3.611 \(\int \frac{\cos ^3(c+d x) (1-\cos ^2(c+d x))}{(a+b \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=268 \[ \frac{a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )}+\frac{a \left (-19 a^2 b^2+12 a^4+6 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left (4 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac{\left (6 a^2-5 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )}-\frac{x \left (12 a^2-b^2\right )}{2 b^5}+\frac{\sin (c+d x) \cos ^3(c+d x)}{2 b d (a+b \cos (c+d x))^2} \]

[Out]

-((12*a^2 - b^2)*x)/(2*b^5) + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a +
b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + (a*(12*a^2 - 11*b^2)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)*d) - ((6*a^2
- 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(2*b*d*(a + b*Cos[c
+ d*x])^2) + ((4*a^2 - 3*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))

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Rubi [A]  time = 0.750621, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3048, 3049, 3023, 2735, 2659, 205} \[ \frac{a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )}+\frac{a \left (-19 a^2 b^2+12 a^4+6 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left (4 a^2-3 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac{\left (6 a^2-5 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )}-\frac{x \left (12 a^2-b^2\right )}{2 b^5}+\frac{\sin (c+d x) \cos ^3(c+d x)}{2 b d (a+b \cos (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]

[Out]

-((12*a^2 - b^2)*x)/(2*b^5) + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a +
b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + (a*(12*a^2 - 11*b^2)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)*d) - ((6*a^2
- 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(2*b*d*(a + b*Cos[c
+ d*x])^2) + ((4*a^2 - 3*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))

Rule 3048

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

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=\frac{\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac{\int \frac{\cos ^2(c+d x) \left (-3 \left (a^2-b^2\right )+4 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac{\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\int \frac{\cos (c+d x) \left (2 \left (4 a^4-7 a^2 b^2+3 b^4\right )-a b \left (a^2-b^2\right ) \cos (c+d x)-2 \left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac{\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\int \frac{-2 a \left (6 a^4-11 a^2 b^2+5 b^4\right )+2 b \left (2 a^4-3 a^2 b^2+b^4\right ) \cos (c+d x)+2 a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}-\frac{\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac{\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\int \frac{-2 a b \left (6 a^4-11 a^2 b^2+5 b^4\right )-2 \left (a^2-b^2\right )^2 \left (12 a^2-b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (12 a^2-b^2\right ) x}{2 b^5}+\frac{a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}-\frac{\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac{\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (a \left (12 a^4-19 a^2 b^2+6 b^4\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )}\\ &=-\frac{\left (12 a^2-b^2\right ) x}{2 b^5}+\frac{a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}-\frac{\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac{\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (a \left (12 a^4-19 a^2 b^2+6 b^4\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^5 \left (a^2-b^2\right ) d}\\ &=-\frac{\left (12 a^2-b^2\right ) x}{2 b^5}+\frac{a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{3/2} b^5 (a+b)^{3/2} d}+\frac{a \left (12 a^2-11 b^2\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}-\frac{\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}+\frac{\left (4 a^2-3 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 5.00148, size = 374, normalized size = 1.4 \[ \frac{\frac{72 a^4 b^2 \sin (2 (c+d x))-80 a^3 b^3 \sin (c+d x)+8 a^3 b^3 \sin (3 (c+d x))-70 a^2 b^4 \sin (2 (c+d x))-a^2 b^4 \sin (4 (c+d x))-16 a b \left (-13 a^2 b^2+12 a^4+b^4\right ) (c+d x) \cos (c+d x)-4 b^2 \left (-13 a^2 b^2+12 a^4+b^4\right ) (c+d x) \cos (2 (c+d x))+56 a^4 b^2 c+44 a^2 b^4 c+56 a^4 b^2 d x+44 a^2 b^4 d x+96 a^5 b \sin (c+d x)-96 a^6 c-96 a^6 d x-8 a b^5 \sin (c+d x)-8 a b^5 \sin (3 (c+d x))+2 b^6 \sin (2 (c+d x))+b^6 \sin (4 (c+d x))-4 b^6 c-4 b^6 d x}{(a+b \cos (c+d x))^2}-\frac{16 a \left (-19 a^2 b^2+12 a^4+6 b^4\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}}{16 b^5 d (a-b) (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]

[Out]

((-16*a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] +
(-96*a^6*c + 56*a^4*b^2*c + 44*a^2*b^4*c - 4*b^6*c - 96*a^6*d*x + 56*a^4*b^2*d*x + 44*a^2*b^4*d*x - 4*b^6*d*x
- 16*a*b*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Cos[c + d*x] - 4*b^2*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Cos
[2*(c + d*x)] + 96*a^5*b*Sin[c + d*x] - 80*a^3*b^3*Sin[c + d*x] - 8*a*b^5*Sin[c + d*x] + 72*a^4*b^2*Sin[2*(c +
d*x)] - 70*a^2*b^4*Sin[2*(c + d*x)] + 2*b^6*Sin[2*(c + d*x)] + 8*a^3*b^3*Sin[3*(c + d*x)] - 8*a*b^5*Sin[3*(c
+ d*x)] - a^2*b^4*Sin[4*(c + d*x)] + b^6*Sin[4*(c + d*x)])/(a + b*Cos[c + d*x])^2)/(16*(a - b)*b^5*(a + b)*d)

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Maple [B]  time = 0.04, size = 704, 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)^3*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x)

[Out]

6/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)^3*a+1/d/b^3/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c
)^3+6/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)*a-1/d/b^3/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2
*c)-12/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^2+1/d/b^3*arctan(tan(1/2*d*x+1/2*c))+6/d*a^4/b^4/(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)*tan(1/2*d*x+1/2*c)^3-1/d*a^3/b^3/(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)*tan(1/2*d*x+1/2*c)^3-6/d*a^2/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)*tan(1/2*d*x+1/2*c)^3+6/d*a^4/b^4/(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)*tan(1/2*d*
x+1/2*c)+1/d*a^3/b^3/(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)*tan(1/2*d*x+1/2*c)-6/d*a^2/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)*tan(1/2*d*x+1/2*c)+12/d*a^5/b^5/(a^2-b^2)/((a+b)
*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))-19/d*a^3/b^3/(a^2-b^2)/((a+b)*(a-b))^(1/2)*
arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))+6/d*a/b/(a^2-b^2)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/
2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.06507, size = 2171, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/4*(2*(12*a^6*b^2 - 25*a^4*b^4 + 14*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2 + 4*(12*a^7*b - 25*a^5*b^3 + 14*a^3*b
^5 - a*b^7)*d*x*cos(d*x + c) + 2*(12*a^8 - 25*a^6*b^2 + 14*a^4*b^4 - a^2*b^6)*d*x - (12*a^7 - 19*a^5*b^2 + 6*a
^3*b^4 + (12*a^5*b^2 - 19*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(12*a^6*b - 19*a^4*b^3 + 6*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*cos(d*x + c) + a^2)) - 2*(12*a^7*b - 23*a^5*b^
3 + 11*a^3*b^5 - (a^4*b^4 - 2*a^2*b^6 + b^8)*cos(d*x + c)^3 + 4*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c)^2 +
(18*a^6*b^2 - 35*a^4*b^4 + 17*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^7 - 2*a^2*b^9 + b^11)*d*cos(d*x +
c)^2 + 2*(a^5*b^6 - 2*a^3*b^8 + a*b^10)*d*cos(d*x + c) + (a^6*b^5 - 2*a^4*b^7 + a^2*b^9)*d), -1/2*((12*a^6*b^2
- 25*a^4*b^4 + 14*a^2*b^6 - b^8)*d*x*cos(d*x + c)^2 + 2*(12*a^7*b - 25*a^5*b^3 + 14*a^3*b^5 - a*b^7)*d*x*cos(
d*x + c) + (12*a^8 - 25*a^6*b^2 + 14*a^4*b^4 - a^2*b^6)*d*x - (12*a^7 - 19*a^5*b^2 + 6*a^3*b^4 + (12*a^5*b^2 -
19*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^2 + 2*(12*a^6*b - 19*a^4*b^3 + 6*a^2*b^5)*cos(d*x + c))*sqrt(a^2 - b^2)*ar
ctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (12*a^7*b - 23*a^5*b^3 + 11*a^3*b^5 - (a^4*b^4 -
2*a^2*b^6 + b^8)*cos(d*x + c)^3 + 4*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c)^2 + (18*a^6*b^2 - 35*a^4*b^4 +
17*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^7 - 2*a^2*b^9 + b^11)*d*cos(d*x + c)^2 + 2*(a^5*b^6 - 2*a^3*b^
8 + a*b^10)*d*cos(d*x + c) + (a^6*b^5 - 2*a^4*b^7 + a^2*b^9)*d)]

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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*(1-cos(d*x+c)**2)/(a+b*cos(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.41852, size = 814, normalized size = 3.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(1-cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(2*(12*a^5 - 19*a^3*b^2 + 6*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^2*b^5 - b^7)*sqrt(a^2 - b^2)) - 2*(12*a^5*tan(1/
2*d*x + 1/2*c)^7 - 18*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 7*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 + 18*a^2*b^3*tan(1/2*d*x
+ 1/2*c)^7 - 4*a*b^4*tan(1/2*d*x + 1/2*c)^7 - b^5*tan(1/2*d*x + 1/2*c)^7 + 36*a^5*tan(1/2*d*x + 1/2*c)^5 - 18
*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 37*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 + 14*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 + 4*a*b^
4*tan(1/2*d*x + 1/2*c)^5 + 3*b^5*tan(1/2*d*x + 1/2*c)^5 + 36*a^5*tan(1/2*d*x + 1/2*c)^3 + 18*a^4*b*tan(1/2*d*x
+ 1/2*c)^3 - 37*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 14*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 4*a*b^4*tan(1/2*d*x + 1/
2*c)^3 - 3*b^5*tan(1/2*d*x + 1/2*c)^3 + 12*a^5*tan(1/2*d*x + 1/2*c) + 18*a^4*b*tan(1/2*d*x + 1/2*c) - 7*a^3*b^
2*tan(1/2*d*x + 1/2*c) - 18*a^2*b^3*tan(1/2*d*x + 1/2*c) - 4*a*b^4*tan(1/2*d*x + 1/2*c) + b^5*tan(1/2*d*x + 1/
2*c))/((a^2*b^4 - b^6)*(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 + 2*a*tan(1/2*d*x + 1/2*c)^2 + a +
b)^2) + (12*a^2 - b^2)*(d*x + c)/b^5)/d