Integrand size = 35, antiderivative size = 583 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {\sqrt {a+b} \left (2 a A b^2-3 A b^3+8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}-\frac {\sqrt {a+b} \left (3 A b^4+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^3 d}-\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{64 a^2 d}+\frac {\left (A b^2+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 a d}+\frac {A b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{8 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d} \]
1/4*A*cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d-1/64*(a-b)*(3*A*b^2 -4*a^2*(13*A+20*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2 ),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+s ec(d*x+c))/(a-b))^(1/2)/a^2/d+1/64*(2*a*A*b^2-3*A*b^3+8*a^3*(3*A+4*C)+a^2* (52*A*b+80*C*b))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),( (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^2/d-1/64*(3*A*b^4+24*a^2*b^2*(A+2*C)+16*a^4*(3*A+4* C))*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^3/d-1/64*b*(3*A*b^2-4*a^2*(13*A+20*C))*sin(d*x+c)*(a+b* sec(d*x+c))^(1/2)/a^2/d+1/32*(A*b^2+4*a^2*(3*A+4*C))*cos(d*x+c)*sin(d*x+c) *(a+b*sec(d*x+c))^(1/2)/a/d+1/8*A*b*cos(d*x+c)^2*sin(d*x+c)*(a+b*sec(d*x+c ))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(1904\) vs. \(2(583)=1166\).
Time = 21.46 (sec) , antiderivative size = 1904, normalized size of antiderivative = 3.27 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \]
(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((3*A*b* Sin[c + d*x])/16 + ((16*a^2*A + A*b^2 + 16*a^2*C)*Sin[2*(c + d*x)])/(32*a) + (3*A*b*Sin[3*(c + d*x)])/16 + (a*A*Sin[4*(c + d*x)])/16))/(d*(b + a*Cos [c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^(3/2)*( A + C*Sec[c + d*x]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x) /2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(52*a^3*A*b*Tan[(c + d*x)/2] + 52*a^2*A*b ^2*Tan[(c + d*x)/2] - 3*a*A*b^3*Tan[(c + d*x)/2] - 3*A*b^4*Tan[(c + d*x)/2 ] + 80*a^3*b*C*Tan[(c + d*x)/2] + 80*a^2*b^2*C*Tan[(c + d*x)/2] - 104*a^3* A*b*Tan[(c + d*x)/2]^3 + 6*a*A*b^3*Tan[(c + d*x)/2]^3 - 160*a^3*b*C*Tan[(c + d*x)/2]^3 + 52*a^3*A*b*Tan[(c + d*x)/2]^5 - 52*a^2*A*b^2*Tan[(c + d*x)/ 2]^5 - 3*a*A*b^3*Tan[(c + d*x)/2]^5 + 3*A*b^4*Tan[(c + d*x)/2]^5 + 80*a^3* b*C*Tan[(c + d*x)/2]^5 - 80*a^2*b^2*C*Tan[(c + d*x)/2]^5 + 96*a^4*A*Ellipt icPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x) /2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]* Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x )/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 128*a^4*C*EllipticPi[-1, ArcSin[ Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(...
Time = 3.15 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4583, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4592, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4583 |
| \(\displaystyle \frac {1}{4} \int \frac {1}{2} \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (3 A+8 C) \sec ^2(c+d x)+2 a (3 A+4 C) \sec (c+d x)+3 A b\right )dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (3 A+8 C) \sec ^2(c+d x)+2 a (3 A+4 C) \sec (c+d x)+3 A b\right )dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (3 A+8 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a (3 A+4 C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 A b\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {3 \cos ^2(c+d x) \left (4 (3 A+4 C) a^2+2 b (11 A+16 C) \sec (c+d x) a+A b^2+b^2 (9 A+16 C) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \int \frac {\cos ^2(c+d x) \left (4 (3 A+4 C) a^2+2 b (11 A+16 C) \sec (c+d x) a+A b^2+b^2 (9 A+16 C) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \int \frac {4 (3 A+4 C) a^2+2 b (11 A+16 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a+A b^2+b^2 (9 A+16 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-b \left (4 (3 A+4 C) a^2+A b^2\right ) \sec ^2(c+d x)-2 a \left (4 (3 A+4 C) a^2+b^2 (19 A+32 C)\right ) \sec (c+d x)+b \left (3 A b^2-4 a^2 (13 A+20 C)\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-b \left (4 (3 A+4 C) a^2+A b^2\right ) \sec ^2(c+d x)-2 a \left (4 (3 A+4 C) a^2+b^2 (19 A+32 C)\right ) \sec (c+d x)+b \left (3 A b^2-4 a^2 (13 A+20 C)\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {-b \left (4 (3 A+4 C) a^2+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a \left (4 (3 A+4 C) a^2+b^2 (19 A+32 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+b \left (3 A b^2-4 a^2 (13 A+20 C)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+2 b \left (4 (3 A+4 C) a^2+A b^2\right ) \sec (c+d x) a+3 A b^4+b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+2 b \left (4 (3 A+4 C) a^2+A b^2\right ) \sec (c+d x) a+3 A b^4+b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+2 b \left (4 (3 A+4 C) a^2+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+3 A b^4+b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+3 A b^4+\left (2 a b \left (4 (3 A+4 C) a^2+A b^2\right )-b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {16 (3 A+4 C) a^4+24 b^2 (A+2 C) a^2+3 A b^4+\left (2 a b \left (4 (3 A+4 C) a^2+A b^2\right )-b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+b \left (8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)+2 a A b^2-3 A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)+2 a A b^2-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)+2 a A b^2-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 \sqrt {a+b} \left (8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)+2 a A b^2-3 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {\left (4 a^2 (3 A+4 C)+A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {b \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-\frac {2 (a-b) \sqrt {a+b} \left (3 A b^2-4 a^2 (13 A+20 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d}-\frac {2 \sqrt {a+b} \left (16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 \sqrt {a+b} \left (8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)+2 a A b^2-3 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}}{4 a}\right )+\frac {A b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\) |
(A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + ((A*b*C os[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d + (((A*b^2 + 4*a^2* (3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d) - (-1/2*((-2*(a - b)*Sqrt[a + b]*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Cot[c + d* x]*EllipticE[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))])/d + (2*Sqrt[a + b]*(2*a*A*b^2 - 3*A*b^3 + 8*a^3*(3*A + 4*C) + a^2*(5 2*A*b + 80*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sq rt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-(( b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*Sqrt[a + b]*(3*A*b^4 + 24*a^2*b^2* (A + 2*C) + 16*a^4*(3*A + 4*C))*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 + (b* (3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a* d))/(4*a))/2)/8
3.8.24.3.1 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_)]*(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_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Co t[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*(C*n + A*(n + 1))*Csc[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, C}, x] && NeQ[a^2 - b^2, 0] && Gt Q[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]
Leaf count of result is larger than twice the leaf count of optimal. \(5198\) vs. \(2(534)=1068\).
Time = 6.37 (sec) , antiderivative size = 5199, normalized size of antiderivative = 8.92
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integral((C*b*cos(d*x + c)^4*sec(d*x + c)^3 + C*a*cos(d*x + c)^4*sec(d*x + c)^2 + A*b*cos(d*x + c)^4*sec(d*x + c) + A*a*cos(d*x + c)^4)*sqrt(b*sec(d *x + c) + a), x)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^4\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]