Integrand size = 35, antiderivative size = 294 \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {4 a b (9 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {8 a b C \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \] Output:
4/15*a*b*(9*A+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+2/231*(11*a^2*(7*A+5*C)+5*b^2*(11*A+9*C))*cos(d*x+c)^(1 /2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+8/99*a*b*C*s in(d*x+c)/d/sec(d*x+c)^(7/2)+2/77*(4*a^2*C+b^2*(11*A+9*C))*sin(d*x+c)/d/se c(d*x+c)^(5/2)+2/11*C*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(5/2)+4/4 5*a*b*(9*A+7*C)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/231*(11*a^2*(7*A+5*C)+5*b^ 2*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)
Time = 5.48 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (14784 a b (9 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+480 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (308 a b (36 A+43 C) \cos (c+d x)+5 \left (132 a^2 (14 A+13 C)+3 b^2 (572 A+531 C)+36 \left (11 A b^2+11 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+308 a b C \cos (3 (c+d x))+63 b^2 C \cos (4 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{55440 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/ 2),x]
Output:
(Sqrt[Sec[c + d*x]]*(14784*a*b*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(308*a*b*(36*A + 43*C)*Cos[c + d*x] + 5*(132*a^2*(14*A + 13*C) + 3*b^2*(572*A + 531*C) + 36*(11*A*b^2 + 11*a^2* C + 16*b^2*C)*Cos[2*(c + d*x)] + 308*a*b*C*Cos[3*(c + d*x)] + 63*b^2*C*Cos [4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)
Time = 1.42 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.89, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {3042, 4709, 3042, 3529, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos (c+d x)^2\right )}{\sec (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (C \cos ^2(c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{11} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (4 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+a (11 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (4 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+a (11 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (4 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (11 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (9 (11 A+5 C) a^2+22 b (9 A+7 C) \cos (c+d x) a+9 \left (4 C a^2+b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )dx+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (9 (11 A+5 C) a^2+22 b (9 A+7 C) \cos (c+d x) a+9 \left (4 C a^2+b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )dx+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (9 (11 A+5 C) a^2+22 b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+9 \left (4 C a^2+b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (9 \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right )+154 a b (9 A+7 C) \cos (c+d x)\right )dx+\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \cos ^{\frac {3}{2}}(c+d x) \left (9 \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right )+154 a b (9 A+7 C) \cos (c+d x)\right )dx+\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (9 \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right )+154 a b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+154 a b (9 A+7 C) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )+\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+154 a b (9 A+7 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )+\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+154 a b (9 A+7 C) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+154 a b (9 A+7 C) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+154 a b (9 A+7 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {18 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {1}{7} \left (9 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+154 a b (9 A+7 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )\right )+\frac {8 a b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )\) |
Input:
Int[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*C*Cos[c + d*x]^(5/2)*(a + b*Cos[ c + d*x])^2*Sin[c + d*x])/(11*d) + ((8*a*b*C*Cos[c + d*x]^(7/2)*Sin[c + d* x])/(9*d) + ((18*(4*a^2*C + b^2*(11*A + 9*C))*Cos[c + d*x]^(5/2)*Sin[c + d *x])/(7*d) + (9*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)) + 154*a* b*(9*A + 7*C)*((6*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*Cos[c + d*x]^(3/2) *Sin[c + d*x])/(5*d)))/7)/9)/11)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(648\) vs. \(2(265)=530\).
Time = 39.45 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.21
| 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 (20160 C \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-24640 a b C -50400 C \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7920 A \,b^{2}+7920 a^{2} C +49280 a b C +56880 C \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-11088 a A b -11880 A \,b^{2}-11880 a^{2} C -45584 a b C -34920 C \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (4620 A \,a^{2}+11088 a A b +9240 A \,b^{2}+9240 a^{2} C +20944 a b C +13860 C \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2310 A \,a^{2}-2772 a A b -2640 A \,b^{2}-2640 a^{2} C -3696 a b C -2790 C \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1155 A \,a^{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 )+825 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 )-4158 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 +825 a^{2} C \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 )+675 C \,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 )-3234 C \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 \right )}{3465 \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}\) | \(649\) |
| parts | \(\text {Expression too large to display}\) | \(1039\) |
Input:
int((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x,method=_RETUR NVERBOSE)
Output:
-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*C*b ^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-24640*C*a*b-50400*C*b^2)*sin (1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^2+7920*C*a^2+49280*C*a*b+5 6880*C*b^2)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-11088*A*a*b-11880*A* b^2-11880*C*a^2-45584*C*a*b-34920*C*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+ 1/2*c)+(4620*A*a^2+11088*A*a*b+9240*A*b^2+9240*C*a^2+20944*C*a*b+13860*C*b ^2)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2310*A*a^2-2772*A*a*b-2640*A *b^2-2640*C*a^2-3696*C*a*b-2790*C*b^2)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/ 2*c)+1155*A*a^2*(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))+825*A*b^2*(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))-4158*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)*E llipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b+825*a^2*C*(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 ))+675*C*b^2*(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))-3234*C*(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*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c) /(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {462 i \, \sqrt {2} {\left (9 \, A + 7 \, C\right )} a b {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 462 i \, \sqrt {2} {\left (9 \, A + 7 \, C\right )} a b {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 15 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{2} + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{2} - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + \frac {2 \, {\left (315 \, C b^{2} \cos \left (d x + c\right )^{5} + 770 \, C a b \cos \left (d x + c\right )^{4} + 154 \, {\left (9 \, A + 7 \, C\right )} a b \cos \left (d x + c\right )^{2} + 45 \, {\left (11 \, C a^{2} + {\left (11 \, A + 9 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (11 \, {\left (7 \, A + 5 \, C\right )} a^{2} + 5 \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \] Input:
integrate((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algori thm="fricas")
Output:
1/3465*(462*I*sqrt(2)*(9*A + 7*C)*a*b*weierstrassZeta(-4, 0, weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 462*I*sqrt(2)*(9*A + 7*C)* a*b*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin (d*x + c))) - 15*sqrt(2)*(11*I*(7*A + 5*C)*a^2 + 5*I*(11*A + 9*C)*b^2)*wei erstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*sqrt(2)*(-11*I *(7*A + 5*C)*a^2 - 5*I*(11*A + 9*C)*b^2)*weierstrassPInverse(-4, 0, cos(d* x + c) - I*sin(d*x + c)) + 2*(315*C*b^2*cos(d*x + c)^5 + 770*C*a*b*cos(d*x + c)^4 + 154*(9*A + 7*C)*a*b*cos(d*x + c)^2 + 45*(11*C*a^2 + (11*A + 9*C) *b^2)*cos(d*x + c)^3 + 15*(11*(7*A + 5*C)*a^2 + 5*(11*A + 9*C)*b^2)*cos(d* x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{2}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*cos(d*x+c))**2*(A+C*cos(d*x+c)**2)/sec(d*x+c)**(3/2),x)
Output:
Integral((A + C*cos(c + d*x)**2)*(a + b*cos(c + d*x))**2/sec(c + d*x)**(3/ 2), x)
\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^2/sec(d*x + c)^(3/2) , x)
\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^2/sec(d*x + c)^(3/2) , x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2)/(1/cos(c + d*x))^(3/2) ,x)
Output:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2)/(1/cos(c + d*x))^(3/2) , x)
\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3}+2 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )^{2}}d x \right ) b^{2} c +2 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{2}}d x \right ) a b c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{2}}d x \right ) a \,b^{2} \] Input:
int((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x)
Output:
int(sqrt(sec(c + d*x))/sec(c + d*x)**2,x)*a**3 + 2*int((sqrt(sec(c + d*x)) *cos(c + d*x))/sec(c + d*x)**2,x)*a**2*b + int((sqrt(sec(c + d*x))*cos(c + d*x)**4)/sec(c + d*x)**2,x)*b**2*c + 2*int((sqrt(sec(c + d*x))*cos(c + d* x)**3)/sec(c + d*x)**2,x)*a*b*c + int((sqrt(sec(c + d*x))*cos(c + d*x)**2) /sec(c + d*x)**2,x)*a**2*c + int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec( c + d*x)**2,x)*a*b**2