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

Optimal. Leaf size=254 \[ \frac{2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 a b (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a b (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]

[Out]

(4*a*b*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[
(c + d*x)/2, 2])/(231*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*
d) + (4*a*b*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(4*a^2*C + b^2*(11*A + 9*C))*Cos[c + d*x]
^(5/2)*Sin[c + d*x])/(77*d) + (8*a*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*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 = 0.479593, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3050, 3033, 3023, 2748, 2635, 2641, 2639} \[ \frac{2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 a b (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a b (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*b*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[
(c + d*x)/2, 2])/(231*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*
d) + (4*a*b*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(4*a^2*C + b^2*(11*A + 9*C))*Cos[c + d*x]
^(5/2)*Sin[c + d*x])/(77*d) + (8*a*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(5/2)*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(11*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 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))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 C \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+5 C)+\frac{1}{2} b (11 A+9 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{8 a b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 C \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+5 C)+\frac{11}{2} a b (9 A+7 C) \cos (c+d x)+\frac{9}{4} \left (4 a^2 C+b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{8 a b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 C \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 (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )+\frac{77}{4} a b (9 A+7 C) \cos (c+d x)\right ) \, dx\\ &=\frac{2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{8 a b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{1}{9} (2 a b (9 A+7 C)) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{77} \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{4 a b (9 A+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{8 a b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 C \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac{1}{15} (2 a b (9 A+7 C)) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a b (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{4 a b (9 A+7 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{8 a b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 C \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 = 1.50415, size = 187, normalized size = 0.74 \[ \frac{240 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (5 \left (36 \left (11 a^2 C+11 A b^2+16 b^2 C\right ) \cos (2 (c+d x))+132 a^2 (14 A+13 C)+308 a b C \cos (3 (c+d x))+3 b^2 (572 A+531 C)+63 b^2 C \cos (4 (c+d x))\right )+308 a b (36 A+43 C) \cos (c+d x)\right )+7392 a b (9 A+7 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{27720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7392*a*b*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 240*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c +
d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(308*a*b*(36*A + 43*C)*Cos[c + d*x] + 5*(132*a^2*(14*A + 13*C) + 3*b^2*(572
*A + 531*C) + 36*(11*A*b^2 + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 308*a*b*C*Cos[3*(c + d*x)] + 63*b^2*C*Cos
[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)

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Maple [B]  time = 0.502, size = 649, normalized size = 2.6 \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))^2*(A+C*cos(d*x+c)^2),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^2*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^12+(-24640*C*a*b-50400*C*b^2)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^2+7920*C*a^2+49280*C*a*b
+56880*C*b^2)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-11088*A*a*b-11880*A*b^2-11880*C*a^2-45584*C*a*b-34920*
C*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(4620*A*a^2+11088*A*a*b+9240*A*b^2+9240*C*a^2+20944*C*a*b+13860
*C*b^2)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2310*A*a^2-2772*A*a*b-2640*A*b^2-2640*C*a^2-3696*C*a*b-2790*
C*b^2)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-4158*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))*a*b+1155*A*a^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))+825*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))-3234*C*(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+825*a^2*C*(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^2*C*(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)))/(-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 (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \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))^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)^2*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 (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{\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))^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(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))**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 )}^{2} \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))^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)^2*cos(d*x + c)^(3/2), x)