Integrand size = 33, antiderivative size = 295 \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 b^2 (9 A b+13 a B) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (27 a A b+22 a^2 B+7 b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 b B (b+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \] Output:
2/15*(15*A*a^3+27*A*a*b^2+27*B*a^2*b+7*B*b^3)*cos(d*x+c)^(1/2)*EllipticE(s in(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/d+2/21*(21*A*a^2*b+5*A*b^3+7*B *a^3+15*B*a*b^2)*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*s ec(d*x+c)^(1/2)/d+2/63*b^2*(9*A*b+13*B*a)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/ 45*b*(27*A*a*b+22*B*a^2+7*B*b^2)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/21*(21*A* a^2*b+5*A*b^3+7*B*a^3+15*B*a*b^2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/9*b*B*(b +a*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(7/2)
Time = 11.30 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {\sec (c+d x)} \left (168 \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+120 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (7 b \left (108 a A b+108 a^2 B+43 b^2 B\right ) \cos (c+d x)+5 \left (252 a^2 A b+78 A b^3+84 a^3 B+234 a b^2 B+18 b^2 (A b+3 a B) \cos (2 (c+d x))+7 b^3 B \cos (3 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1260 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]] ,x]
Output:
(Sqrt[Sec[c + d*x]]*(168*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sq rt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*(21*a^2*A*b + 5*A*b^3 + 7 *a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*b*( 108*a*A*b + 108*a^2*B + 43*b^2*B)*Cos[c + d*x] + 5*(252*a^2*A*b + 78*A*b^3 + 84*a^3*B + 234*a*b^2*B + 18*b^2*(A*b + 3*a*B)*Cos[2*(c + d*x)] + 7*b^3* B*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)
Time = 2.00 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.95, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3439, 3042, 4513, 27, 3042, 4562, 27, 3042, 4535, 3042, 4256, 3042, 4258, 3042, 3120, 4533, 3042, 4258, 3042, 3119}
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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3439 |
| \(\displaystyle \int \frac {(a \sec (c+d x)+b)^3 (A \sec (c+d x)+B)}{\sec ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )^3 \left (A \csc \left (c+d x+\frac {\pi }{2}\right )+B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int -\frac {(b+a \sec (c+d x)) \left (3 a (3 a A+b B) \sec ^2(c+d x)+\left (9 B a^2+18 A b a+7 b^2 B\right ) \sec (c+d x)+b (9 A b+13 a B)\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(b+a \sec (c+d x)) \left (3 a (3 a A+b B) \sec ^2(c+d x)+\left (9 B a^2+18 A b a+7 b^2 B\right ) \sec (c+d x)+b (9 A b+13 a B)\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (3 a (3 a A+b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (9 B a^2+18 A b a+7 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+b (9 A b+13 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2}{7} \int -\frac {21 a^2 (3 a A+b B) \sec ^2(c+d x)+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \sec (c+d x)+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {21 a^2 (3 a A+b B) \sec ^2(c+d x)+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \sec (c+d x)+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {21 a^2 (3 a A+b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 a^2 (3 a A+b B) \sec ^2(c+d x)+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 a^2 (3 a A+b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 a^2 (3 a A+b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 a^2 (3 a A+b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 a^2 (3 a A+b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 a^2 (3 a A+b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 a^2 (3 a A+b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {21}{5} \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {42 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+9 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]
Output:
(2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2 *b^2*(9*A*b + 13*a*B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((42*(15*a^ 3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (14*b*(27*a*A*b + 22*a^2*B + 7*b^2 *B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + 9*(21*a^2*A*b + 5*A*b^3 + 7*a ^3*B + 15*a*b^2*B)*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[S ec[c + d*x]])/(3*d) + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])))/7)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[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[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* (x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim p[g^(m + n) Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(d + c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] + Sim p[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[ a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] & & LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
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 _)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si mp[1/(d*n) Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(744\) vs. \(2(270)=540\).
Time = 24.90 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-1120 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} b^{3}+\left (720 A \,b^{3}+2160 B a \,b^{2}+2240 b^{3} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1512 A a \,b^{2}-1080 A \,b^{3}-1512 B \,a^{2} b -3240 B a \,b^{2}-2072 b^{3} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1260 A \,a^{2} b +1512 A a \,b^{2}+840 A \,b^{3}+420 a^{3} B +1512 B \,a^{2} b +2520 B a \,b^{2}+952 b^{3} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-630 A \,a^{2} b -378 A a \,b^{2}-240 A \,b^{3}-210 a^{3} B -378 B \,a^{2} b -720 B a \,b^{2}-168 b^{3} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+315 A \,a^{2} b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+75 A \,b^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-315 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}-567 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}+105 a^{3} B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+225 B a \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-567 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b -147 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{3}\right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(745\) |
| parts | \(\text {Expression too large to display}\) | \(971\) |
Input:
int((a+cos(d*x+c)*b)^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x,method=_RETURNV ERBOSE)
Output:
-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*B*co s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^3+(720*A*b^3+2160*B*a*b^2+2240*B* b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-1512*A*a*b^2-1080*A*b^3-151 2*B*a^2*b-3240*B*a*b^2-2072*B*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c) +(1260*A*a^2*b+1512*A*a*b^2+840*A*b^3+420*B*a^3+1512*B*a^2*b+2520*B*a*b^2+ 952*B*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-630*A*a^2*b-378*A*a*b ^2-240*A*b^3-210*B*a^3-378*B*a^2*b-720*B*a*b^2-168*B*b^3)*sin(1/2*d*x+1/2* c)^2*cos(1/2*d*x+1/2*c)+315*A*a^2*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/ 2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+75*A*b^3*(si n(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1 /2*d*x+1/2*c),2^(1/2))-315*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(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))*a^3-567*A*(sin(1/2* d*x+1/2*c)^2)^(1/2)*(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))*a*b^2+105*a^3*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d *x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+225*B*a*b^2*(si n(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1 /2*d*x+1/2*c),2^(1/2))-567*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(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))*a^2*b-147*B*(sin(1/ 2*d*x+1/2*c)^2)^(1/2)*(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))*b^3)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=-\frac {15 \, \sqrt {2} {\left (7 i \, B a^{3} + 21 i \, A a^{2} b + 15 i \, B a b^{2} + 5 i \, A b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-7 i \, B a^{3} - 21 i \, A a^{2} b - 15 i \, B a b^{2} - 5 i \, A b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, A a^{3} - 27 i \, B a^{2} b - 27 i \, A a b^{2} - 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, A a^{3} + 27 i \, B a^{2} b + 27 i \, A a b^{2} + 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, B b^{3} \cos \left (d x + c\right )^{4} + 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, B a^{2} b + 27 \, A a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (7 \, B a^{3} + 21 \, A a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x, algorith m="fricas")
Output:
-1/315*(15*sqrt(2)*(7*I*B*a^3 + 21*I*A*a^2*b + 15*I*B*a*b^2 + 5*I*A*b^3)*w eierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-7* I*B*a^3 - 21*I*A*a^2*b - 15*I*B*a*b^2 - 5*I*A*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(-15*I*A*a^3 - 27*I*B*a^2* b - 27*I*A*a*b^2 - 7*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(- 4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(15*I*A*a^3 + 27*I*B*a^ 2*b + 27*I*A*a*b^2 + 7*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse (-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(35*B*b^3*cos(d*x + c)^4 + 45* (3*B*a*b^2 + A*b^3)*cos(d*x + c)^3 + 7*(27*B*a^2*b + 27*A*a*b^2 + 7*B*b^3) *cos(d*x + c)^2 + 15*(7*B*a^3 + 21*A*a^2*b + 15*B*a*b^2 + 5*A*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
\[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {\left (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{3}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx \] Input:
integrate((a+b*cos(d*x+c))**3*(A+B*cos(d*x+c))/sec(d*x+c)**(1/2),x)
Output:
Integral((A + B*cos(c + d*x))*(a + b*cos(c + d*x))**3/sqrt(sec(c + d*x)), x)
\[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/sqrt(sec(d*x + c)), x)
\[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/sqrt(sec(d*x + c)), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3)/(1/cos(c + d*x))^(1/2),x )
Output:
int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3)/(1/cos(c + d*x))^(1/2), x)
\[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a^{4}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )}d x \right ) a^{3} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )}d x \right ) b^{4}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) a \,b^{3}+6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) a^{2} b^{2} \] Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x)
Output:
int(sqrt(sec(c + d*x))/sec(c + d*x),x)*a**4 + 4*int((sqrt(sec(c + d*x))*co s(c + d*x))/sec(c + d*x),x)*a**3*b + int((sqrt(sec(c + d*x))*cos(c + d*x)* *4)/sec(c + d*x),x)*b**4 + 4*int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec( c + d*x),x)*a*b**3 + 6*int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c + d* x),x)*a**2*b**2