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

Optimal. Leaf size=189 \[ -\frac {2 a^2 \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d \sqrt {a-b} \sqrt {a+b}}+\frac {a x \left (4 a^2-b^2\right )}{b^5}-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{b d (a+b \cos (c+d x))}-\frac {4 \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.62, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3048, 3050, 3049, 3023, 2735, 2659, 205} \[ -\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac {2 a^2 \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d \sqrt {a-b} \sqrt {a+b}}+\frac {a x \left (4 a^2-b^2\right )}{b^5}+\frac {2 a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{b d (a+b \cos (c+d x))}-\frac {4 \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(4*a^2 - b^2)*x)/b^5 - (2*a^2*(4*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a -
b]*b^5*Sqrt[a + b]*d) - ((12*a^2 - b^2)*Sin[c + d*x])/(3*b^4*d) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(b^3*d) - (
4*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(b*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 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 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 3050

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*
x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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])))

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))^2} \, dx &=\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\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)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (a^2-b^2\right ) \cos (c+d x)-12 a \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {-12 a^2 \left (a^2-b^2\right )+4 a b \left (a^2-b^2\right ) \cos (c+d x)+2 \left (a^2-b^2\right ) \left (12 a^2-b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )}\\ &=-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {-12 a^2 b \left (a^2-b^2\right )-6 a \left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )}\\ &=\frac {a \left (4 a^2-b^2\right ) x}{b^5}-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\left (a^2 \left (4 a^2-3 b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^5}\\ &=\frac {a \left (4 a^2-b^2\right ) x}{b^5}-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\left (2 a^2 \left (4 a^2-3 b^2\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 d}\\ &=\frac {a \left (4 a^2-b^2\right ) x}{b^5}-\frac {2 a^2 \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^5 \sqrt {a+b} d}-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 2.38, size = 217, normalized size = 1.15 \[ \frac {\frac {48 a^2 \left (4 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}+\frac {96 a^4 c+96 a^4 d x-24 a^2 b^2 \sin (2 (c+d x))+12 a b \left (b^2-8 a^2\right ) \sin (c+d x)+24 a b \left (4 a^2-b^2\right ) (c+d x) \cos (c+d x)-24 a^2 b^2 c-24 a^2 b^2 d x+4 a b^3 \sin (3 (c+d x))+2 b^4 \sin (2 (c+d x))-b^4 \sin (4 (c+d x))}{a+b \cos (c+d x)}}{24 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((48*a^2*(4*a^2 - 3*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (96*a^4*c -
24*a^2*b^2*c + 96*a^4*d*x - 24*a^2*b^2*d*x + 24*a*b*(4*a^2 - b^2)*(c + d*x)*Cos[c + d*x] + 12*a*b*(-8*a^2 + b^
2)*Sin[c + d*x] - 24*a^2*b^2*Sin[2*(c + d*x)] + 2*b^4*Sin[2*(c + d*x)] + 4*a*b^3*Sin[3*(c + d*x)] - b^4*Sin[4*
(c + d*x)])/(a + b*Cos[c + d*x]))/(24*b^5*d)

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fricas [A]  time = 0.51, size = 629, normalized size = 3.33 \[ \left [\frac {6 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x \cos \left (d x + c\right ) + 6 \, {\left (4 \, a^{6} - 5 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x + 3 \, {\left (4 \, a^{5} - 3 \, a^{3} b^{2} + {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (12 \, a^{5} b - 13 \, a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{5} - a b^{7}\right )} d\right )}}, \frac {3 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (4 \, a^{6} - 5 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x - 3 \, {\left (4 \, a^{5} - 3 \, a^{3} b^{2} + {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (12 \, a^{5} b - 13 \, a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left ({\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{5} - a b^{7}\right )} d\right )}}\right ] \]

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))^2,x, algorithm="fricas")

[Out]

[1/6*(6*(4*a^5*b - 5*a^3*b^3 + a*b^5)*d*x*cos(d*x + c) + 6*(4*a^6 - 5*a^4*b^2 + a^2*b^4)*d*x + 3*(4*a^5 - 3*a^
3*b^2 + (4*a^4*b - 3*a^2*b^3)*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^5*b - 13*a^3*b^3 + a*b^5 + (a^2*b^4 - b^6)*cos(d*x + c)^3 - 2*(a^3*b^3 - a*b^5)*cos(d
*x + c)^2 + (6*a^4*b^2 - 7*a^2*b^4 + b^6)*cos(d*x + c))*sin(d*x + c))/((a^2*b^6 - b^8)*d*cos(d*x + c) + (a^3*b
^5 - a*b^7)*d), 1/3*(3*(4*a^5*b - 5*a^3*b^3 + a*b^5)*d*x*cos(d*x + c) + 3*(4*a^6 - 5*a^4*b^2 + a^2*b^4)*d*x -
3*(4*a^5 - 3*a^3*b^2 + (4*a^4*b - 3*a^2*b^3)*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))) - (12*a^5*b - 13*a^3*b^3 + a*b^5 + (a^2*b^4 - b^6)*cos(d*x + c)^3 - 2*(a^3*b^3 - a*b
^5)*cos(d*x + c)^2 + (6*a^4*b^2 - 7*a^2*b^4 + b^6)*cos(d*x + c))*sin(d*x + c))/((a^2*b^6 - b^8)*d*cos(d*x + c)
+ (a^3*b^5 - a*b^7)*d)]

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giac [A]  time = 0.87, size = 280, normalized size = 1.48 \[ -\frac {\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )} b^{4}} - \frac {3 \, {\left (4 \, a^{3} - a b^{2}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {6 \, {\left (4 \, a^{4} - 3 \, a^{2} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{5}} + \frac {2 \, {\left (9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{4}}}{3 \, d} \]

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))^2,x, algorithm="giac")

[Out]

-1/3*(6*a^3*tan(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)*b^4) - 3*(4*a^
3 - a*b^2)*(d*x + c)/b^5 - 6*(4*a^4 - 3*a^2*b^2)*(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)))/(sqrt(a^2 - b^2)*b^5) + 2*(9*a^2*tan(1/2*d*
x + 1/2*c)^5 + 3*a*b*tan(1/2*d*x + 1/2*c)^5 + 18*a^2*tan(1/2*d*x + 1/2*c)^3 - 4*b^2*tan(1/2*d*x + 1/2*c)^3 + 9
*a^2*tan(1/2*d*x + 1/2*c) - 3*a*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^4))/d

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maple [B]  time = 0.10, size = 403, normalized size = 2.13 \[ -\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{4} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}-\frac {8 a^{4} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,b^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {6 a^{2} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {12 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{5}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{3}} \]

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))^2,x)

[Out]

-2/d*a^3/b^4*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)-8/d*a^4/b^5/((a-b)*(a+b))^
(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))+6/d*a^2/b^3/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+
1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))-6/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*a^2-2/d/b^3/(1+tan(1
/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*a-12/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3*a^2+8/3/d/b
^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3-6/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)*a^2+2
/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)*a+8/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^3-2/d/b^3*arctan(t
an(1/2*d*x+1/2*c))*a

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

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mupad [B]  time = 3.73, size = 1652, normalized size = 8.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*tan(c/2 + (d*x)/2)^7*(a*b^2 + 2*a^2*b - 4*a^3))/b^4 - (2*tan(c/2 + (d*x)/2)*(2*a^2*b - a*b^2 + 4*a^3))/b^4
+ (2*tan(c/2 + (d*x)/2)^3*(a*b^2 - 6*a^2*b - 36*a^3 + 4*b^3))/(3*b^4) + (2*tan(c/2 + (d*x)/2)^5*(a*b^2 + 6*a^
2*b - 36*a^3 - 4*b^3))/(3*b^4))/(d*(a + b + tan(c/2 + (d*x)/2)^8*(a - b) + tan(c/2 + (d*x)/2)^2*(4*a + 2*b) +
tan(c/2 + (d*x)/2)^6*(4*a - 2*b) + 6*a*tan(c/2 + (d*x)/2)^4)) + (atan((256*a^5*tan(c/2 + (d*x)/2))/(64*a^4*b +
256*a^5 - 64*a^3*b^2 - (256*a^6)/b) - (64*a^4*tan(c/2 + (d*x)/2))/(64*a^3*b - 64*a^4 - (256*a^5)/b + (256*a^6
)/b^2) - (256*a^6*tan(c/2 + (d*x)/2))/(256*a^5*b - 256*a^6 - 64*a^3*b^3 + 64*a^4*b^2) + (64*a^3*tan(c/2 + (d*x
)/2))/(64*a^3 - (64*a^4)/b - (256*a^5)/b^2 + (256*a^6)/b^3))*(a*b^2*1i - a^3*4i)*2i)/(b^5*d) + (a^2*atan(((a^2
*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((32*tan(c/2 + (d*x)/2)*(64*a^8*b - 32*a^9 + a^2*b^7 - 3*a^3*b^6 + 4
*a^4*b^5 + 14*a^5*b^4 - 32*a^6*b^3 - 16*a^7*b^2))/b^8 + (a^2*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((32*(a*
b^14 - 4*a^2*b^13 + a^3*b^12 + 6*a^4*b^11 - 4*a^5*b^10))/b^12 - (32*a^2*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^
(1/2)*(4*a^2 - 3*b^2)*(2*a*b^12 - 4*a^2*b^11 + 2*a^3*b^10))/(b^8*(b^7 - a^2*b^5))))/(b^7 - a^2*b^5))*1i)/(b^7
- a^2*b^5) + (a^2*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((32*tan(c/2 + (d*x)/2)*(64*a^8*b - 32*a^9 + a^2*b^
7 - 3*a^3*b^6 + 4*a^4*b^5 + 14*a^5*b^4 - 32*a^6*b^3 - 16*a^7*b^2))/b^8 - (a^2*(-(a + b)*(a - b))^(1/2)*(4*a^2
- 3*b^2)*((32*(a*b^14 - 4*a^2*b^13 + a^3*b^12 + 6*a^4*b^11 - 4*a^5*b^10))/b^12 + (32*a^2*tan(c/2 + (d*x)/2)*(-
(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*(2*a*b^12 - 4*a^2*b^11 + 2*a^3*b^10))/(b^8*(b^7 - a^2*b^5))))/(b^7 - a^
2*b^5))*1i)/(b^7 - a^2*b^5))/((64*(96*a^10*b - 64*a^11 - 3*a^4*b^7 - 3*a^5*b^6 + 34*a^6*b^5 + 4*a^7*b^4 - 112*
a^8*b^3 + 48*a^9*b^2))/b^12 + (a^2*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((32*tan(c/2 + (d*x)/2)*(64*a^8*b
- 32*a^9 + a^2*b^7 - 3*a^3*b^6 + 4*a^4*b^5 + 14*a^5*b^4 - 32*a^6*b^3 - 16*a^7*b^2))/b^8 + (a^2*(-(a + b)*(a -
b))^(1/2)*(4*a^2 - 3*b^2)*((32*(a*b^14 - 4*a^2*b^13 + a^3*b^12 + 6*a^4*b^11 - 4*a^5*b^10))/b^12 - (32*a^2*tan(
c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*(2*a*b^12 - 4*a^2*b^11 + 2*a^3*b^10))/(b^8*(b^7 - a^2*
b^5))))/(b^7 - a^2*b^5)))/(b^7 - a^2*b^5) - (a^2*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((32*tan(c/2 + (d*x)
/2)*(64*a^8*b - 32*a^9 + a^2*b^7 - 3*a^3*b^6 + 4*a^4*b^5 + 14*a^5*b^4 - 32*a^6*b^3 - 16*a^7*b^2))/b^8 - (a^2*(
-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((32*(a*b^14 - 4*a^2*b^13 + a^3*b^12 + 6*a^4*b^11 - 4*a^5*b^10))/b^12
+ (32*a^2*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*(2*a*b^12 - 4*a^2*b^11 + 2*a^3*b^10))/(b
^8*(b^7 - a^2*b^5))))/(b^7 - a^2*b^5)))/(b^7 - a^2*b^5)))*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*2i)/(d*(b^7
- a^2*b^5))

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

[Out]

Timed out

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