3.1479 \(\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx\)

Optimal. Leaf size=419 \[ \frac{2 b^2 \sin (c+d x) \left (a^2 (-(315 A-123 C))+162 a b B+7 b^2 (9 A+7 C)\right )}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left (a^3 (-(126 A-62 C))+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right )}{15 d}-\frac{2 b \sin (c+d x) (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^4}{d} \]

[Out]

(2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*El
lipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C
) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*(162
*a*b*B - a^2*(315*A - 123*C) + 7*b^2*(9*A + 7*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*b*(117*a^2*b*B
+ 15*b^3*B - a^3*(126*A - 62*C) + 12*a*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) - (2*b*(21*a*
A - 3*b*B - 5*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(9*A - C)*(a + b*Cos[
c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sqrt[Sec[c + d*x]]*Sin[c + d*
x])/d

________________________________________________________________________________________

Rubi [A]  time = 1.44445, antiderivative size = 419, 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, 3047, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac{2 b^2 \sin (c+d x) \left (a^2 (-(315 A-123 C))+162 a b B+7 b^2 (9 A+7 C)\right )}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left (a^3 (-(126 A-62 C))+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right )}{15 d}-\frac{2 b \sin (c+d x) (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^4}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]

[Out]

(2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*El
lipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C
) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*(162
*a*b*B - a^2*(315*A - 123*C) + 7*b^2*(9*A + 7*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*b*(117*a^2*b*B
+ 15*b^3*B - a^3*(126*A - 62*C) + 12*a*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) - (2*b*(21*a*
A - 3*b*B - 5*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(9*A - C)*(a + b*Cos[
c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sqrt[Sec[c + d*x]]*Sin[c + d*
x])/d

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 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + 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] + Dist[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*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(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, B, 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]

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 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 (a+b \cos (c+d x))^4 \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 \frac{(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+b \cos (c+d x))^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^3 \left (\frac{1}{2} (8 A b+a B)+\frac{1}{2} (b B-a (A-C)) \cos (c+d x)-\frac{1}{2} b (9 A-C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \cos (c+d x))^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{9} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^2 \left (\frac{1}{4} a (63 A b+9 a B+b C)+\frac{1}{4} \left (18 a b B-9 a^2 (A-C)+b^2 (9 A+7 C)\right ) \cos (c+d x)-\frac{3}{4} b (21 a A-3 b B-5 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b (21 a A-3 b B-5 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \cos (c+d x))^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{63} \left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{8} a \left (378 a A b+63 a^2 B+9 b^2 B+22 a b C\right )+\frac{1}{8} \left (189 a^2 b B+45 b^3 B-63 a^3 (A-C)+a b^2 (189 A+131 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b (21 a A-3 b B-5 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \cos (c+d x))^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{315} \left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{16} a^2 \left (378 a A b+63 a^2 B+9 b^2 B+22 a b C\right )+\frac{21}{16} \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \cos (c+d x)+\frac{15}{16} b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}-\frac{2 b (21 a A-3 b B-5 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \cos (c+d x))^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{945} \left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{45}{32} \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right )+\frac{63}{32} \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}-\frac{2 b (21 a A-3 b B-5 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \cos (c+d x))^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{21} \left (\left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left (\left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 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 (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}-\frac{2 b (21 a A-3 b B-5 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+b \cos (c+d x))^4 \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 2.79265, size = 327, normalized size = 0.78 \[ \frac{\sqrt{\sec (c+d x)} \left (2 \sin (c+d x) \left (30 b \cos (c+d x) \left (168 a^2 b B+112 a^3 C+4 a b^2 (28 A+29 C)+29 b^3 B\right )+84 b^2 \cos (2 (c+d x)) \left (18 a^2 C+12 a b B+3 A b^2+4 b^2 C\right )+2520 a^4 A+1512 a^2 b^2 C+1008 a b^3 B+360 a b^3 C \cos (3 (c+d x))+252 A b^4+90 b^4 B \cos (3 (c+d x))+35 b^4 C \cos (4 (c+d x))+301 b^4 C\right )+240 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )-336 \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-18 a^2 b^2 (5 A+3 C)+15 a^4 (A-C)-60 a^3 b B-36 a b^3 B-b^4 (9 A+7 C)\right )\right )}{2520 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[c + d*x])^4*(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]]*(-336*(-60*a^3*b*B - 36*a*b^3*B + 15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C
))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C)
+ 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(2520*a^4*A + 252*A*b^4 + 1008*a*b^3*B
+ 1512*a^2*b^2*C + 301*b^4*C + 30*b*(168*a^2*b*B + 29*b^3*B + 112*a^3*C + 4*a*b^2*(28*A + 29*C))*Cos[c + d*x]
+ 84*b^2*(3*A*b^2 + 12*a*b*B + 18*a^2*C + 4*b^2*C)*Cos[2*(c + d*x)] + 90*b^4*B*Cos[3*(c + d*x)] + 360*a*b^3*C
*Cos[3*(c + d*x)] + 35*b^4*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(2520*d)

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Maple [B]  time = 2.28, size = 1652, normalized size = 3.9 \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))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(3/2),x)

[Out]

-2/315*(-1120*C*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)
^10+80*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b^3*(9*B*b+36*C*a+28*C*b)*sin(1/2*d*x+1/2*c)^8*cos
(1/2*d*x+1/2*c)-8*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b^2*(63*A*b^2+252*B*a*b+135*B*b^2+378*C
*a^2+540*C*a*b+259*C*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+56*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*
c)^2)^(1/2)*b*(30*A*a*b^2+9*A*b^3+45*B*a^2*b+36*B*a*b^2+15*B*b^3+30*C*a^3+54*C*a^2*b+60*C*a*b^2+17*C*b^3)*sin(
1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-6*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(105*A*a^4+140*A*a*
b^3+21*A*b^4+210*B*a^2*b^2+84*B*a*b^3+40*B*b^4+140*C*a^3*b+126*C*a^2*b^2+160*C*a*b^3+28*C*b^4)*sin(1/2*d*x+1/2
*c)^2*cos(1/2*d*x+1/2*c)+1260*A*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+420*a*A*b^3*(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))*(-2*sin(1/2*d*x+1/2*c)^4+si
n(1/2*d*x+1/2*c)^2)^(1/2)+315*A*(-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)^(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^4-1890*A*(-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)^(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^2-189*A*(-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)^(
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^4+315*a^4*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1
/2*d*x+1/2*c)^2)^(1/2)+630*a^2*b^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(c
os(1/2*d*x+1/2*c),2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+75*b^4*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/
2*d*x+1/2*c)^2)^(1/2)-1260*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)^(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*b-756*B*(-2*sin(1/2*d*x+1/2*c)^4+s
in(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x
+1/2*c),2^(1/2))*a*b^3+420*a^3*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+300*C*a*b^3*(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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1
/2*d*x+1/2*c)^2)^(1/2)-315*C*(-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)^(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^4-1134*C*(-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)^(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^2-147*C*(-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)^(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^4)/(-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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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 )}^{4} \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))^4*(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)^4*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 ({\left (C b^{4} \cos \left (d x + c\right )^{6} +{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} +{\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\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))^4*(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^4*cos(d*x + c)^6 + (4*C*a*b^3 + B*b^4)*cos(d*x + c)^5 + A*a^4 + (6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4
)*cos(d*x + c)^4 + 2*(2*C*a^3*b + 3*B*a^2*b^2 + 2*A*a*b^3)*cos(d*x + c)^3 + (C*a^4 + 4*B*a^3*b + 6*A*a^2*b^2)*
cos(d*x + c)^2 + (B*a^4 + 4*A*a^3*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))**4*(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{\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 )}^{4} \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))^4*(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)^4*sec(d*x + c)^(3/2), x)