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

Optimal. Leaf size=264 \[ \frac{2 \left (9 a^2 B+14 a b C+7 b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (9 a^2 B+14 a b C+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left (11 a (a C+2 b B)+9 b^2 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (11 a (a C+2 b B)+9 b^2 C\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left (11 a (a C+2 b B)+9 b^2 C\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d} \]

[Out]

(2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(9*b^2*C + 11*a*(2*b*B + a*C))*Ellip
ticF[(c + d*x)/2, 2])/(231*d) + (10*(9*b^2*C + 11*a*(2*b*B + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) +
(2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(9*b^2*C + 11*a*(2*b*B + a*C))*
Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*b*B + 13*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*
b*C*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)

________________________________________________________________________________________

Rubi [A]  time = 0.461829, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3029, 2990, 3023, 2748, 2635, 2639, 2641} \[ \frac{2 \left (9 a^2 B+14 a b C+7 b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (9 a^2 B+14 a b C+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left (11 a (a C+2 b B)+9 b^2 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 \left (11 a (a C+2 b B)+9 b^2 C\right ) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left (11 a (a C+2 b B)+9 b^2 C\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(9*b^2*C + 11*a*(2*b*B + a*C))*Ellip
ticF[(c + d*x)/2, 2])/(231*d) + (10*(9*b^2*C + 11*a*(2*b*B + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) +
(2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(9*b^2*C + 11*a*(2*b*B + a*C))*
Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*b*B + 13*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*
b*C*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)

Rule 3029

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] :> Dist[1/b^2, Int[(a + b*Sin[e + f*x])
^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]

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 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 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]

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]

Rubi steps

\begin{align*} \int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \, dx\\ &=\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{2}{11} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{1}{2} a (11 a B+7 b C)+\frac{1}{2} \left (9 b^2 C+11 a (2 b B+a C)\right ) \cos (c+d x)+\frac{1}{2} b (11 b B+13 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (11 b B+13 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{4}{99} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{11}{4} \left (9 a^2 B+7 b^2 B+14 a b C\right )+\frac{9}{4} \left (9 b^2 C+11 a (2 b B+a C)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b (11 b B+13 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{1}{9} \left (9 a^2 B+7 b^2 B+14 a b C\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx+\frac{1}{11} \left (9 b^2 C+11 a (2 b B+a C)\right ) \int \cos ^{\frac{7}{2}}(c+d x) \, dx\\ &=\frac{2 \left (9 a^2 B+7 b^2 B+14 a b C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 \left (9 b^2 C+11 a (2 b B+a C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b (11 b B+13 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{1}{15} \left (9 a^2 B+7 b^2 B+14 a b C\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{77} \left (5 \left (9 b^2 C+11 a (2 b B+a C)\right )\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (9 a^2 B+7 b^2 B+14 a b C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 \left (9 b^2 C+11 a (2 b B+a C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (9 a^2 B+7 b^2 B+14 a b C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 \left (9 b^2 C+11 a (2 b B+a C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b (11 b B+13 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac{1}{231} \left (5 \left (9 b^2 C+11 a (2 b B+a C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (9 a^2 B+7 b^2 B+14 a b C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 \left (9 b^2 C+11 a (2 b B+a C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{10 \left (9 b^2 C+11 a (2 b B+a C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 \left (9 a^2 B+7 b^2 B+14 a b C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 \left (9 b^2 C+11 a (2 b B+a C)\right ) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 b (11 b B+13 a C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}\\ \end{align*}

Mathematica [A]  time = 1.67936, size = 196, normalized size = 0.74 \[ \frac{1200 \left (11 a^2 C+22 a b B+9 b^2 C\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+3696 \left (9 a^2 B+14 a b C+7 b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (154 \left (36 a^2 B+86 a b C+43 b^2 B\right ) \cos (c+d x)+180 \left (11 a^2 C+22 a b B+16 b^2 C\right ) \cos (2 (c+d x))+15 \left (572 a^2 C+1144 a b B+21 b^2 C \cos (4 (c+d x))+531 b^2 C\right )+770 b (2 a C+b B) \cos (3 (c+d x))\right )}{27720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3696*(9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(c + d*x)/2, 2] + 1200*(22*a*b*B + 11*a^2*C + 9*b^2*C)*Elliptic
F[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(36*a^2*B + 43*b^2*B + 86*a*b*C)*Cos[c + d*x] + 180*(22*a*b*B +
11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 770*b*(b*B + 2*a*C)*Cos[3*(c + d*x)] + 15*(1144*a*b*B + 572*a^2*C + 53
1*b^2*C + 21*b^2*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)

________________________________________________________________________________________

Maple [B]  time = 0.647, size = 666, normalized size = 2.5 \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*(B*cos(d*x+c)+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+(-12320*B*b^2-24640*C*a*b-50400*C*b^2)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(15840*B*a*b+24640*B*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)+(-5544*B*a^2-23760*B*a*b-22792*
B*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)+(5544*B*a^2+18480*B*a*b+104
72*B*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)+(-1386*B*a^2-5280*B*a*b-1
848*B*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)+1650*a*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))-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^2-1617*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))*b^2+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))-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)/(-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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\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*(B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*(b*cos(d*x + c) + a)^2*cos(d*x + c)^(3/2), 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} + B a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (C a^{2} + 2 \, B a b\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*(B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

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))^2*(B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

Timed out