3.296 \(\int \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx\)

Optimal. Leaf size=386 \[ -\frac{2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}+\frac{2 \left (24 a^2 A b-16 a^3 B-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left (a^2-b^2\right ) \left (24 a^2 A b-16 a^3 B-36 a b^2 B+75 A b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (24 a^3 A b-24 a^2 b^2 B-16 a^4 B+57 a A b^3+147 b^4 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d} \]

[Out]

(2*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x
)/2, (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(24*a^2*A*b + 75*A*b^3 -
16*a^3*B - 36*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^4*d*Sq
rt[a + b*Cos[c + d*x]]) + (2*(24*a^2*A*b + 75*A*b^3 - 16*a^3*B - 36*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c +
d*x])/(315*b^3*d) - (2*(36*a*A*b - 24*a^2*B - 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^3*d) +
(2*(3*A*b - 2*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*B*Cos[c + d*x]^2*(a
+ b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*b*d)

________________________________________________________________________________________

Rubi [A]  time = 0.785479, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2990, 3049, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (-24 a^2 B+36 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}+\frac{2 \left (24 a^2 A b-16 a^3 B-36 a b^2 B+75 A b^3\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left (a^2-b^2\right ) \left (24 a^2 A b-16 a^3 B-36 a b^2 B+75 A b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (24 a^3 A b-24 a^2 b^2 B-16 a^4 B+57 a A b^3+147 b^4 B\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 B \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]

[Out]

(2*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x
)/2, (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(24*a^2*A*b + 75*A*b^3 -
16*a^3*B - 36*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^4*d*Sq
rt[a + b*Cos[c + d*x]]) + (2*(24*a^2*A*b + 75*A*b^3 - 16*a^3*B - 36*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c +
d*x])/(315*b^3*d) - (2*(36*a*A*b - 24*a^2*B - 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^3*d) +
(2*(3*A*b - 2*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*B*Cos[c + d*x]^2*(a
+ b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*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 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 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 ^3(c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx &=\frac{2 B \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d}+\frac{2 \int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left (2 a B+\frac{7}{2} b B \cos (c+d x)+\frac{3}{2} (3 A b-2 a B) \cos ^2(c+d x)\right ) \, dx}{9 b}\\ &=\frac{2 (3 A b-2 a B) \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 B \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d}+\frac{4 \int \sqrt{a+b \cos (c+d x)} \left (\frac{3}{2} a (3 A b-2 a B)+\frac{1}{4} b (45 A b-2 a B) \cos (c+d x)-\frac{1}{4} \left (36 a A b-24 a^2 B-49 b^2 B\right ) \cos ^2(c+d x)\right ) \, dx}{63 b^2}\\ &=-\frac{2 \left (36 a A b-24 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^3 d}+\frac{2 (3 A b-2 a B) \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 B \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d}+\frac{8 \int \sqrt{a+b \cos (c+d x)} \left (-\frac{3}{8} b \left (6 a A b-4 a^2 B-49 b^2 B\right )+\frac{3}{8} \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 B\right ) \cos (c+d x)\right ) \, dx}{315 b^3}\\ &=\frac{2 \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{2 \left (36 a A b-24 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^3 d}+\frac{2 (3 A b-2 a B) \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 B \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d}+\frac{16 \int \frac{\frac{3}{16} b \left (6 a^2 A b+75 A b^3-4 a^3 B+111 a b^2 B\right )+\frac{3}{16} \left (24 a^3 A b+57 a A b^3-16 a^4 B-24 a^2 b^2 B+147 b^4 B\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{945 b^3}\\ &=\frac{2 \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{2 \left (36 a A b-24 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^3 d}+\frac{2 (3 A b-2 a B) \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 B \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d}-\frac{\left (\left (a^2-b^2\right ) \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 B\right )\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{315 b^4}+\frac{\left (24 a^3 A b+57 a A b^3-16 a^4 B-24 a^2 b^2 B+147 b^4 B\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{315 b^4}\\ &=\frac{2 \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{2 \left (36 a A b-24 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^3 d}+\frac{2 (3 A b-2 a B) \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 B \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d}+\frac{\left (\left (24 a^3 A b+57 a A b^3-16 a^4 B-24 a^2 b^2 B+147 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}{315 b^4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left (\left (a^2-b^2\right ) \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 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}{315 b^4 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (24 a^3 A b+57 a A b^3-16 a^4 B-24 a^2 b^2 B+147 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 )}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left (a^2-b^2\right ) \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 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 )}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (24 a^2 A b+75 A b^3-16 a^3 B-36 a b^2 B\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac{2 \left (36 a A b-24 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^3 d}+\frac{2 (3 A b-2 a B) \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 B \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{9 b d}\\ \end{align*}

Mathematica [A]  time = 1.47992, size = 292, normalized size = 0.76 \[ \frac{8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (b^2 \left (6 a^2 A b-4 a^3 B+111 a b^2 B+75 A b^3\right ) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )+\left (24 a^3 A b-24 a^2 b^2 B-16 a^4 B+57 a A b^3+147 b^4 B\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )\right )-b (a+b \cos (c+d x)) \left (-2 \left (-48 a^2 A b+32 a^3 B+57 a b^2 B+345 A b^3\right ) \sin (c+d x)-b \left (\left (-24 a^2 B+36 a A b+266 b^2 B\right ) \sin (2 (c+d x))+5 b (2 (a B+9 A b) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right )\right )}{1260 b^4 d \sqrt{a+b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]

[Out]

(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(6*a^2*A*b + 75*A*b^3 - 4*a^3*B + 111*a*b^2*B)*EllipticF[(c + d*x)/
2, (2*b)/(a + b)] + (24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*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])*(-2*(-48*a^2*A*b + 3
45*A*b^3 + 32*a^3*B + 57*a*b^2*B)*Sin[c + d*x] - b*((36*a*A*b - 24*a^2*B + 266*b^2*B)*Sin[2*(c + d*x)] + 5*b*(
2*(9*A*b + a*B)*Sin[3*(c + d*x)] + 7*b*B*Sin[4*(c + d*x)]))))/(1260*b^4*d*Sqrt[a + b*Cos[c + d*x]])

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Maple [B]  time = 4.869, size = 1635, normalized size = 4.2 \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)^3*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)),x)

[Out]

-2/315*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*B*b^5*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^10+(720*A*b^5+640*B*a*b^4+2240*B*b^5)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-432*A*a*b^4-1080*A*b^5
+8*B*a^2*b^3-960*B*a*b^4-2072*B*b^5)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(-12*A*a^2*b^3+432*A*a*b^4+840*A*
b^5+8*B*a^3*b^2-8*B*a^2*b^3+728*B*a*b^4+952*B*b^5)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(24*A*a^3*b^2+6*A*a
^2*b^3-258*A*a*b^4-240*A*b^5-16*B*a^4*b-4*B*a^3*b^2-24*B*a^2*b^3-204*B*a*b^4-168*B*b^5)*sin(1/2*d*x+1/2*c)^2*c
os(1/2*d*x+1/2*c)-24*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)*Ellipt
icF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b-51*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^2*b^3+75*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))
+24*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-24*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+57*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^3-57*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+16*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)*Ellip
ticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5+20*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^3*b^2-36*a*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))*b^
4-16*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+16*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-24*B*(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^2+24*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^2*b^3+147*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)*Ellip
ticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^4-147*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))*b^5)/b^4/(-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 )} \sqrt{b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*cos(d*x + c)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B \cos \left (d x + c\right )^{4} + A \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)^3*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

integral((B*cos(d*x + c)^4 + A*cos(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)**3*(a+b*cos(d*x+c))**(1/2)*(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 )} \sqrt{b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*cos(d*x + c)^3, x)