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

Optimal. Leaf size=170 \[ \frac{10 (a C+b B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 (9 a B+7 b C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 (9 a B+7 b C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d} \]

[Out]

(2*(9*a*B + 7*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(b*B + a*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (10
*(b*B + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a*B + 7*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45
*d) + (2*(b*B + a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)

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Rubi [A]  time = 0.267145, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {3029, 2968, 3023, 2748, 2635, 2639, 2641} \[ \frac{10 (a C+b B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 (9 a B+7 b C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 (9 a B+7 b C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(9*a*B + 7*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*(b*B + a*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (10
*(b*B + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a*B + 7*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45
*d) + (2*(b*B + a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*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 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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)) \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)) (B+C \cos (c+d x)) \, dx\\ &=\int \cos ^{\frac{5}{2}}(c+d x) \left (a B+(b B+a C) \cos (c+d x)+b C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2}{9} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{1}{2} (9 a B+7 b C)+\frac{9}{2} (b B+a C) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+(b B+a C) \int \cos ^{\frac{7}{2}}(c+d x) \, dx+\frac{1}{9} (9 a B+7 b C) \int \cos ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 (9 a B+7 b C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 (b B+a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{7} (5 (b B+a C)) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} (9 a B+7 b C) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 (9 a B+7 b C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 (b B+a C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 (9 a B+7 b C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 (b B+a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{21} (5 (b B+a C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (9 a B+7 b C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 (b B+a C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{10 (b B+a C) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 (9 a B+7 b C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 (b B+a C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{2 b C \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 1.34683, size = 125, normalized size = 0.74 \[ \frac{300 (a C+b B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+84 (9 a B+7 b C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} (7 (36 a B+43 b C) \cos (c+d x)+5 (18 (a C+b B) \cos (2 (c+d x))+78 a C+78 b B+7 b C \cos (3 (c+d x))))}{630 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(84*(9*a*B + 7*b*C)*EllipticE[(c + d*x)/2, 2] + 300*(b*B + a*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]
*(7*(36*a*B + 43*b*C)*Cos[c + d*x] + 5*(78*b*B + 78*a*C + 18*(b*B + a*C)*Cos[2*(c + d*x)] + 7*b*C*Cos[3*(c + d
*x)]))*Sin[c + d*x])/(630*d)

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Maple [B]  time = 0.59, size = 451, 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))*(B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*C*b*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^10+(720*B*b+720*C*a+2240*C*b)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-504*B*a-1080*B*b-1080*C*a-2072*C*b)*
sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(504*B*a+840*B*b+840*C*a+952*C*b)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2
*c)+(-126*B*a-240*B*b-240*C*a-168*C*b)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+75*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))-189*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+75*a*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))-147*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*s
in(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*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} + B \cos \left (d x + c\right )\right )}{\left (b \cos \left (d x + c\right ) + a\right )} \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))*(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)*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 \cos \left (d x + c\right )^{4} + B a \cos \left (d x + c\right )^{2} +{\left (C a + B 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))*(B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

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

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*(b*cos(d*x + c) + a)*cos(d*x + c)^(3/2), x)