Integrand size = 43, antiderivative size = 271 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {4 a^3 (5 A+9 B+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (35 A+21 B+13 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (140 A+147 B+106 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {2 (7 B+6 C) \sqrt {\sec (c+d x)} \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 a d}+\frac {2 (5 A+9 B+7 C) \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d} \] Output:
-4/5*a^3*(5*A+9*B+7*C)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/ 2))*sec(d*x+c)^(1/2)/d+4/21*a^3*(35*A+21*B+13*C)*cos(d*x+c)^(1/2)*InverseJ acobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+4/105*a^3*(140*A+147*B+1 06*C)*sec(d*x+c)^(1/2)*sin(d*x+c)/d+2/7*C*sec(d*x+c)^(1/2)*(a+a*sec(d*x+c) )^3*sin(d*x+c)/d+2/35*(7*B+6*C)*sec(d*x+c)^(1/2)*(a^2+a^2*sec(d*x+c))^2*si n(d*x+c)/a/d+2/15*(5*A+9*B+7*C)*sec(d*x+c)^(1/2)*(a^3+a^3*sec(d*x+c))*sin( d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.18 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.32 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {a^3 e^{-i d x} \sec ^{\frac {7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (-630 i A-1134 i B-882 i C-840 i A \cos (2 (c+d x))-1512 i B \cos (2 (c+d x))-1176 i C \cos (2 (c+d x))-210 i A \cos (4 (c+d x))-378 i B \cos (4 (c+d x))-294 i C \cos (4 (c+d x))+80 (35 A+21 B+13 C) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+14 i (5 A+9 B+7 C) e^{-2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+70 A \sin (c+d x)+210 B \sin (c+d x)+380 C \sin (c+d x)+630 A \sin (2 (c+d x))+840 B \sin (2 (c+d x))+840 C \sin (2 (c+d x))+70 A \sin (3 (c+d x))+210 B \sin (3 (c+d x))+260 C \sin (3 (c+d x))+315 A \sin (4 (c+d x))+378 B \sin (4 (c+d x))+294 C \sin (4 (c+d x))\right )}{420 d} \] Input:
Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Sqrt[Sec[c + d*x]],x]
Output:
(a^3*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])*((-630*I)*A - (1134*I)*B - (882*I)*C - (840*I)*A*Cos[2*(c + d*x)] - (1512*I)*B*Cos[2*(c + d*x)] - (1 176*I)*C*Cos[2*(c + d*x)] - (210*I)*A*Cos[4*(c + d*x)] - (378*I)*B*Cos[4*( c + d*x)] - (294*I)*C*Cos[4*(c + d*x)] + 80*(35*A + 21*B + 13*C)*Cos[c + d *x]^(7/2)*EllipticF[(c + d*x)/2, 2] + ((14*I)*(5*A + 9*B + 7*C)*(1 + E^((2 *I)*(c + d*x)))^(7/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x) )])/E^((2*I)*(c + d*x)) + 70*A*Sin[c + d*x] + 210*B*Sin[c + d*x] + 380*C*S in[c + d*x] + 630*A*Sin[2*(c + d*x)] + 840*B*Sin[2*(c + d*x)] + 840*C*Sin[ 2*(c + d*x)] + 70*A*Sin[3*(c + d*x)] + 210*B*Sin[3*(c + d*x)] + 260*C*Sin[ 3*(c + d*x)] + 315*A*Sin[4*(c + d*x)] + 378*B*Sin[4*(c + d*x)] + 294*C*Sin [4*(c + d*x)]))/(420*d*E^(I*d*x))
Time = 1.71 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4576, 27, 3042, 4506, 27, 3042, 4506, 3042, 4485, 27, 3042, 4274, 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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4576 |
| \(\displaystyle \frac {2 \int \frac {(\sec (c+d x) a+a)^3 (a (7 A-C)+a (7 B+6 C) \sec (c+d x))}{2 \sqrt {\sec (c+d x)}}dx}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\sec (c+d x) a+a)^3 (a (7 A-C)+a (7 B+6 C) \sec (c+d x))}{\sqrt {\sec (c+d x)}}dx}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (7 A-C)+a (7 B+6 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {2}{5} \int \frac {(\sec (c+d x) a+a)^2 \left ((35 A-7 B-11 C) a^2+7 (5 A+9 B+7 C) \sec (c+d x) a^2\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {(\sec (c+d x) a+a)^2 \left ((35 A-7 B-11 C) a^2+7 (5 A+9 B+7 C) \sec (c+d x) a^2\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((35 A-7 B-11 C) a^2+7 (5 A+9 B+7 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {(\sec (c+d x) a+a) \left ((35 A-42 B-41 C) a^3+(140 A+147 B+106 C) \sec (c+d x) a^3\right )}{\sqrt {\sec (c+d x)}}dx+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((35 A-42 B-41 C) a^3+(140 A+147 B+106 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (2 \int -\frac {21 a^4 (5 A+9 B+7 C)-5 a^4 (35 A+21 B+13 C) \sec (c+d x)}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}\right )+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {21 a^4 (5 A+9 B+7 C)-5 a^4 (35 A+21 B+13 C) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx\right )+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {21 a^4 (5 A+9 B+7 C)-5 a^4 (35 A+21 B+13 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (-21 a^4 (5 A+9 B+7 C) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+5 a^4 (35 A+21 B+13 C) \int \sqrt {\sec (c+d x)}dx+\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}\right )+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (-21 a^4 (5 A+9 B+7 C) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 a^4 (35 A+21 B+13 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}\right )+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (5 a^4 (35 A+21 B+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx-21 a^4 (5 A+9 B+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}\right )+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (5 a^4 (35 A+21 B+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-21 a^4 (5 A+9 B+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}\right )+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (5 a^4 (35 A+21 B+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {42 a^4 (5 A+9 B+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {14 (5 A+9 B+7 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 \sec (c+d x)+a^4\right )}{3 d}+\frac {2}{3} \left (\frac {2 a^4 (140 A+147 B+106 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}+\frac {10 a^4 (35 A+21 B+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {42 a^4 (5 A+9 B+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 (7 B+6 C) \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{7 a}+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}\) |
Input:
Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[ Sec[c + d*x]],x]
Output:
(2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + ((2*( 7*B + 6*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(5* d) + ((14*(5*A + 9*B + 7*C)*Sqrt[Sec[c + d*x]]*(a^4 + a^4*Sec[c + d*x])*Si n[c + d*x])/(3*d) + (2*((-42*a^4*(5*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*Elli pticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (10*a^4*(35*A + 21*B + 13*C) *Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a ^4*(140*A + 147*B + 106*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d))/3)/5)/(7*a )
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*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_)]*(B_.) + 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*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Cs c[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b* B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, 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. \(1071\) vs. \(2(246)=492\).
Time = 13.41 (sec) , antiderivative size = 1072, normalized size of antiderivative = 3.96
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1072\) |
| parts | \(\text {Expression too large to display}\) | \(1326\) |
Input:
int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2),x, method=_RETURNVERBOSE)
Output:
-a^3*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4*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))+ 2*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin (1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c) ,2^(1/2))+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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))+2*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+s in(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)))+16/5*( 1/8*B+3/8*C)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x +1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2 *c)-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/ 2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c )*sin(1/2*d*x+1/2*c)^4+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1 /2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+8 *sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(2*sin(1/2*d*x+1/2*c)^2-1)^(...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.04 \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (35 \, A + 21 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (35 \, A + 21 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (5 \, A + 9 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (5 \, A + 9 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (21 \, {\left (15 \, A + 18 \, B + 14 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 21 \, B + 26 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 21 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 15 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d \cos \left (d x + c\right )^{3}} \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 /2),x, algorithm="fricas")
Output:
-2/105*(5*I*sqrt(2)*(35*A + 21*B + 13*C)*a^3*cos(d*x + c)^3*weierstrassPIn verse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*I*sqrt(2)*(35*A + 21*B + 1 3*C)*a^3*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d* x + c)) + 21*I*sqrt(2)*(5*A + 9*B + 7*C)*a^3*cos(d*x + c)^3*weierstrassZet a(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I *sqrt(2)*(5*A + 9*B + 7*C)*a^3*cos(d*x + c)^3*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (21*(15*A + 18*B + 14*C)*a^3*cos(d*x + c)^3 + 5*(7*A + 21*B + 26*C)*a^3*cos(d*x + c)^2 + 21* (B + 3*C)*a^3*cos(d*x + c) + 15*C*a^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d *cos(d*x + c)^3)
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)* *(1/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 /2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^3/s qrt(sec(d*x + c)), x)
\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 /2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^3/s qrt(sec(d*x + c)), x)
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(1/2),x)
Output:
int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(1/2), x)
\[ \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=a^{3} \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a +3 \left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) a +\left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}d x \right ) b +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) a +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) b +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) a +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) c \right ) \] Input:
int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2),x)
Output:
a**3*(int(sqrt(sec(c + d*x))/sec(c + d*x),x)*a + 3*int(sqrt(sec(c + d*x)), x)*a + int(sqrt(sec(c + d*x)),x)*b + int(sqrt(sec(c + d*x))*sec(c + d*x)** 4,x)*c + int(sqrt(sec(c + d*x))*sec(c + d*x)**3,x)*b + 3*int(sqrt(sec(c + d*x))*sec(c + d*x)**3,x)*c + int(sqrt(sec(c + d*x))*sec(c + d*x)**2,x)*a + 3*int(sqrt(sec(c + d*x))*sec(c + d*x)**2,x)*b + 3*int(sqrt(sec(c + d*x))* sec(c + d*x)**2,x)*c + 3*int(sqrt(sec(c + d*x))*sec(c + d*x),x)*a + 3*int( sqrt(sec(c + d*x))*sec(c + d*x),x)*b + int(sqrt(sec(c + d*x))*sec(c + d*x) ,x)*c)