3.311 \(\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx\)

Optimal. Leaf size=462 \[ -\frac{2 \left (-8 a^2 B+22 a A b-81 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left (110 a^2 A b-40 a^3 B-335 a b^2 B-539 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left (110 a^3 A b-285 a^2 b^2 B-40 a^4 B-1254 a A b^3-675 b^4 B\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left (a^2-b^2\right ) \left (110 a^3 A b-285 a^2 b^2 B-40 a^4 B-1254 a A b^3-675 b^4 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-3069 a^2 A b^3+110 a^4 A b-255 a^3 b^2 B-40 a^5 B-3705 a b^4 B-1617 A b^5\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]

[Out]

(-2*(110*a^4*A*b - 3069*a^2*A*b^3 - 1617*A*b^5 - 40*a^5*B - 255*a^3*b^2*B - 3705*a*b^4*B)*Sqrt[a + b*Cos[c + d
*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(
110*a^3*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Elliptic
F[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(110*a^3*A*b - 1254*a*A*b^3 - 40*a^4
*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*A*b - 539*A*
b^3 - 40*a^3*B - 335*a*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) - (2*(22*a*A*b - 8*a^2*B -
81*b^2*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(7/
2)*Sin[c + d*x])/(99*b^2*d) + (2*B*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)

________________________________________________________________________________________

Rubi [A]  time = 0.931401, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {2990, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (-8 a^2 B+22 a A b-81 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left (110 a^2 A b-40 a^3 B-335 a b^2 B-539 A b^3\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left (110 a^3 A b-285 a^2 b^2 B-40 a^4 B-1254 a A b^3-675 b^4 B\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left (a^2-b^2\right ) \left (110 a^3 A b-285 a^2 b^2 B-40 a^4 B-1254 a A b^3-675 b^4 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (-3069 a^2 A b^3+110 a^4 A b-255 a^3 b^2 B-40 a^5 B-3705 a b^4 B-1617 A b^5\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]

[Out]

(-2*(110*a^4*A*b - 3069*a^2*A*b^3 - 1617*A*b^5 - 40*a^5*B - 255*a^3*b^2*B - 3705*a*b^4*B)*Sqrt[a + b*Cos[c + d
*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(
110*a^3*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Elliptic
F[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(110*a^3*A*b - 1254*a*A*b^3 - 40*a^4
*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*A*b - 539*A*
b^3 - 40*a^3*B - 335*a*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) - (2*(22*a*A*b - 8*a^2*B -
81*b^2*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(7/
2)*Sin[c + d*x])/(99*b^2*d) + (2*B*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)

Rule 2990

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x
])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*(m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c -
b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m,
1] &&  !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

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 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx &=\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac{2 \int (a+b \cos (c+d x))^{5/2} \left (a B+\frac{9}{2} b B \cos (c+d x)+\frac{1}{2} (11 A b-4 a B) \cos ^2(c+d x)\right ) \, dx}{11 b}\\ &=\frac{2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac{4 \int (a+b \cos (c+d x))^{5/2} \left (\frac{1}{4} b (77 A b-10 a B)-\frac{1}{4} \left (22 a A b-8 a^2 B-81 b^2 B\right ) \cos (c+d x)\right ) \, dx}{99 b^2}\\ &=-\frac{2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac{8 \int (a+b \cos (c+d x))^{3/2} \left (\frac{3}{8} b \left (143 a A b-10 a^2 B+135 b^2 B\right )-\frac{1}{8} \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) \cos (c+d x)\right ) \, dx}{693 b^2}\\ &=-\frac{2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac{16 \int \sqrt{a+b \cos (c+d x)} \left (\frac{3}{16} b \left (605 a^2 A b+539 A b^3-10 a^3 B+1010 a b^2 B\right )-\frac{3}{16} \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \cos (c+d x)\right ) \, dx}{3465 b^2}\\ &=-\frac{2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac{32 \int \frac{\frac{3}{32} b \left (1705 a^3 A b+2871 a A b^3+10 a^4 B+3315 a^2 b^2 B+675 b^4 B\right )-\frac{3}{32} \left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{10395 b^2}\\ &=-\frac{2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac{\left (\left (a^2-b^2\right ) \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{3465 b^3}-\frac{\left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{3465 b^3}\\ &=-\frac{2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}-\frac{\left (\left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{3465 b^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (a^2-b^2\right ) \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}}} \, dx}{3465 b^3 \sqrt{a+b \cos (c+d x)}}\\ &=-\frac{2 \left (110 a^4 A b-3069 a^2 A b^3-1617 A b^5-40 a^5 B-255 a^3 b^2 B-3705 a b^4 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (a^2-b^2\right ) \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (110 a^3 A b-1254 a A b^3-40 a^4 B-285 a^2 b^2 B-675 b^4 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (110 a^2 A b-539 A b^3-40 a^3 B-335 a b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^2 d}-\frac{2 \left (22 a A b-8 a^2 B-81 b^2 B\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}+\frac{2 (11 A b-4 a B) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 B \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}\\ \end{align*}

Mathematica [A]  time = 1.98594, size = 357, normalized size = 0.77 \[ \frac{b (a+b \cos (c+d x)) \left (\left (880 a^3 A b+18660 a^2 b^2 B-320 a^4 B+32868 a A b^3+13050 b^4 B\right ) \sin (c+d x)+b \left (4 \left (1650 a^2 A b+30 a^3 B+3095 a b^2 B+1463 A b^3\right ) \sin (2 (c+d x))+5 b \left (\left (452 a^2 B+836 a A b+513 b^2 B\right ) \sin (3 (c+d x))+7 b ((46 a B+22 A b) \sin (4 (c+d x))+9 b B \sin (5 (c+d x)))\right )\right )\right )+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b^2 \left (1705 a^3 A b+3315 a^2 b^2 B+10 a^4 B+2871 a A b^3+675 b^4 B\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (3069 a^2 A b^3-110 a^4 A b+255 a^3 b^2 B+40 a^5 B+3705 a b^4 B+1617 A b^5\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{27720 b^3 d \sqrt{a+b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]

[Out]

(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(1705*a^3*A*b + 2871*a*A*b^3 + 10*a^4*B + 3315*a^2*b^2*B + 675*b^4
*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b
^2*B + 3705*a*b^4*B)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])
) + b*(a + b*Cos[c + d*x])*((880*a^3*A*b + 32868*a*A*b^3 - 320*a^4*B + 18660*a^2*b^2*B + 13050*b^4*B)*Sin[c +
d*x] + b*(4*(1650*a^2*A*b + 1463*A*b^3 + 30*a^3*B + 3095*a*b^2*B)*Sin[2*(c + d*x)] + 5*b*((836*a*A*b + 452*a^2
*B + 513*b^2*B)*Sin[3*(c + d*x)] + 7*b*((22*A*b + 46*a*B)*Sin[4*(c + d*x)] + 9*b*B*Sin[5*(c + d*x)])))))/(2772
0*b^3*d*Sqrt[a + b*Cos[c + d*x]])

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Maple [B]  time = 4.483, size = 1983, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x)

[Out]

-2/3465*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-245*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b
/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2+20160*
B*b^6*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+40*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c
)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6-40*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*
(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6-1617*
A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c
),(-2*b/(a-b))^(1/2))*b^6+675*B*b^6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))
^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+(-12320*A*b^6-35840*B*a*b^5-50400*B*b^6)*sin(1/2*d*x+1
/2*c)^10*cos(1/2*d*x+1/2*c)+(22880*A*a*b^5+24640*A*b^6+21920*B*a^2*b^4+71680*B*a*b^5+56880*B*b^6)*sin(1/2*d*x+
1/2*c)^8*cos(1/2*d*x+1/2*c)+(-14960*A*a^2*b^4-34320*A*a*b^5-22792*A*b^6-4640*B*a^3*b^3-32880*B*a^2*b^4-66160*B
*a*b^5-34920*B*b^6)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(3520*A*a^3*b^3+14960*A*a^2*b^4+26488*A*a*b^5+1047
2*A*b^6-20*B*a^4*b^2+4640*B*a^3*b^3+25120*B*a^2*b^4+30320*B*a*b^5+13860*B*b^6)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*
x+1/2*c)+(-110*A*a^4*b^2-1760*A*a^3*b^3-7326*A*a^2*b^4-7524*A*a*b^5-1848*A*b^6+40*B*a^5*b+10*B*a^4*b^2-3210*B*
a^3*b^3-7080*B*a^2*b^4-6690*B*a*b^5-2790*B*b^6)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1254*A*a*b^5*(sin(1/2*
d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-
b))^(1/2))+110*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(co
s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2-3069*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c
)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^4+1617*A*(sin(1/2*d*x+1/2*c)^2)^
(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b
^5-390*a^2*b^4*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(co
s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+255*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)
/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2+3069*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2
*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^3-110*
A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c
),(-2*b/(a-b))^(1/2))*a^5*b-1364*A*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-
b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3-255*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*
sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^3+110*A*(sin(1/
2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(
a-b))^(1/2))*a^5*b+3705*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ell
ipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^4-40*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*
x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b-3705*B*(sin(1/2*d*x+1/2*c
)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2)
)*a*b^5)/b^3/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1
/2*c)^2*b+a+b)^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*cos(d*x + c)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{5} + A a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt{b \cos \left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

integral((B*b^2*cos(d*x + c)^5 + A*a^2*cos(d*x + c)^2 + (2*B*a*b + A*b^2)*cos(d*x + c)^4 + (B*a^2 + 2*A*a*b)*c
os(d*x + c)^3)*sqrt(b*cos(d*x + c) + a), 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(cos(d*x+c)**2*(a+b*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/2)*cos(d*x + c)^2, x)