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

Optimal. Leaf size=305 \[ \frac{2 \left (77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (26 a^2 B+33 a A b+9 b^2 B\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left (27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left (77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]

[Out]

(2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*a^3*A + 165*a*A*b^
2 + 165*a^2*b*B + 45*b^3*B)*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45
*b^3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*Cos[c + d*
x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(33*a*A*b + 26*a^2*B + 9*b^2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) +
(2*b^2*(11*A*b + 15*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d
*x])^2*Sin[c + d*x])/(11*d)

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Rubi [A]  time = 0.5446, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2990, 3033, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 \left (77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (26 a^2 B+33 a A b+9 b^2 B\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left (27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left (77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*a^3*A + 165*a*A*b^
2 + 165*a^2*b*B + 45*b^3*B)*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45
*b^3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*Cos[c + d*
x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(33*a*A*b + 26*a^2*B + 9*b^2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) +
(2*b^2*(11*A*b + 15*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d
*x])^2*Sin[c + d*x])/(11*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 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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

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 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx &=\frac{2 b B \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{2}{11} \int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\frac{1}{2} a (11 a A+5 b B)+\frac{1}{2} \left (9 b^2 B+11 a (2 A b+a B)\right ) \cos (c+d x)+\frac{1}{2} b (11 A b+15 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b^2 (11 A b+15 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{4}{99} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{4} a^2 (11 a A+5 b B)+\frac{11}{4} \left (27 a^2 A b+7 A b^3+9 a^3 B+21 a b^2 B\right ) \cos (c+d x)+\frac{9}{4} b \left (33 a A b+26 a^2 B+9 b^2 B\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b \left (33 a A b+26 a^2 B+9 b^2 B\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b^2 (11 A b+15 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{8}{693} \int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{9}{8} \left (77 a^3 A+165 a A b^2+165 a^2 b B+45 b^3 B\right )+\frac{77}{8} \left (27 a^2 A b+7 A b^3+9 a^3 B+21 a b^2 B\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b \left (33 a A b+26 a^2 B+9 b^2 B\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b^2 (11 A b+15 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{1}{9} \left (27 a^2 A b+7 A b^3+9 a^3 B+21 a b^2 B\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{77} \left (77 a^3 A+165 a A b^2+165 a^2 b B+45 b^3 B\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (77 a^3 A+165 a A b^2+165 a^2 b B+45 b^3 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (27 a^2 A b+7 A b^3+9 a^3 B+21 a b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b \left (33 a A b+26 a^2 B+9 b^2 B\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b^2 (11 A b+15 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{1}{15} \left (27 a^2 A b+7 A b^3+9 a^3 B+21 a b^2 B\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (77 a^3 A+165 a A b^2+165 a^2 b B+45 b^3 B\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (27 a^2 A b+7 A b^3+9 a^3 B+21 a b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (77 a^3 A+165 a A b^2+165 a^2 b B+45 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (77 a^3 A+165 a A b^2+165 a^2 b B+45 b^3 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (27 a^2 A b+7 A b^3+9 a^3 B+21 a b^2 B\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 b \left (33 a A b+26 a^2 B+9 b^2 B\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b^2 (11 A b+15 a B) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b B \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 1.90529, size = 235, normalized size = 0.77 \[ \frac{240 \left (77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3696 \left (27 a^2 A b+9 a^3 B+21 a b^2 B+7 A b^3\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (180 b \left (33 a^2 B+33 a A b+16 b^2 B\right ) \cos (2 (c+d x))+154 \left (108 a^2 A b+36 a^3 B+129 a b^2 B+43 A b^3\right ) \cos (c+d x)+15 \left (616 a^3 A+1716 a^2 b B+1716 a A b^2+21 b^3 B \cos (4 (c+d x))+531 b^3 B\right )+770 b^2 (3 a B+A b) \cos (3 (c+d x))\right )}{27720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3696*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*EllipticE[(c + d*x)/2, 2] + 240*(77*a^3*A + 165*a*A*b^2 +
165*a^2*b*B + 45*b^3*B)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(108*a^2*A*b + 43*A*b^3 + 36*a^3
*B + 129*a*b^2*B)*Cos[c + d*x] + 180*b*(33*a*A*b + 33*a^2*B + 16*b^2*B)*Cos[2*(c + d*x)] + 770*b^2*(A*b + 3*a*
B)*Cos[3*(c + d*x)] + 15*(616*a^3*A + 1716*a*A*b^2 + 1716*a^2*b*B + 531*b^3*B + 21*b^3*B*Cos[4*(c + d*x)]))*Si
n[c + d*x])/(27720*d)

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Maple [B]  time = 3.362, size = 825, normalized size = 2.7 \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)^(3/2)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x)

[Out]

-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*B*b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12+(-12320*A*b^3-36960*B*a*b^2-50400*B*b^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(23760*A*a*b^2+24640
*A*b^3+23760*B*a^2*b+73920*B*a*b^2+56880*B*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-16632*A*a^2*b-35640*
A*a*b^2-22792*A*b^3-5544*B*a^3-35640*B*a^2*b-68376*B*a*b^2-34920*B*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c
)+(4620*A*a^3+16632*A*a^2*b+27720*A*a*b^2+10472*A*b^3+5544*B*a^3+27720*B*a^2*b+31416*B*a*b^2+13860*B*b^3)*sin(
1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2310*A*a^3-4158*A*a^2*b-7920*A*a*b^2-1848*A*b^3-1386*B*a^3-7920*B*a^2*b-
5544*B*a*b^2-2790*B*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1155*A*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si
n(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+2475*A*a*b^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-6237*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si
n(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-1617*A*(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^3+2475*a^2*b*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))+675*B*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))-2079*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3-4851*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2)/(-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 (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3*cos(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 (B b^{3} \cos \left (d x + c\right )^{5} + A a^{3} \cos \left (d x + c\right ) +{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

[Out]

Timed out