrt {a+b \sec (c+d x)} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+B \csc \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{9} \int \frac {3 b (2 A+3 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+A b+9 a B}{2 \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {3 b (2 A+3 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+A b+9 a B}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {3 b (2 A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(7 a A+9 b B+9 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+A b+9 a B}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \int \frac {-7 (7 A+9 C) a^2-9 b B a-(47 A b+63 C b+45 a B) \sec (c+d x) a+6 A b^2-4 b (A b+9 a B) \sec ^2(c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-7 (7 A+9 C) a^2-9 b B a-(47 A b+63 C b+45 a B) \sec (c+d x) a+6 A b^2-4 b (A b+9 a B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-7 (7 A+9 C) a^2-9 b B a-(47 A b+63 C b+45 a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a+6 A b^2-4 b (A b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {-2 b \left (-7 (7 A+9 C) a^2-9 b B a+6 A b^2\right ) \sec ^2(c+d x)+a \left (21 (7 A+9 C) a^2+207 b B a+2 A b^2\right ) \sec (c+d x)+3 \left (75 B a^3+b (13 A+21 C) a^2-12 b^2 B a+8 A b^3\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (-7 (7 A+9 C) a^2-9 b B a+6 A b^2\right ) \sec ^2(c+d x)+a \left (21 (7 A+9 C) a^2+207 b B a+2 A b^2\right ) \sec (c+d x)+3 \left (75 B a^3+b (13 A+21 C) a^2-12 b^2 B a+8 A b^3\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (-7 (7 A+9 C) a^2-9 b B a+6 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (21 (7 A+9 C) a^2+207 b B a+2 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (75 B a^3+b (13 A+21 C) a^2-12 b^2 B a+8 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {3 \left (-21 (7 A+9 C) a^4-57 b B a^3+6 b^2 (4 A+7 C) a^2-24 b^3 B a+\left (-75 B a^3-3 b (37 A+49 C) a^2-6 b^2 B a+4 A b^3\right ) \sec (c+d x) a+16 A b^4\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-21 (7 A+9 C) a^4-57 b B a^3+6 b^2 (4 A+7 C) a^2-24 b^3 B a+\left (-75 B a^3-3 b (37 A+49 C) a^2-6 b^2 B a+4 A b^3\right ) \sec (c+d x) a+16 A b^4}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-21 (7 A+9 C) a^4-57 b B a^3+6 b^2 (4 A+7 C) a^2-24 b^3 B a+\left (-75 B a^3-3 b (37 A+49 C) a^2-6 b^2 B a+4 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+16 A b^4}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\) |
Input:
Int[Cos[c + d*x]^(9/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2*(A*b + 9*a*B)*Sqrt[a + b*Sec[c + d *x]]*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((2*(6*A*b^2 - 9*a*b*B - 7 *a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d* x]^(3/2)) - (-(((2*(a^2 - b^2)*(16*A*b^3 - 75*a^3*B - 24*a*b^2*B + 6*a^2*b *(6*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2 *a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(16*A *b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C ))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sq rt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]))/a) + (2*(8*A*b^3 + 7 5*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d*Sqrt[Sec[c + d*x]]))/(5*a))/(7*a))/9)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 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] && GtQ[m, 0] && LeQ[n, -1]
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]
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x ]
Leaf count of result is larger than twice the leaf count of optimal. \(3632\) vs. \(2(426)=852\).
Time = 32.31 (sec) , antiderivative size = 3633, normalized size of antiderivative = 7.95
Input:
int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x,method=_RETURNVERBOSE)
Output:
2/315/d*(-16*A*((a-b)/(a+b))^(1/2)*b^5*sin(d*x+c)+(-147*cos(d*x+c)^2-294*c os(d*x+c)-147)*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d *x+c)+1))^(1/2)*a^5*EllipticF(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)), (-(a+b)/(a-b))^(1/2))+(75*cos(d*x+c)^2+150*cos(d*x+c)+75)*B*(1/(a+b)*(b+a* cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*EllipticF(( (a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-189*cos (d*x+c)^2-378*cos(d*x+c)-189)*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^ (1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*EllipticF(((a-b)/(a+b))^(1/2)*(csc(d*x+ c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(147*cos(d*x+c)^2+294*cos(d*x+c)+147) *A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2 )*a^5*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b)) ^(1/2))+(16*cos(d*x+c)^2+32*cos(d*x+c)+16)*A*(1/(a+b)*(b+a*cos(d*x+c))/(co s(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*b^5*EllipticE(((a-b)/(a+b))^(1 /2)*(csc(d*x+c)-cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(189*cos(d*x+c)^2+378*co s(d*x+c)+189)*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d* x+c)+1))^(1/2)*a^5*EllipticE(((a-b)/(a+b))^(1/2)*(csc(d*x+c)-cot(d*x+c)),( -(a+b)/(a-b))^(1/2))+(40*cos(d*x+c)^4+40*cos(d*x+c)^3+62*cos(d*x+c)^2+62*c os(d*x+c)+147)*sin(d*x+c)*A*((a-b)/(a+b))^(1/2)*a^4*b+(-cos(d*x+c)^3-cos(d *x+c)^2-11*cos(d*x+c)+13)*sin(d*x+c)*A*((a-b)/(a+b))^(1/2)*a^3*b^2+(2*cos( d*x+c)^2+2*cos(d*x+c)-24)*sin(d*x+c)*A*((a-b)/(a+b))^(1/2)*a^2*b^3+(-8*...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.57 \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="fricas")
Output:
2/945*(3*(35*A*a^5*cos(d*x + c)^3 + 75*B*a^5 + (13*A + 21*C)*a^4*b - 12*B* a^3*b^2 + 8*A*a^2*b^3 + 5*(9*B*a^5 + A*a^4*b)*cos(d*x + c)^2 + (7*(7*A + 9 *C)*a^5 + 9*B*a^4*b - 6*A*a^3*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b) /cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - sqrt(1/2)*(225*I*B*a^5 + 3*I*(13*A + 21*C)*a^4*b - 96*I*B*a^3*b^2 + 12*I*(3*A + 7*C)*a^2*b^3 - 48*I *B*a*b^4 + 32*I*A*b^5)*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(1/2)*(-225*I*B*a^5 - 3*I*(13*A + 21*C)*a^4*b + 96*I*B*a^3 *b^2 - 12*I*(3*A + 7*C)*a^2*b^3 + 48*I*B*a*b^4 - 32*I*A*b^5)*sqrt(a)*weier strassPInverse(-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(1/2)*(-21*I*(7*A + 9*C)*a^5 - 57*I*B*a^4*b + 6*I*(4*A + 7*C)*a^3*b^2 - 24*I*B*a^2*b^3 + 16* I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a)) - 3*s qrt(1/2)*(21*I*(7*A + 9*C)*a^5 + 57*I*B*a^4*b - 6*I*(4*A + 7*C)*a^3*b^2 + 24*I*B*a^2*b^3 - 16*I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2 )/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2 )/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a)))/(a^5*d)
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(9/2)*(a+b*sec(d*x+c))**(1/2)*(A+B*sec(d*x+c)+C*sec( d*x+c)**2),x)
Output:
Timed out
\[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(b*sec(d*x + c) + a) *cos(d*x + c)^(9/2), x)
\[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d* x+c)^2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(b*sec(d*x + c) + a) *cos(d*x + c)^(9/2), x)
Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:
int(cos(c + d*x)^(9/2)*(a + b/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/ cos(c + d*x)^2),x)
Output:
int(cos(c + d*x)^(9/2)*(a + b/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/ cos(c + d*x)^2), x)
\[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a \] Input:
int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d* x)**2,x)*c + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)* *4*sec(c + d*x),x)*b + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos (c + d*x)**4,x)*a