\(\int \frac {\cos (c+d x) (B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx\) [842]

Optimal result

Integrand size = 40, antiderivative size = 262 \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {2 \left (6 a^2 b B-3 b^3 B-8 a^3 C+5 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (6 a b B-8 a^2 C-b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d} \]

[Out]

-2*a^2*(B*b-C*a)*sin(d*x+c)/b^2/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)+2/3*C*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2
/d+2/3*(6*B*a^2*b-3*B*b^3-8*C*a^3+5*C*a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2
*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/b^3/(a^2-b^2)/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/3
*(6*B*a*b-8*C*a^2-C*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*
(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/b^3/d/(a+b*cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3108, 3067, 3102, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {2 a^2 (b B-a C) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^2 C+6 a b B-b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-8 a^3 C+6 a^2 b B+5 a b^2 C-3 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 C \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b^2 d} \]

[In]

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

[Out]

(2*(6*a^2*b*B - 3*b^3*B - 8*a^3*C + 5*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])
/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(6*a*b*B - 8*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c +
d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*(b*B - a*C)
*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b
^2*d)

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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 3067

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(B*c - A*d)*(b*c - a*d)^2*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(
f*d^2*(n + 1)*(c^2 - d^2))), x] - Dist[1/(d^2*(n + 1)*(c^2 - d^2)), Int[(c + d*Sin[e + f*x])^(n + 1)*Simp[d*(n
+ 1)*(B*(b*c - a*d)^2 - A*d*(a^2*c + b^2*c - 2*a*b*d)) - ((B*c - A*d)*(a^2*d^2*(n + 2) + b^2*(c^2 + d^2*(n +
1))) + 2*a*b*d*(A*c*d*(n + 2) - B*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b^2*B*d*(n + 1)*(c^2 - d^2)*Sin[e + f*x
]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && LtQ[n, -1]

Rule 3102

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 3108

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) (B+C \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx \\ & = -\frac {2 a^2 (b B-a C) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \int \frac {\frac {1}{2} a b (b B-a C)+\frac {1}{2} \left (2 a^2-b^2\right ) (b B-a C) \cos (c+d x)+\frac {1}{2} b \left (a^2-b^2\right ) C \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{b^2 \left (a^2-b^2\right )} \\ & = -\frac {2 a^2 (b B-a C) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {4 \int \frac {\frac {1}{4} b^2 \left (3 a b B-2 a^2 C-b^2 C\right )+\frac {1}{4} b \left (6 a^2 b B-3 b^3 B-8 a^3 C+5 a b^2 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )} \\ & = -\frac {2 a^2 (b B-a C) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}-\frac {\left (6 a b B-8 a^2 C-b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 b^3}+\frac {\left (6 a^2 b B-3 b^3 B-8 a^3 C+5 a b^2 C\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3 b^3 \left (a^2-b^2\right )} \\ & = -\frac {2 a^2 (b B-a C) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d}+\frac {\left (\left (6 a^2 b B-3 b^3 B-8 a^3 C+5 a 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}{3 b^3 \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (6 a b B-8 a^2 C-b^2 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}{3 b^3 \sqrt {a+b \cos (c+d x)}} \\ & = \frac {2 \left (6 a^2 b B-3 b^3 B-8 a^3 C+5 a 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 )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (6 a b B-8 a^2 C-b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 C \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 b^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.29 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.72 \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {2 \left (\frac {\sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (\left (6 a^2 b B-3 b^3 B-8 a^3 C+5 a b^2 C\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+(a-b) \left (-6 a b B+8 a^2 C+b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )}{a-b}+b \left (\frac {a \left (3 a b B-4 a^2 C+b^2 C\right )}{-a^2+b^2}+b C \cos (c+d x)\right ) \sin (c+d x)\right )}{3 b^3 d \sqrt {a+b \cos (c+d x)}} \]

[In]

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

[Out]

(2*((Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((6*a^2*b*B - 3*b^3*B - 8*a^3*C + 5*a*b^2*C)*EllipticE[(c + d*x)/2, (2
*b)/(a + b)] + (a - b)*(-6*a*b*B + 8*a^2*C + b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b) + b*((a*(3
*a*b*B - 4*a^2*C + b^2*C))/(-a^2 + b^2) + b*C*Cos[c + d*x])*Sin[c + d*x]))/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1274\) vs. \(2(302)=604\).

Time = 14.57 (sec) , antiderivative size = 1275, normalized size of antiderivative = 4.87

method	result	size
parts	\(\text {Expression too large to display}\)	\(1275\)
default	\(\text {Expression too large to display}\)	\(1336\)

[In]

int(cos(d*x+c)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+cos(d*x+c)*b)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2*B*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^2*b-2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1
/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+2*a*b^2*(sin(1/2*d*x+1/2*c)^2)
^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+2*
(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),
(-2*b/(a-b))^(1/2))*a^3-2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ell
ipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c
)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^2+(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2
*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3)/b^2/(a-
b)/(a+b)/sin(1/2*d*x+1/2*c)/(-2*b*sin(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d-2/3*C*(4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^4*a^2*b^2-4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*b^4-8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^3*b-2
*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^2*b^2+2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a*b^3+2*cos(1/2*d*x
+1/2*c)*sin(1/2*d*x+1/2*c)^2*b^4+8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^
(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4-7*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*
d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-(sin(1/2*d*x+1/2*c)^2
)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b
^4-8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/
2*c),(-2*b/(a-b))^(1/2))*a^4+8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2
)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*
x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-5*(sin(1/2*d*x+1/2*c)^2
)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a
*b^3)/b^3/(a-b)/(a+b)/sin(1/2*d*x+1/2*c)/(-2*b*sin(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 789, normalized size of antiderivative = 3.01 \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {6 \, {\left (4 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - C a b^{4} + {\left (C a^{2} b^{3} - C b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) - {\left (\sqrt {2} {\left (16 i \, C a^{4} b - 12 i \, B a^{3} b^{2} - 16 i \, C a^{2} b^{3} + 15 i \, B a b^{4} - 3 i \, C b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (16 i \, C a^{5} - 12 i \, B a^{4} b - 16 i \, C a^{3} b^{2} + 15 i \, B a^{2} b^{3} - 3 i \, C a b^{4}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - {\left (\sqrt {2} {\left (-16 i \, C a^{4} b + 12 i \, B a^{3} b^{2} + 16 i \, C a^{2} b^{3} - 15 i \, B a b^{4} + 3 i \, C b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-16 i \, C a^{5} + 12 i \, B a^{4} b + 16 i \, C a^{3} b^{2} - 15 i \, B a^{2} b^{3} + 3 i \, C a b^{4}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + 3 \, {\left (\sqrt {2} {\left (-8 i \, C a^{3} b^{2} + 6 i \, B a^{2} b^{3} + 5 i \, C a b^{4} - 3 i \, B b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-8 i \, C a^{4} b + 6 i \, B a^{3} b^{2} + 5 i \, C a^{2} b^{3} - 3 i \, B a b^{4}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, {\left (\sqrt {2} {\left (8 i \, C a^{3} b^{2} - 6 i \, B a^{2} b^{3} - 5 i \, C a b^{4} + 3 i \, B b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (8 i \, C a^{4} b - 6 i \, B a^{3} b^{2} - 5 i \, C a^{2} b^{3} + 3 i \, B a b^{4}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right )}{9 \, {\left ({\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{4} - a b^{6}\right )} d\right )}} \]

[In]

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

[Out]

1/9*(6*(4*C*a^3*b^2 - 3*B*a^2*b^3 - C*a*b^4 + (C*a^2*b^3 - C*b^5)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d
*x + c) - (sqrt(2)*(16*I*C*a^4*b - 12*I*B*a^3*b^2 - 16*I*C*a^2*b^3 + 15*I*B*a*b^4 - 3*I*C*b^5)*cos(d*x + c) +
sqrt(2)*(16*I*C*a^5 - 12*I*B*a^4*b - 16*I*C*a^3*b^2 + 15*I*B*a^2*b^3 - 3*I*C*a*b^4))*sqrt(b)*weierstrassPInver
se(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)
- (sqrt(2)*(-16*I*C*a^4*b + 12*I*B*a^3*b^2 + 16*I*C*a^2*b^3 - 15*I*B*a*b^4 + 3*I*C*b^5)*cos(d*x + c) + sqrt(2)
*(-16*I*C*a^5 + 12*I*B*a^4*b + 16*I*C*a^3*b^2 - 15*I*B*a^2*b^3 + 3*I*C*a*b^4))*sqrt(b)*weierstrassPInverse(4/3
*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*(s
qrt(2)*(-8*I*C*a^3*b^2 + 6*I*B*a^2*b^3 + 5*I*C*a*b^4 - 3*I*B*b^5)*cos(d*x + c) + sqrt(2)*(-8*I*C*a^4*b + 6*I*B
*a^3*b^2 + 5*I*C*a^2*b^3 - 3*I*B*a*b^4))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b
^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I
*b*sin(d*x + c) + 2*a)/b)) + 3*(sqrt(2)*(8*I*C*a^3*b^2 - 6*I*B*a^2*b^3 - 5*I*C*a*b^4 + 3*I*B*b^5)*cos(d*x + c)
+ sqrt(2)*(8*I*C*a^4*b - 6*I*B*a^3*b^2 - 5*I*C*a^2*b^3 + 3*I*B*a*b^4))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3
*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b
^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)))/((a^2*b^5 - b^7)*d*cos(d*x + c) + (a^3*b^4 - a*b^6)
*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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