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

Optimal. Leaf size=203 \[ \frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {C x}{b^3} \]

[Out]

C*x/b^3-a*(3*A*b^4+(2*a^4-5*a^2*b^2+6*b^4)*C)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b
^3/(a+b)^(5/2)/d+1/2*a*(A*b^2+C*a^2)*sin(d*x+c)/b^2/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/2*(2*A*b^4-3*a^4*C+a^2*b^
2*(A+6*C))*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))

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Rubi [A]  time = 0.46, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3032, 3021, 2735, 2659, 205} \[ -\frac {a \left (C \left (-5 a^2 b^2+2 a^4+6 b^4\right )+3 A b^4\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {\left (a^2 b^2 (A+6 C)-3 a^4 C+2 A b^4\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {a \left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {C x}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*x)/b^3 - (a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(
(a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 + a^2*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]
)^2) + ((2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2735

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

Rule 3021

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

Rule 3032

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

Rubi steps

\begin {align*} \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {\int \frac {2 b \left (A b^2+a^2 C\right )-a \left (A b^2-\left (a^2-2 b^2\right ) C\right ) \cos (c+d x)-2 b \left (a^2-b^2\right ) C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (2 A b^4-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {-a b^2 \left (3 A b^2-\left (a^2-4 b^2\right ) C\right )+2 b \left (a^2-b^2\right )^2 C \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {C x}{b^3}+\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (2 A b^4-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\left (a \left (3 A b^4+\left (2 a^4-5 a^2 b^2+6 b^4\right ) C\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {C x}{b^3}+\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (2 A b^4-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\left (a \left (3 A b^4+\left (2 a^4-5 a^2 b^2+6 b^4\right ) C\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^3 \left (a^2-b^2\right )^2 d}\\ &=\frac {C x}{b^3}-\frac {a \left (3 A b^4+2 a^4 C-5 a^2 b^2 C+6 b^4 C\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^3 (a+b)^{5/2} d}+\frac {a \left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (2 A b^4-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 1.29, size = 194, normalized size = 0.96 \[ \frac {\frac {a b \left (a^2 C+A b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac {b \left (-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac {2 a \left (C \left (2 a^4-5 a^2 b^2+6 b^4\right )+3 A b^4\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{5/2}}+2 C (c+d x)}{2 b^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*C*(c + d*x) + (2*a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 +
b^2]])/(-a^2 + b^2)^(5/2) + (a*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (b*
(2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^3*d)

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fricas [B]  time = 1.25, size = 1051, normalized size = 5.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(C*a^6*b^2 - 3*C*a^4*b^4 + 3*C*a^2*b^6 - C*b^8)*d*x*cos(d*x + c)^2 + 8*(C*a^7*b - 3*C*a^5*b^3 + 3*C*a^
3*b^5 - C*a*b^7)*d*x*cos(d*x + c) + 4*(C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*d*x - (2*C*a^7 - 5*C*a^5
*b^2 + 3*(A + 2*C)*a^3*b^4 + (2*C*a^5*b^2 - 5*C*a^3*b^4 + 3*(A + 2*C)*a*b^6)*cos(d*x + c)^2 + 2*(2*C*a^6*b - 5
*C*a^4*b^3 + 3*(A + 2*C)*a^2*b^5)*cos(d*x + c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d
*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*co
s(d*x + c) + a^2)) - 2*(2*C*a^7*b - (2*A + 7*C)*a^5*b^3 + (A + 5*C)*a^3*b^5 + A*a*b^7 + (3*C*a^6*b^2 - (A + 9*
C)*a^4*b^4 - (A - 6*C)*a^2*b^6 + 2*A*b^8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11
)*d*cos(d*x + c)^2 + 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*d*cos(d*x + c) + (a^8*b^3 - 3*a^6*b^5 + 3*a^
4*b^7 - a^2*b^9)*d), 1/2*(2*(C*a^6*b^2 - 3*C*a^4*b^4 + 3*C*a^2*b^6 - C*b^8)*d*x*cos(d*x + c)^2 + 4*(C*a^7*b -
3*C*a^5*b^3 + 3*C*a^3*b^5 - C*a*b^7)*d*x*cos(d*x + c) + 2*(C*a^8 - 3*C*a^6*b^2 + 3*C*a^4*b^4 - C*a^2*b^6)*d*x
- (2*C*a^7 - 5*C*a^5*b^2 + 3*(A + 2*C)*a^3*b^4 + (2*C*a^5*b^2 - 5*C*a^3*b^4 + 3*(A + 2*C)*a*b^6)*cos(d*x + c)^
2 + 2*(2*C*a^6*b - 5*C*a^4*b^3 + 3*(A + 2*C)*a^2*b^5)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) +
b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (2*C*a^7*b - (2*A + 7*C)*a^5*b^3 + (A + 5*C)*a^3*b^5 + A*a*b^7 + (3*C*a^6
*b^2 - (A + 9*C)*a^4*b^4 - (A - 6*C)*a^2*b^6 + 2*A*b^8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^5 - 3*a^4*b^7 + 3*
a^2*b^9 - b^11)*d*cos(d*x + c)^2 + 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*d*cos(d*x + c) + (a^8*b^3 - 3*
a^6*b^5 + 3*a^4*b^7 - a^2*b^9)*d)]

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giac [B]  time = 7.82, size = 479, normalized size = 2.36 \[ -\frac {\frac {{\left (2 \, C a^{5} - 5 \, C a^{3} b^{2} + 3 \, A a b^{4} + 6 \, C a b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \sqrt {a^{2} - b^{2}}} - \frac {{\left (d x + c\right )} C}{b^{3}} + \frac {2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{2}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((2*C*a^5 - 5*C*a^3*b^2 + 3*A*a*b^4 + 6*C*a*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a
*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^4*b^3 - 2*a^2*b^5 + b^7)*sqrt(a^2 - b^2)
) - (d*x + c)*C/b^3 + (2*C*a^5*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^4*b*tan(1/2*d*x + 1/2*c)^3 - 2*A*a^3*b^2*tan(1/2
*d*x + 1/2*c)^3 - 5*C*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + A*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*b^3*tan(1/2*
d*x + 1/2*c)^3 - A*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^5*tan(1/2*d*x + 1/2*c)^3 + 2*C*a^5*tan(1/2*d*x + 1/2*c
) + 3*C*a^4*b*tan(1/2*d*x + 1/2*c) - 2*A*a^3*b^2*tan(1/2*d*x + 1/2*c) - 5*C*a^3*b^2*tan(1/2*d*x + 1/2*c) - A*a
^2*b^3*tan(1/2*d*x + 1/2*c) - 6*C*a^2*b^3*tan(1/2*d*x + 1/2*c) - A*a*b^4*tan(1/2*d*x + 1/2*c) - 2*A*b^5*tan(1/
2*d*x + 1/2*c))/((a^4*b^2 - 2*a^2*b^4 + b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^2))
/d

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maple [B]  time = 0.12, size = 1093, normalized size = 5.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+1/d
*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*a/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+2/d*b^
2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-2/d*a^4/b
^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+1/d/b/(a
*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*a^3/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+6/d/(a*ta
n(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*a^2+2/d*a^2/(a*t
an(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-1/d*b/(a*tan(1/2*d*x+1/2*
c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*a*A+2/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/
2*d*x+1/2*c)^2*b+a+b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-2/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2
*c)^2*b+a+b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C-1/d/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/
(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*a^3*C+6/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)/(a-b)^2
*tan(1/2*d*x+1/2*c)*C*a^2-3/d*a*b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-
b)*(a+b))^(1/2))*A-2/d*a^5/b^3/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*
(a+b))^(1/2))*C+5/d*a^3/b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b)
)^(1/2))*C-6/d*b*a/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2)
)*C+2/d/b^3*arctan(tan(1/2*d*x+1/2*c))*C

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 10.42, size = 6587, normalized size = 32.45 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)^3*(2*A*b^4 - 2*C*a^4 + 2*A*a^2*b^2 + 6*C*a^2*b^2 + A*a*b^3 + C*a^3*b))/((a*b^2 - b^3)*(a
+ b)^2) + (tan(c/2 + (d*x)/2)*(2*A*b^4 - 2*C*a^4 + 2*A*a^2*b^2 + 6*C*a^2*b^2 - A*a*b^3 - C*a^3*b))/((a + b)*(b
^4 - 2*a*b^3 + a^2*b^2)))/(d*(2*a*b + tan(c/2 + (d*x)/2)^2*(2*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^4*(a^2 - 2*a*b
+ b^2) + a^2 + b^2)) - (2*C*atan(((C*((C*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*b^11 - 6*A*a
^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9 + 18*C*a^7*b
^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b
^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^14 - 32*a^3*b^13 + 32*a^4*b^12
+ 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6)*8i)/(b^3*(a*b^10 + b^11 - 3*a
^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))*1i)/b^3 + (8*tan(c/2 + (d*x)/2)*(8*C^2*a^10
+ 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 9*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 -
48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*A*C*a^4*b^6 + 12*A*C*a
^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))/b^3 - (C*((C*((
8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*
a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^14
- 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (C*tan
(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^14 - 32*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 +
32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6)*8i)/(b^3*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 -
a^6*b^5 - a^7*b^4)))*1i)/b^3 - (8*tan(c/2 + (d*x)/2)*(8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 9
*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7
*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*A*C*a^4*b^6 + 12*A*C*a^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*
b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))/b^3)/((C*((C*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 -
12*A*a^4*b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*
C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11
- 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^14 - 32*
a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6)*8i)/(b^
3*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))*1i)/b^3 + (8*tan(c/2 +
(d*x)/2)*(8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 9*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b
^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*
A*C*a^4*b^6 + 12*A*C*a^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*
b^4))*1i)/b^3 - (16*(4*C^3*a^9 + 12*C^3*a*b^8 - 2*C^3*a^8*b + 24*C^3*a^2*b^7 - 34*C^3*a^3*b^6 - 26*C^3*a^4*b^5
+ 36*C^3*a^5*b^4 + 13*C^3*a^6*b^3 - 18*C^3*a^7*b^2 + 6*A*C^2*a*b^8 + 30*A*C^2*a^2*b^7 - 12*A*C^2*a^3*b^6 - 18
*A*C^2*a^4*b^5 + 6*A*C^2*a^5*b^4 + 6*A*C^2*a^6*b^3 + 9*A^2*C*a^2*b^7))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^1
0 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (C*((C*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^
4*b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^
9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*
b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (C*tan(c/2 + (d*x)/2)*(8*a*b^15 - 8*a^2*b^14 - 32*a^3*b^13
+ 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6)*8i)/(b^3*(a*b^1
0 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))*1i)/b^3 - (8*tan(c/2 + (d*x)/2
)*(8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 9*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*
C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*A*C*a^4*
b^6 + 12*A*C*a^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4))*1i
)/b^3)))/(b^3*d) + (a*atan(((a*((8*tan(c/2 + (d*x)/2)*(8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 9
*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7
*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*A*C*a^4*b^6 + 12*A*C*a^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*
b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4) + (a*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*
b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9
+ 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^
10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (4*a*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^
4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2)*(8*a*b^15 - 8*a^2*b^14 - 32*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6
*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/((b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a
^8*b^5 - a^10*b^3)*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))*(-(a
+ b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2))/(2*(b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a
^6*b^7 + 5*a^8*b^5 - a^10*b^3)))*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2)*1i)/
(2*(b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3)) + (a*((8*tan(c/2 + (d*x)/2)*(8*C^2*a^
10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 9*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6
- 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*A*C*a^4*b^6 + 12*A*
C*a^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4) - (a*((8*(4*C*
b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^1
2 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^14 - 12*C*
a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (4*a*tan(c/2
+ (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2)*(8*a*b^15 - 8*a^2*b^14 - 3
2*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/((b^
13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3)*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a
^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*
b^2))/(2*(b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3)))*(-(a + b)^5*(a - b)^5)^(1/2)*(
3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2)*1i)/(2*(b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a
^10*b^3)))/((16*(4*C^3*a^9 + 12*C^3*a*b^8 - 2*C^3*a^8*b + 24*C^3*a^2*b^7 - 34*C^3*a^3*b^6 - 26*C^3*a^4*b^5 + 3
6*C^3*a^5*b^4 + 13*C^3*a^6*b^3 - 18*C^3*a^7*b^2 + 6*A*C^2*a*b^8 + 30*A*C^2*a^2*b^7 - 12*A*C^2*a^3*b^6 - 18*A*C
^2*a^4*b^5 + 6*A*C^2*a^5*b^4 + 6*A*C^2*a^6*b^3 + 9*A^2*C*a^2*b^7))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 +
3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (a*((8*tan(c/2 + (d*x)/2)*(8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9
- 8*C^2*a^9*b + 9*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 - 52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6
*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*A*C*a^4*b^6 + 12*A*C*a^6*b^4))/(a*b^10 + b^11 - 3
*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4) + (a*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3
*b^12 - 12*A*a^4*b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13 + 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^
10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^
2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (4*a*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)
^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2)*(8*a*b^15 - 8*a^2*b^14 - 32*a^3*b^13 + 32*a^4*b^12 + 48*
a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a^10*b^6))/((b^13 - 5*a^2*b^11 + 10*a^4*b^9 -
10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3)*(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 -
a^7*b^4)))*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2))/(2*(b^13 - 5*a^2*b^11 +
10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3)))*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 -
5*C*a^2*b^2))/(2*(b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3)) + (a*((8*tan(c/2 + (d*x
)/2)*(8*C^2*a^10 + 4*C^2*b^10 - 8*C^2*a*b^9 - 8*C^2*a^9*b + 9*A^2*a^2*b^8 + 24*C^2*a^2*b^8 + 32*C^2*a^3*b^7 -
52*C^2*a^4*b^6 - 48*C^2*a^5*b^5 + 57*C^2*a^6*b^4 + 32*C^2*a^7*b^3 - 32*C^2*a^8*b^2 + 36*A*C*a^2*b^8 - 30*A*C*a
^4*b^6 + 12*A*C*a^6*b^4))/(a*b^10 + b^11 - 3*a^2*b^9 - 3*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)
- (a*((8*(4*C*b^15 + 6*A*a^2*b^13 + 12*A*a^3*b^12 - 12*A*a^4*b^11 - 6*A*a^5*b^10 + 6*A*a^6*b^9 - 8*C*a^2*b^13
+ 34*C*a^3*b^12 + 6*C*a^4*b^11 - 36*C*a^5*b^10 - 4*C*a^6*b^9 + 18*C*a^7*b^8 + 2*C*a^8*b^7 - 4*C*a^9*b^6 - 6*A*
a*b^14 - 12*C*a*b^14))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) +
(4*a*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2)*(8*a*b^15 -
8*a^2*b^14 - 32*a^3*b^13 + 32*a^4*b^12 + 48*a^5*b^11 - 48*a^6*b^10 - 32*a^7*b^9 + 32*a^8*b^8 + 8*a^9*b^7 - 8*a
^10*b^6))/((b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3)*(a*b^10 + b^11 - 3*a^2*b^9 - 3
*a^3*b^8 + 3*a^4*b^7 + 3*a^5*b^6 - a^6*b^5 - a^7*b^4)))*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*
b^4 - 5*C*a^2*b^2))/(2*(b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3)))*(-(a + b)^5*(a -
b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2))/(2*(b^13 - 5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*
a^8*b^5 - a^10*b^3))))*(-(a + b)^5*(a - b)^5)^(1/2)*(3*A*b^4 + 2*C*a^4 + 6*C*b^4 - 5*C*a^2*b^2)*1i)/(d*(b^13 -
5*a^2*b^11 + 10*a^4*b^9 - 10*a^6*b^7 + 5*a^8*b^5 - a^10*b^3))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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