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

Optimal. Leaf size=435 \[ \frac{2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left (3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 b^2 d}-\frac{2 \left (a^2-b^2\right ) \left (3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left (3 a^2 b^2 (33 A+17 C)+8 a^4 C+3 b^4 (319 A+247 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]

[Out]

(2*a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/
2, (2*b)/(a + b)])/(693*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(8*a^4*C + 15*b^4*(11*A + 9
*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(693*
b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Cos[
c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*a*(99*A*b^2 + 8*a^2*C + 67*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c +
d*x])/(693*b^2*d) + (2*(8*a^2*C + 9*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) - (
8*a*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c
+ d*x])/(11*b*d)

________________________________________________________________________________________

Rubi [A]  time = 0.879467, antiderivative size = 435, 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 = {3050, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left (3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 b^2 d}-\frac{2 \left (a^2-b^2\right ) \left (3 a^2 b^2 (33 A+19 C)+8 a^4 C+15 b^4 (11 A+9 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left (3 a^2 b^2 (33 A+17 C)+8 a^4 C+3 b^4 (319 A+247 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/
2, (2*b)/(a + b)])/(693*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(8*a^4*C + 15*b^4*(11*A + 9
*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(693*
b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Cos[
c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*a*(99*A*b^2 + 8*a^2*C + 67*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c +
d*x])/(693*b^2*d) + (2*(8*a^2*C + 9*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) - (
8*a*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c
+ d*x])/(11*b*d)

Rule 3050

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] + 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)) + (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])))

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 (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \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 C+\frac{1}{2} b (11 A+9 C) \cos (c+d x)-2 a C \cos ^2(c+d x)\right ) \, dx}{11 b}\\ &=-\frac{8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \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{5}{2} a b C+\frac{1}{4} \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{99 b^2}\\ &=\frac{2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \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{15}{8} b \left (33 A b^2-2 a^2 C+27 b^2 C\right )+\frac{5}{8} a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) \cos (c+d x)\right ) \, dx}{693 b^2}\\ &=\frac{2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac{2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \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{15}{8} a b \left (132 A b^2-\left (a^2-101 b^2\right ) C\right )+\frac{15}{16} \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \cos (c+d x)\right ) \, dx}{3465 b^2}\\ &=\frac{2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac{2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac{2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac{32 \int \frac{\frac{15}{32} b \left (2 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (297 A+221 C)\right )+\frac{15}{32} a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{10395 b^2}\\ &=\frac{2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac{2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac{2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \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 (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{693 b^3}+\frac{\left (a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right )\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{693 b^3}\\ &=\frac{2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac{2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac{2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}+\frac{\left (a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{693 b^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\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}{693 b^3 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left (a^2-b^2\right ) \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\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 )}{693 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac{2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac{2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d}\\ \end{align*}

Mathematica [A]  time = 1.64698, size = 328, normalized size = 0.75 \[ \frac{b (a+b \cos (c+d x)) \left (\left (12 a^2 b^2 (396 A+311 C)-64 a^4 C+6 b^4 (506 A+435 C)\right ) \sin (c+d x)+b \left (4 a \left (6 a^2 C+594 A b^2+619 b^2 C\right ) \sin (2 (c+d x))+b \left (\left (452 a^2 C+396 A b^2+513 b^2 C\right ) \sin (3 (c+d x))+7 b C (46 a \sin (4 (c+d x))+9 b \sin (5 (c+d x)))\right )\right )\right )+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b \left (3 a^2 b^3 (297 A+221 C)+2 a^4 b C+15 b^5 (11 A+9 C)\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+a \left (3 a^2 b^2 (33 A+17 C)+8 a^4 C+3 b^4 (319 A+247 C)\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{5544 b^3 d \sqrt{a+b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(2*a^4*b*C + 15*b^5*(11*A + 9*C) + 3*a^2*b^3*(297*A + 221*C))*Ellipt
icF[(c + d*x)/2, (2*b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*((a + b)*Ellip
ticE[(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])*((-64*a^
4*C + 12*a^2*b^2*(396*A + 311*C) + 6*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4*a*(594*A*b^2 + 6*a^2*C + 619*b^2
*C)*Sin[2*(c + d*x)] + b*((396*A*b^2 + 452*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7*b*C*(46*a*Sin[4*(c + d*x)]
+ 9*b*Sin[5*(c + d*x)])))))/(5544*b^3*d*Sqrt[a + b*Cos[c + d*x]])

________________________________________________________________________________________

Maple [B]  time = 0.434, size = 1791, normalized size = 4.1 \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)*(a+b*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2),x)

[Out]

-2/693*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4032*C*b^6*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+
1/2*c)^12+(-7168*C*a*b^5-10080*C*b^6)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(1584*A*b^6+4384*C*a^2*b^4+1433
6*C*a*b^5+11376*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-3168*A*a*b^5-2376*A*b^6-928*C*a^3*b^3-6576*C*
a^2*b^4-13232*C*a*b^5-6984*C*b^6)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(2376*A*a^2*b^4+3168*A*a*b^5+1848*A*
b^6-4*C*a^4*b^2+928*C*a^3*b^3+5024*C*a^2*b^4+6064*C*a*b^5+2772*C*b^6)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+
(-594*A*a^3*b^3-1188*A*a^2*b^4-1122*A*a*b^5-528*A*b^6+8*C*a^5*b+2*C*a^4*b^2-642*C*a^3*b^3-1416*C*a^2*b^4-1338*
C*a*b^5-558*C*b^6)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-99*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^4*b^2-66*A*a^2*(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^4+165*A*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)*Elli
pticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+99*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^4*b^2-99*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
+957*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^4-957*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^5-8*C*(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^6-49*C*(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^2-78*a^2*C*(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)*El
lipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^4+135*b^6*C*(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))+8*C*(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^6-8
*C*(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+51*C*(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^2-51*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*si
n(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+741*C*(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^4-741*C*(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)*Elli
pticE(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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{5} + 2 \, C a b \cos \left (d x + c\right )^{4} + 2 \, A a b \cos \left (d x + c\right )^{2} + A a^{2} \cos \left (d x + c\right ) +{\left (C a^{2} + A b^{2}\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)*(a+b*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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)*(a+b*cos(d*x+c))**(5/2)*(A+C*cos(d*x+c)**2),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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