Integrand size = 35, antiderivative size = 587 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}+\frac {\sqrt {a+b} \left (15 A b^3+24 a^3 (3 A+4 C)+4 a^2 b (71 A+108 C)+2 a b^2 (59 A+192 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}+\frac {\sqrt {a+b} \left (5 A b^4-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (5 A b^2+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {5 A b \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d} \]
5/24*A*b*cos(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/4*A*cos(d*x+c) ^3*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d+1/192*(a-b)*(15*A*b^2+4*a^2*(71*A+1 08*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+sec(d*x+c))/(a -b))^(1/2)/a/d+1/192*(15*A*b^3+24*a^3*(3*A+4*C)+4*a^2*b*(71*A+108*C)+2*a*b ^2*(59*A+192*C))*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/d+1/64*(5*A*b^4-120*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^2/d+1/192*b*(15*A*b^2+4*a^2*(71*A+108*C))*sin(d*x+c)*(a+ b*sec(d*x+c))^(1/2)/a/d+1/32*(5*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)/d
Leaf count is larger than twice the leaf count of optimal. \(4992\) vs. \(2(587)=1174\).
Time = 28.06 (sec) , antiderivative size = 4992, normalized size of antiderivative = 8.50 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]
(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((17*a*A *b*Sin[c + d*x])/48 + ((48*a^2*A + 59*A*b^2 + 48*a^2*C)*Sin[2*(c + d*x)])/ 96 + (17*a*A*b*Sin[3*(c + d*x)])/48 + (a^2*A*Sin[4*(c + d*x)])/16))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + (Sqrt[Cos[c + d*x]*S ec[(c + d*x)/2]^2]*((3*a^3*A)/(4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x ]]) + (161*a*A*b^2)/(48*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^ 3*C)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a*b^2*C)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (89*a^2*A*b*Sqrt[Sec[c + d*x]])/(48 *Sqrt[b + a*Cos[c + d*x]]) + (133*A*b^3*Sqrt[Sec[c + d*x]])/(192*Sqrt[b + a*Cos[c + d*x]]) + (11*a^2*b*C*Sqrt[Sec[c + d*x]])/(4*Sqrt[b + a*Cos[c + d *x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (71*a^2*A* b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(48*Sqrt[b + a*Cos[c + d*x]]) + (5* A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(64*Sqrt[b + a*Cos[c + d*x]]) + (9*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(4*Sqrt[b + a*Cos[c + d*x ]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(a*b*(a + b)*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Elliptic E[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-30*a*A*b^2 + 1 5*A*b^3 - 24*a^3*(3*A + 4*C) - 4*a^2*b*(53*A + 84*C))*EllipticF[ArcSin[Tan [(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c ...
Time = 3.11 (sec) , antiderivative size = 595, 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, 4582, 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.
\(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
\(\Big \downarrow \) 4583 |
\(\displaystyle \frac {1}{4} \int \frac {1}{2} \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (b (A+8 C) \sec ^2(c+d x)+2 a (3 A+4 C) \sec (c+d x)+5 A b\right )dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (b (A+8 C) \sec ^2(c+d x)+2 a (3 A+4 C) \sec (c+d x)+5 A b\right )dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (A+8 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a (3 A+4 C) \csc \left (c+d x+\frac {\pi }{2}\right )+5 A b\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 4582 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {1}{2} \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (b^2 (11 A+48 C) \sec ^2(c+d x)+2 a b (31 A+48 C) \sec (c+d x)+3 \left (4 (3 A+4 C) a^2+5 A b^2\right )\right )dx+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (b^2 (11 A+48 C) \sec ^2(c+d x)+2 a b (31 A+48 C) \sec (c+d x)+3 \left (4 (3 A+4 C) a^2+5 A b^2\right )\right )dx+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b^2 (11 A+48 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (31 A+48 C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (4 (3 A+4 C) a^2+5 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 4582 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {\cos (c+d x) \left (b \left (12 (3 A+4 C) a^2+b^2 (59 A+192 C)\right ) \sec ^2(c+d x)+2 a \left (12 (3 A+4 C) a^2+b^2 (161 A+288 C)\right ) \sec (c+d x)+b \left (4 (71 A+108 C) a^2+15 A b^2\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {\cos (c+d x) \left (b \left (12 (3 A+4 C) a^2+b^2 (59 A+192 C)\right ) \sec ^2(c+d x)+2 a \left (12 (3 A+4 C) a^2+b^2 (161 A+288 C)\right ) \sec (c+d x)+b \left (4 (71 A+108 C) a^2+15 A b^2\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {b \left (12 (3 A+4 C) a^2+b^2 (59 A+192 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (12 (3 A+4 C) a^2+b^2 (161 A+288 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+b \left (4 (71 A+108 C) a^2+15 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b^2 \left (4 (71 A+108 C) a^2+15 A b^2\right ) \sec ^2(c+d x)-2 a b \left (12 (3 A+4 C) a^2+b^2 (59 A+192 C)\right ) \sec (c+d x)+3 \left (-16 (3 A+4 C) a^4-120 b^2 (A+2 C) a^2+5 A b^4\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b^2 \left (4 (71 A+108 C) a^2+15 A b^2\right ) \sec ^2(c+d x)-2 a b \left (12 (3 A+4 C) a^2+b^2 (59 A+192 C)\right ) \sec (c+d x)+3 \left (-16 (3 A+4 C) a^4-120 b^2 (A+2 C) a^2+5 A b^4\right )}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b^2 \left (4 (71 A+108 C) a^2+15 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (12 (3 A+4 C) a^2+b^2 (59 A+192 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-16 (3 A+4 C) a^4-120 b^2 (A+2 C) a^2+5 A b^4\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 4546 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {3 \left (-16 (3 A+4 C) a^4-120 b^2 (A+2 C) a^2+5 A b^4\right )+\left (-\left (\left (4 (71 A+108 C) a^2+15 A b^2\right ) b^2\right )-2 a \left (12 (3 A+4 C) a^2+b^2 (59 A+192 C)\right ) b\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {3 \left (-16 (3 A+4 C) a^4-120 b^2 (A+2 C) a^2+5 A b^4\right )+\left (-\left (\left (4 (71 A+108 C) a^2+15 A b^2\right ) b^2\right )-2 a \left (12 (3 A+4 C) a^2+b^2 (59 A+192 C)\right ) b\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 4409 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx-b \left (24 a^3 (3 A+4 C)+4 a^2 b (71 A+108 C)+2 a b^2 (59 A+192 C)+15 A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (24 a^3 (3 A+4 C)+4 a^2 b (71 A+108 C)+2 a b^2 (59 A+192 C)+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 4271 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (24 a^3 (3 A+4 C)+4 a^2 b (71 A+108 C)+2 a b^2 (59 A+192 C)+15 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 4319 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b^2 \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}-\frac {2 \sqrt {a+b} \left (24 a^3 (3 A+4 C)+4 a^2 b (71 A+108 C)+2 a b^2 (59 A+192 C)+15 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}\right )+\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
\(\Big \downarrow \) 4492 |
\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {3 \left (4 a^2 (3 A+4 C)+5 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}+\frac {1}{4} \left (\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-\frac {2 (a-b) \sqrt {a+b} \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \left (-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}-\frac {2 \sqrt {a+b} \left (24 a^3 (3 A+4 C)+4 a^2 b (71 A+108 C)+2 a b^2 (59 A+192 C)+15 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}\right )\right )+\frac {5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}\) |
(A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d) + ((5*A*b *Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*(5*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*d) + (-1/2*((-2*(a - b)*Sqrt[a + b]*(15*A*b^2 + 4*a^2*(71*A + 108*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]*(15*A*b^3 + 24*a^3*(3*A + 4*C) + 4*a^ 2*b*(71*A + 108*C) + 2*a*b^2*(59*A + 192*C))*Cot[c + d*x]*EllipticF[ArcSin [Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[ c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (6*Sqrt[a + b]*(5*A*b^4 - 120*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*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b* Sec[c + d*x]]*Sin[c + d*x])/(a*d))/4)/6)/8
3.8.32.3.1 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. \(5461\) vs. \(2(538)=1076\).
Time = 495.72 (sec) , antiderivative size = 5462, normalized size of antiderivative = 9.30
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integral((C*b^2*cos(d*x + c)^4*sec(d*x + c)^4 + 2*C*a*b*cos(d*x + c)^4*sec (d*x + c)^3 + 2*A*a*b*cos(d*x + c)^4*sec(d*x + c) + A*a^2*cos(d*x + c)^4 + (C*a^2 + A*b^2)*cos(d*x + c)^4*sec(d*x + c)^2)*sqrt(b*sec(d*x + c) + a), x)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^4\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]