Integrand size = 35, antiderivative size = 286 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 a^3 (27 A+17 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (21 A+11 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (27 A+17 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a^3 (21 A+16 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d} \]
8/105*a^3*(21*A+16*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/9*C*sec(d*x+c)^(3/2) *(a+a*sec(d*x+c))^3*sin(d*x+c)/d+4/21*C*sec(d*x+c)^(3/2)*(a^2+a^2*sec(d*x+ c))^2*sin(d*x+c)/a/d+2/315*(63*A+73*C)*sec(d*x+c)^(3/2)*(a^3+a^3*sec(d*x+c ))*sin(d*x+c)/d+4/15*a^3*(27*A+17*C)*sin(d*x+c)*sec(d*x+c)^(1/2)/d-4/15*a^ 3*(27*A+17*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(si n(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+4/21*a^3*(21 *A+11*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)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.82 (sec) , antiderivative size = 818, normalized size of antiderivative = 2.86 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{5 \sqrt {2} d (A+2 C+A \cos (2 c+2 d x))}+\frac {17 C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{45 \sqrt {2} d (A+2 C+A \cos (2 c+2 d x))}+\frac {A \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{d (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {11 C \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{21 d (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {\sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \left (\frac {(27 A+17 C) \cos (d x) \csc (c)}{15 d}+\frac {C \sec (c) \sec ^4(c+d x) \sin (d x)}{18 d}+\frac {\sec (c) \sec ^3(c+d x) (7 C \sin (c)+27 C \sin (d x))}{126 d}+\frac {\sec (c) \sec ^2(c+d x) (135 C \sin (c)+63 A \sin (d x)+238 C \sin (d x))}{630 d}+\frac {\sec (c) \sec (c+d x) (63 A \sin (c)+238 C \sin (c)+315 A \sin (d x)+330 C \sin (d x))}{630 d}+\frac {(21 A+22 C) \tan (c)}{42 d}\right )}{(A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]
(3*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I )*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2) )/(5*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (17*C*Sqrt[E^(I *(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E ^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/ 2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(45*Sqrt[2]* d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Sqrt[Cos[c + d*x]]*Ellipt icF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec [c + d*x]^2))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (11* C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a *Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + A*Cos[2*c + 2*d* x])*Sec[c + d*x]^(9/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(((27*A + 17*C)*Cos[d*x]*Csc[c])/(15*d) + (C*Sec[c]*Se c[c + d*x]^4*Sin[d*x])/(18*d) + (Sec[c]*Sec[c + d*x]^3*(7*C*Sin[c] + 27*C* Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^2*(135*C*Sin[c] + 63*A*Sin[d*x] + 238*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x]*(63*A*Sin[c] + 238*C*Sin [c] + 315*A*Sin[d*x] + 330*C*Sin[d*x]))/(630*d) + ((21*A + 22*C)*Tan[c]...
Time = 1.72 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4577, 27, 3042, 4506, 27, 3042, 4506, 27, 3042, 4485, 27, 3042, 4274, 3042, 4255, 3042, 4258, 3042, 3119, 3120}
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.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4577 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^3 (a (9 A+C)+6 a C \sec (c+d x))dx}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^3 (a (9 A+C)+6 a C \sec (c+d x))dx}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (9 A+C)+6 a C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {1}{2} \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^2 \left ((63 A+13 C) a^2+(63 A+73 C) \sec (c+d x) a^2\right )dx+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^2 \left ((63 A+13 C) a^2+(63 A+73 C) \sec (c+d x) a^2\right )dx+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((63 A+13 C) a^2+(63 A+73 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int 3 \sqrt {\sec (c+d x)} (\sec (c+d x) a+a) \left ((63 A+23 C) a^3+6 (21 A+16 C) \sec (c+d x) a^3\right )dx+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a) \left ((63 A+23 C) a^3+6 (21 A+16 C) \sec (c+d x) a^3\right )dx+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((63 A+23 C) a^3+6 (21 A+16 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\frac {2}{3} \int \frac {3}{2} \sqrt {\sec (c+d x)} \left (5 (21 A+11 C) a^4+7 (27 A+17 C) \sec (c+d x) a^4\right )dx+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \sqrt {\sec (c+d x)} \left (5 (21 A+11 C) a^4+7 (27 A+17 C) \sec (c+d x) a^4\right )dx+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (21 A+11 C) a^4+7 (27 A+17 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (7 a^4 (27 A+17 C) \int \sec ^{\frac {3}{2}}(c+d x)dx+5 a^4 (21 A+11 C) \int \sqrt {\sec (c+d x)}dx+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+11 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+7 a^4 (27 A+17 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+11 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+7 a^4 (27 A+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+11 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+7 a^4 (27 A+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 a^4 (27 A+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (27 A+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}+7 a^4 (27 A+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}+\frac {6}{5} \left (\frac {4 a^4 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}+\frac {10 a^4 (21 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+7 a^4 (27 A+17 C) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {12 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}\) |
(2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + ((12* C*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2* (63*A + 73*C)*Sec[c + d*x]^(3/2)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(5 *d) + (6*((10*a^4*(21*A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^4*(21*A + 16*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/d + 7*a^4*(27*A + 17*C)*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x) /2, 2]*Sqrt[Sec[c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d)))/5) /7)/(9*a)
3.3.23.3.1 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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
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_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[ e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Simp[1/(n + 1) Int[(d*Csc [e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x ], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && !LeQ[ n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x] )^n*Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[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[(-C) *Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n *Simp[A*b*(m + n + 1) + b*C*n + a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1219\) vs. \(2(306)=612\).
Time = 5.23 (sec) , antiderivative size = 1220, normalized size of antiderivative = 4.27
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1220\) |
| parts | \(\text {Expression too large to display}\) | \(1548\) |
-a^3*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A*(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))+ 6*C*(-1/56*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+5/21 *(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)))+6*A/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*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1 /2*c)^2-(sin(1/2*d*x+1/2*c)^2)^(1/2)*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))+2*C*(-1/144*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^5- 7/180*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/ 2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(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))- 7/15*(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*si n(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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.98 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (21 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (21 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (27 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (27 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (42 \, {\left (27 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (21 \, A + 22 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (9 \, A + 34 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 135 \, C a^{3} \cos \left (d x + c\right ) + 35 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d \cos \left (d x + c\right )^{4}} \]
-2/315*(15*I*sqrt(2)*(21*A + 11*C)*a^3*cos(d*x + c)^4*weierstrassPInverse( -4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(21*A + 11*C)*a^3*cos (d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21 *I*sqrt(2)*(27*A + 17*C)*a^3*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierst rassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I*sqrt(2)*(27*A + 17*C)*a^3*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 , cos(d*x + c) - I*sin(d*x + c))) - (42*(27*A + 17*C)*a^3*cos(d*x + c)^4 + 15*(21*A + 22*C)*a^3*cos(d*x + c)^3 + 7*(9*A + 34*C)*a^3*cos(d*x + c)^2 + 135*C*a^3*cos(d*x + c) + 35*C*a^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*co s(d*x + c)^4)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )} \,d x } \]
\[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )} \,d x } \]
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}} \,d x \]