Integrand size = 35, antiderivative size = 206 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=-\frac {2 \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a b (3 A+C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 d}-\frac {2 b^2 (5 A-C) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 a b (3 A-C) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{d} \] Output:
-2/5*(5*a^2*(A-C)-b^2*(5*A+3*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+4/3*a*b*(3*A+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-2/5*b^2*(5*A-C)*sin(d*x+ c)/d/sec(d*x+c)^(3/2)-4/3*a*b*(3*A-C)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2*A*(a +b*cos(d*x+c))^2*sec(d*x+c)^(1/2)*sin(d*x+c)/d
Time = 3.76 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (12 \left (-5 a^2 (A-C)+b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 a b (3 A+C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {2 \left (30 a^2 A+3 b^2 C+20 a b C \cos (c+d x)+3 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )}{30 d} \] 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[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(12*(-5*a^2*(A - C) + b^2*(5*A + 3* C))*EllipticE[(c + d*x)/2, 2] + 40*a*b*(3*A + C)*EllipticF[(c + d*x)/2, 2] + (2*(30*a^2*A + 3*b^2*C + 20*a*b*C*Cos[c + d*x] + 3*b^2*C*Cos[2*(c + d*x )])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(30*d)
Time = 1.24 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.94, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {3042, 4709, 3042, 3527, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 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.
| \(\displaystyle \int \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{3/2} (a+b \cos (c+d x))^2 \left (A+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^2 \left (C \cos ^2(c+d x)+A\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (2 \int \frac {(a+b \cos (c+d x)) \left (-b (5 A-C) \cos ^2(c+d x)-a (A-C) \cos (c+d x)+4 A b\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\int \frac {(a+b \cos (c+d x)) \left (-b (5 A-C) \cos ^2(c+d x)-a (A-C) \cos (c+d x)+4 A b\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b (5 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a (A-C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 A b\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{5} \int \frac {-10 a b (3 A-C) \cos ^2(c+d x)-\left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x)+20 a A b}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \int \frac {-10 a b (3 A-C) \cos ^2(c+d x)-\left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x)+20 a A b}{\sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \int \frac {-10 a b (3 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (b^2 (5 A+3 C)-5 a^2 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+20 a A b}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {10 a b (3 A+C)-3 \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx-\frac {20 a b (3 A-C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {10 a b (3 A+C)-3 \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {20 a b (3 A-C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {10 a b (3 A+C)-3 \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {20 a b (3 A-C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \left (\frac {1}{3} \left (10 a b (3 A+C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-3 \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )-\frac {20 a b (3 A-C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \left (\frac {1}{3} \left (10 a b (3 A+C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-3 \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )-\frac {20 a b (3 A-C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \left (\frac {1}{3} \left (10 a b (3 A+C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-\frac {20 a b (3 A-C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {20 a b (3 A+C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {6 \left (5 a^2 (A-C)-b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-\frac {20 a b (3 A-C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt {\cos (c+d x)}}-\frac {2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\) |
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*b^2*(5*A - C)*Cos[c + d*x]^(3/2 )*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[ Cos[c + d*x]]) + (((-6*(5*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(c + d* x)/2, 2])/d + (20*a*b*(3*A + C)*EllipticF[(c + d*x)/2, 2])/d)/3 - (20*a*b* (3*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d))/5)
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[((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^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
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. \(480\) vs. \(2(187)=374\).
Time = 8.51 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.33
| method | result | size |
| default | \(\frac {\frac {16 C \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5}-\frac {16 C b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{3}-\frac {16 C \,b^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}+4 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2}-4 a A 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 )-2 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^{2}+2 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 ) b^{2}+\frac {8 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a b}{3}+\frac {4 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{2}}{5}-\frac {4 a b 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 )}{3}+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 {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}+\frac {6 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 ) b^{2}}{5}}{\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}\) | \(481\) |
| parts | \(\text {Expression too large to display}\) | \(851\) |
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:
2/15*(24*C*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-40*C*b*sin(1/2*d*x+ 1/2*c)^4*cos(1/2*d*x+1/2*c)*a-24*C*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 2*c)+30*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^2-30*a*A*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))-15*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^2+15*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))*b^2+20*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a*b+6*C*cos(1/2*d* x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b^2-10*a*b*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))+15*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(co s(1/2*d*x+1/2*c),2^(1/2))*a^2+9*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))*b^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 = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.03 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {-10 i \, \sqrt {2} {\left (3 \, A + C\right )} a b {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 10 i \, \sqrt {2} {\left (3 \, A + C\right )} a b {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 3 \, \sqrt {2} {\left (5 i \, {\left (A - C\right )} a^{2} - i \, {\left (5 \, A + 3 \, C\right )} b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 \, \sqrt {2} {\left (-5 i \, {\left (A - C\right )} a^{2} + i \, {\left (5 \, A + 3 \, C\right )} b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + \frac {2 \, {\left (3 \, C b^{2} \cos \left (d x + c\right )^{2} + 10 \, C a b \cos \left (d x + c\right ) + 15 \, A a^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{15 \, d} \] 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:
1/15*(-10*I*sqrt(2)*(3*A + C)*a*b*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 10*I*sqrt(2)*(3*A + C)*a*b*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 3*sqrt(2)*(5*I*(A - C)*a^2 - I*(5*A + 3*C )*b^2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I* sin(d*x + c))) - 3*sqrt(2)*(-5*I*(A - C)*a^2 + I*(5*A + 3*C)*b^2)*weierstr assZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(3*C*b^2*cos(d*x + c)^2 + 10*C*a*b*cos(d*x + c) + 15*A*a^2)*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**2*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(3/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \] 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)
\[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \] 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. \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2 \,d x \] Input:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(3/2)*(a + b*cos(c + d*x))^2,x )
Output:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(3/2)*(a + b*cos(c + d*x))^2, x)
\[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right ) a^{2} c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )d x \right ) a \,b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(3/2),x)
Output:
2*int(sqrt(sec(c + d*x))*cos(c + d*x)*sec(c + d*x),x)*a**2*b + int(sqrt(se c(c + d*x))*cos(c + d*x)**4*sec(c + d*x),x)*b**2*c + 2*int(sqrt(sec(c + d* x))*cos(c + d*x)**3*sec(c + d*x),x)*a*b*c + int(sqrt(sec(c + d*x))*cos(c + d*x)**2*sec(c + d*x),x)*a**2*c + int(sqrt(sec(c + d*x))*cos(c + d*x)**2*s ec(c + d*x),x)*a*b**2 + int(sqrt(sec(c + d*x))*sec(c + d*x),x)*a**3