3.1465 \(\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=345 \[ \frac{2 \sin (c+d x) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right )}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{15 d}+\frac{2 b (4 a C+11 b B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac{5}{2}}(c+d x)} \]

[Out]

(2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/
(15*d) + (2*(110*a*b*B + 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]
*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(11*b*B + 4*a*C)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(11*A*b^2 +
22*a*b*B + 4*a^2*C + 9*b^2*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*
x])/(11*d*Sec[c + d*x]^(5/2)) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^
(3/2)) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.749051, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4221, 3049, 3033, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 \sin (c+d x) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right )}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{15 d}+\frac{2 b (4 a C+11 b B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]

[Out]

(2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/
(15*d) + (2*(110*a*b*B + 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]
*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(11*b*B + 4*a*C)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(11*A*b^2 +
22*a*b*B + 4*a^2*C + 9*b^2*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*
x])/(11*d*Sec[c + d*x]^(5/2)) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^
(3/2)) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (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] + Dist[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)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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])))

Rule 3033

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
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[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]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[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]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2635

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] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{1}{11} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\frac{1}{2} a (11 A+5 C)+\frac{1}{2} (11 A b+11 a B+9 b C) \cos (c+d x)+\frac{1}{2} (11 b B+4 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (11 b B+4 a C) \sin (c+d x)}{99 d \sec ^{\frac{7}{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{1}{99} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{4} a^2 (11 A+5 C)+\frac{11}{4} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)+\frac{9}{4} \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (11 b B+4 a C) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 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{1}{693} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{8} \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )+\frac{77}{8} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b (11 b B+4 a C) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 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{1}{9} \left (\left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{77} \left (\left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 b (11 b B+4 a C) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 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{2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (110 a b B+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)}}+\frac{1}{15} \left (\left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (\left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\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 (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{2 b (11 b B+4 a C) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 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{2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (110 a b B+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)}}\\ \end{align*}

Mathematica [A]  time = 2.07236, size = 259, normalized size = 0.75 \[ \frac{\sqrt{\sec (c+d x)} \left (2 \sin (2 (c+d x)) \left (154 \cos (c+d x) \left (36 a^2 B+72 a A b+86 a b C+43 b^2 B\right )+5 \left (36 \cos (2 (c+d x)) \left (11 a^2 C+22 a b B+11 A b^2+16 b^2 C\right )+132 a^2 (14 A+13 C)+154 b (2 a C+b B) \cos (3 (c+d x))+3432 a b B+3 b^2 (572 A+531 C)+63 b^2 C \cos (4 (c+d x))\right )\right )+480 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )+7392 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (9 a^2 B+2 a b (9 A+7 C)+7 b^2 B\right )\right )}{55440 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]

[Out]

(Sqrt[Sec[c + d*x]]*(7392*(9*a^2*B + 7*b^2*B + 2*a*b*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]
+ 480*(110*a*b*B + 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*
(154*(72*a*A*b + 36*a^2*B + 43*b^2*B + 86*a*b*C)*Cos[c + d*x] + 5*(3432*a*b*B + 132*a^2*(14*A + 13*C) + 3*b^2*
(572*A + 531*C) + 36*(11*A*b^2 + 22*a*b*B + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b*(b*B + 2*a*C)*Cos[3*
(c + d*x)] + 63*b^2*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)

________________________________________________________________________________________

Maple [B]  time = 1.498, size = 863, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x)

[Out]

-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*b^2*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12+(-12320*B*b^2-24640*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+15840*B*a*
b+24640*B*b^2+7920*C*a^2+49280*C*a*b+56880*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^2-5544*B*a^2-23760*B*a*b-22792*B*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+5544*B*a^2+18480*B*a*b+10472*B*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^2-1386*B*a^2-5280*B*a*b-1848*B*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b+1650*a*b*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))-2079*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^2-1617*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^2+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*b^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))-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+si
n(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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2/sec(d*x + c)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C b^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac{3}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

integral((C*b^2*cos(d*x + c)^4 + (2*C*a*b + B*b^2)*cos(d*x + c)^3 + A*a^2 + (C*a^2 + 2*B*a*b + A*b^2)*cos(d*x
+ c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c))/sec(d*x + c)^(3/2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**2*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/sec(d*x+c)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2/sec(d*x + c)^(3/2), x)