3.631 \(\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=356 \[ \frac{2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}-\frac{2 a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (3 a^2 b^2 (21 A+11 C)+8 a^4 C+21 b^4 (9 A+7 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]

[Out]

(2*(8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2
*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(a^2 - b^2)*(63*A*b^2 + 8*a^2*C + 39*b^2*C
)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]
]) + (2*a*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(8*a^2*C + 7
*b^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) - (8*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin
[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)

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Rubi [A]  time = 0.652903, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}-\frac{2 a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (3 a^2 b^2 (21 A+11 C)+8 a^4 C+21 b^4 (9 A+7 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2
*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(a^2 - b^2)*(63*A*b^2 + 8*a^2*C + 39*b^2*C
)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]
]) + (2*a*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(8*a^2*C + 7
*b^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) - (8*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin
[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*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))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{2 \int (a+b \cos (c+d x))^{3/2} \left (a C+\frac{1}{2} b (9 A+7 C) \cos (c+d x)-2 a C \cos ^2(c+d x)\right ) \, dx}{9 b}\\ &=-\frac{8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{4 \int (a+b \cos (c+d x))^{3/2} \left (-\frac{3}{2} a b C+\frac{1}{4} \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x)\right ) \, dx}{63 b^2}\\ &=\frac{2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{8 \int \sqrt{a+b \cos (c+d x)} \left (\frac{3}{8} b \left (63 A b^2-2 a^2 C+49 b^2 C\right )+\frac{3}{8} a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \cos (c+d x)\right ) \, dx}{315 b^2}\\ &=\frac{2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac{2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{16 \int \frac{\frac{3}{8} a b \left (126 A b^2+\left (a^2+93 b^2\right ) C\right )+\frac{3}{16} \left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 C)\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{945 b^2}\\ &=\frac{2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac{2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}-\frac{\left (a \left (a^2-b^2\right ) \left (63 A b^2+8 a^2 C+39 b^2 C\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{315 b^3}+\frac{\left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 C)\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{315 b^3}\\ &=\frac{2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac{2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac{\left (\left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 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}{315 b^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (a \left (a^2-b^2\right ) \left (63 A b^2+8 a^2 C+39 b^2 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}{315 b^3 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 C)\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 a \left (a^2-b^2\right ) \left (63 A b^2+8 a^2 C+39 b^2 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 )}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac{2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac{8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac{2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}\\ \end{align*}

Mathematica [A]  time = 1.3193, size = 269, normalized size = 0.76 \[ \frac{b (a+b \cos (c+d x)) \left (b \left (2 \left (6 a^2 C+126 A b^2+133 b^2 C\right ) \sin (2 (c+d x))+5 b C (20 a \sin (3 (c+d x))+7 b \sin (4 (c+d x)))\right )-4 a \left (8 a^2 C-252 A b^2-201 b^2 C\right ) \sin (c+d x)\right )+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (2 a b^2 \left (C \left (a^2+93 b^2\right )+126 A b^2\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (3 a^2 b^2 (21 A+11 C)+8 a^4 C+21 b^4 (9 A+7 C)\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )}{1260 b^3 d \sqrt{a+b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(2*a*b^2*(126*A*b^2 + (a^2 + 93*b^2)*C)*EllipticF[(c + d*x)/2, (2*b)/(a
+ b)] + (8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*((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])*(-4*a*(-252*A*b^2 + 8*a^2*C - 201*b^2*C
)*Sin[c + d*x] + b*(2*(126*A*b^2 + 6*a^2*C + 133*b^2*C)*Sin[2*(c + d*x)] + 5*b*C*(20*a*Sin[3*(c + d*x)] + 7*b*
Sin[4*(c + d*x)]))))/(1260*b^3*d*Sqrt[a + b*Cos[c + d*x]])

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Maple [B]  time = 0.378, size = 1527, 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)*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x)

[Out]

-2/315*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*C*b^5*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^10+(1360*C*a*b^4+2240*C*b^5)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-504*A*b^5-424*C*a^2*b^3-2040*C*
a*b^4-2072*C*b^5)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(756*A*a*b^4+504*A*b^5-4*C*a^3*b^2+424*C*a^2*b^3+156
8*C*a*b^4+952*C*b^5)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-252*A*a^2*b^3-378*A*a*b^4-126*A*b^5+8*C*a^4*b+2
*C*a^3*b^2-282*C*a^2*b^3-444*C*a*b^4-168*C*b^5)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+63*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^2-63*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^3+189*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^4-189*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^5-63*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^3*b^2+63*a*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))*b^4+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^5-8*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^4*b+33*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^3*b^2-33*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^3+147*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*b^4-147*
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))*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^5-31*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^3*b^2+39*a*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))*b^4)/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 (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{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))^(3/2)*(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)^(3/2)*cos(d*x + c), x)

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

[Out]

integral((C*b*cos(d*x + c)^4 + C*a*cos(d*x + c)^3 + A*b*cos(d*x + c)^2 + A*a*cos(d*x + c))*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)*(a+b*cos(d*x+c))**(3/2)*(A+C*cos(d*x+c)**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} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{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))^(3/2)*(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)^(3/2)*cos(d*x + c), x)