3.1041 \(\int \frac{\cos (c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx\)
Optimal. Leaf size=258 \[ \frac{2 \left (10 a^2 b B-8 a^3 C-a b^2 (15 A+7 C)+5 b^3 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-4 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d} \]
[Out]
(2*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(
15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C - a*b^2*(15*A + 7*C))*Sqrt[(
a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(
5*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*
Sin[c + d*x])/(5*b*d)
________________________________________________________________________________________
Rubi [A] time = 0.423549, antiderivative size = 258, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used =
{3049, 3023, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 \left (10 a^2 b B-8 a^3 C-a b^2 (15 A+7 C)+5 b^3 B\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-4 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]
[Out]
(2*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(
15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C - a*b^2*(15*A + 7*C))*Sqrt[(
a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(
5*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*
Sin[c + d*x])/(5*b*d)
Rule 3049
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] :> -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)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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 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 2752
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Rule 2663
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] && !GtQ[a + b, 0]
Rule 2661
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2655
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] && !GtQ[a + b, 0]
Rule 2653
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{a+b \cos (c+d x)}} \, dx &=\frac{2 C \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac{2 \int \frac{a C+\frac{1}{2} b (5 A+3 C) \cos (c+d x)+\frac{1}{2} (5 b B-4 a C) \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{5 b}\\ &=\frac{2 (5 b B-4 a C) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac{2 C \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac{4 \int \frac{\frac{1}{4} b (5 b B+2 a C)+\frac{1}{4} \left (15 A b^2-10 a b B+8 a^2 C+9 b^2 C\right ) \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^2}\\ &=\frac{2 (5 b B-4 a C) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac{2 C \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac{\left (15 A b^2-10 a b B+8 a^2 C+9 b^2 C\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{15 b^3}+\frac{\left (10 a^2 b B+5 b^3 B-8 a^3 C-a b^2 (15 A+7 C)\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{15 b^3}\\ &=\frac{2 (5 b B-4 a C) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac{2 C \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}+\frac{\left (\left (15 A b^2-10 a b B+8 a^2 C+9 b^2 C\right ) \sqrt{a+b \cos (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}} \, dx}{15 b^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (\left (10 a^2 b B+5 b^3 B-8 a^3 C-a b^2 (15 A+7 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \cos (c+d x)}{a+b}}} \, dx}{15 b^3 \sqrt{a+b \cos (c+d x)}}\\ &=\frac{2 \left (15 A b^2-10 a b B+8 a^2 C+9 b^2 C\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left (10 a^2 b B+5 b^3 B-8 a^3 C-a b^2 (15 A+7 C)\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (5 b B-4 a C) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{15 b^2 d}+\frac{2 C \cos (c+d x) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{5 b d}\\ \end{align*}
Mathematica [A] time = 1.06969, size = 186, normalized size = 0.72 \[ \frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left (\left (8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right ) \left ((a+b) E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )-a F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )+b^2 (2 a C+5 b B) F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )\right )+2 b \sin (c+d x) (a+b \cos (c+d x)) (-4 a C+5 b B+3 b C \cos (c+d x))}{15 b^3 d \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]
[Out]
(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(5*b*B + 2*a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (15*A*b^2 -
10*a*b*B + 8*a^2*C + 9*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)
/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(5*b*B - 4*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(15*b^3*d*Sqrt[a + b
*Cos[c + d*x]])
________________________________________________________________________________________
Maple [B] time = 1.251, size = 1258, normalized size = 4.9 \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)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(1/2),x)
[Out]
2/15*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(8*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2
*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b-9*C*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^
2+7*a*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c)
,(-2*b/(a-b))^(1/2))*b^2+10*B*cos(1/2*d*x+1/2*c)*a*b^2-10*B*cos(1/2*d*x+1/2*c)^3*a*b^2-6*C*cos(1/2*d*x+1/2*c)^
3*a*b^2-8*C*cos(1/2*d*x+1/2*c)*a^2*b+2*C*cos(1/2*d*x+1/2*c)*a*b^2+4*C*cos(1/2*d*x+1/2*c)^5*a*b^2+8*C*cos(1/2*d
*x+1/2*c)^3*a^2*b-15*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticE(cos
(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^2-5*b^3*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)
/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+15*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d
*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3+8*C*(sin(1/2*d*x+1/2*c)^2
)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3-8*C*
(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(
a-b))^(1/2))*a^3+9*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticE(cos(1
/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3-10*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^2-24*C*cos(1/2*d*x+1/2*c)^7*b^3+48*C*cos(1/2*d*x+1
/2*c)^5*b^3-30*C*cos(1/2*d*x+1/2*c)^3*b^3+6*C*cos(1/2*d*x+1/2*c)*b^3-10*B*cos(1/2*d*x+1/2*c)*b^3-20*B*cos(1/2*
d*x+1/2*c)^5*b^3+30*B*cos(1/2*d*x+1/2*c)^3*b^3-10*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-
b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b+10*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*c
os(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b+15*a*A*b^2*(sin
(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b)
)^(1/2)))/b^3/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+
1/2*c)^2*b+a+b)^(1/2)/d
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{\sqrt{b \cos \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)/sqrt(b*cos(d*x + c) + a), x)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C \cos \left (d x + c\right )^{3} + B \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )}{\sqrt{b \cos \left (d x + c\right ) + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
integral((C*cos(d*x + c)^3 + B*cos(d*x + c)^2 + A*cos(d*x + c))/sqrt(b*cos(d*x + c) + a), 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)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**(1/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{\sqrt{b \cos \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)/sqrt(b*cos(d*x + c) + a), x)