3.431 \(\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)
Optimal. Leaf size=211 \[ \frac{10 a (11 A+11 B+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 a (9 A+7 (B+C)) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (11 A+11 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 a (9 A+7 (B+C)) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (11 A+11 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d} \]
[Out]
(2*a*(9*A + 7*(B + C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*a*(11*A + 11*B + 9*C)*EllipticF[(c + d*x)/2, 2]
)/(231*d) + (10*a*(11*A + 11*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*(B + C))*Cos[c
+ d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(11*A + 11*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*(B
+ C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*a*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)
________________________________________________________________________________________
Rubi [A] time = 0.287296, antiderivative size = 211, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used =
{3033, 3023, 2748, 2635, 2639, 2641} \[ \frac{10 a (11 A+11 B+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{2 a (9 A+7 (B+C)) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 a (11 A+11 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 a (9 A+7 (B+C)) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (11 A+11 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
[Out]
(2*a*(9*A + 7*(B + C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*a*(11*A + 11*B + 9*C)*EllipticF[(c + d*x)/2, 2]
)/(231*d) + (10*a*(11*A + 11*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*(B + C))*Cos[c
+ d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(11*A + 11*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*(B
+ C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*a*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)
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 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{5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{2 a C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{2}{11} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{11 a A}{2}+\frac{1}{2} a (11 A+11 B+9 C) \cos (c+d x)+\frac{11}{2} a (B+C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 a (B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 a C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{4}{99} \int \cos ^{\frac{5}{2}}(c+d x) \left (\frac{11}{4} a (9 A+7 (B+C))+\frac{9}{4} a (11 A+11 B+9 C) \cos (c+d x)\right ) \, dx\\ &=\frac{2 a (B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 a C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{11} (a (11 A+11 B+9 C)) \int \cos ^{\frac{7}{2}}(c+d x) \, dx+\frac{1}{9} (a (9 A+7 (B+C))) \int \cos ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 a (9 A+7 (B+C)) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a (11 A+11 B+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 a (B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 a C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{77} (5 a (11 A+11 B+9 C)) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} (a (9 A+7 (B+C))) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 a (9 A+7 (B+C)) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 a (11 A+11 B+9 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 a (9 A+7 (B+C)) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a (11 A+11 B+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 a (B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 a C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{231} (5 a (11 A+11 B+9 C)) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a (9 A+7 (B+C)) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{10 a (11 A+11 B+9 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{10 a (11 A+11 B+9 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{231 d}+\frac{2 a (9 A+7 (B+C)) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac{2 a (11 A+11 B+9 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac{2 a (B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{2 a C \cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [C] time = 6.43207, size = 1344, normalized size = 6.37 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
[Out]
a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-((9*A + 7*B + 7*C)*Cot[c])/(15*d) + ((506*A +
506*B + 435*C)*Cos[d*x]*Sin[c])/(1848*d) + ((18*A + 19*B + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((44*A + 44*B
+ 57*C)*Cos[3*d*x]*Sin[3*c])/(1232*d) + ((B + C)*Cos[4*d*x]*Sin[4*c])/(72*d) + (C*Cos[5*d*x]*Sin[5*c])/(176*d)
+ ((506*A + 506*B + 435*C)*Cos[c]*Sin[d*x])/(1848*d) + ((18*A + 19*B + 19*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + (
(44*A + 44*B + 57*C)*Cos[3*c]*Sin[3*d*x])/(1232*d) + ((B + C)*Cos[4*c]*Sin[4*d*x])/(72*d) + (C*Cos[5*c]*Sin[5*
d*x])/(176*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]
^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^
2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (5*B*(1
+ Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2
*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcT
an[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (15*C*(1 + Cos[c + d*x])*Csc[c]
*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c
]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 +
Sin[d*x - ArcTan[Cot[c]]]])/(77*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((
HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1
- Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1
+ Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*
x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 +
Tan[c]^2]]))/(10*d) - (7*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3
/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[
1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) -
((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c
]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d) - (7*C*(1 + Co
s[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*
Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqr
t[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c
])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[
Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d))
________________________________________________________________________________________
Maple [B] time = 0.2, size = 543, 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)^(5/2)*(a+a*cos(d*x+c))*(A+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)*a*(20160*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*
c)^12+(-12320*B-62720*C)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A+32560*B+81520*C)*sin(1/2*d*x+1/2*c)^
8*cos(1/2*d*x+1/2*c)+(-17424*A-34672*B-57712*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(14784*A+19712*B+24332
*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-4026*A-4488*B-4638*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+82
5*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-2079*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))+825*B*(si
n(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))-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))+675*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))-1617*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)))/(-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 ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+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) + A)*(a*cos(d*x + c) + a)*cos(d*x + c)^(5/2), x)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a \cos \left (d x + c\right )^{5} +{\left (B + C\right )} a \cos \left (d x + c\right )^{4} +{\left (A + B\right )} a \cos \left (d x + c\right )^{3} + A a \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)^(5/2)*(a+a*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
[Out]
integral((C*a*cos(d*x + c)^5 + (B + C)*a*cos(d*x + c)^4 + (A + B)*a*cos(d*x + c)^3 + A*a*cos(d*x + c)^2)*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)**(5/2)*(a+a*cos(d*x+c))*(A+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)^(5/2)*(a+a*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")
[Out]
Timed out