3.1013 \(\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^5} \, dx\)

Optimal. Leaf size=249 \[ \frac{\left (-7 a^2 b^2 C+2 a^3 b B-2 a^4 C+3 a b^3 B-b^4 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left (11 a^2 b B-13 a^3 C-17 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{b \left (-7 a^2 C+5 a b B-3 b^2 C\right ) \sin (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{b (b B-2 a C) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

[Out]

((2*a^3*b*B + 3*a*b^3*B - 2*a^4*C - 7*a^2*b^2*C - b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(
(a - b)^(7/2)*(a + b)^(7/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (b*
(5*a*b*B - 7*a^2*C - 3*b^2*C)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (b*(11*a^2*b*B + 4*b^
3*B - 13*a^3*C - 17*a*b^2*C)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.746219, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.104, Rules used = {24, 2754, 12, 2659, 205} \[ \frac{\left (-7 a^2 b^2 C+2 a^3 b B-2 a^4 C+3 a b^3 B-b^4 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left (11 a^2 b B-13 a^3 C-17 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{b \left (-7 a^2 C+5 a b B-3 b^2 C\right ) \sin (c+d x)}{6 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{b (b B-2 a C) \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^5,x]

[Out]

((2*a^3*b*B + 3*a*b^3*B - 2*a^4*C - 7*a^2*b^2*C - b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(
(a - b)^(7/2)*(a + b)^(7/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (b*
(5*a*b*B - 7*a^2*C - 3*b^2*C)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (b*(11*a^2*b*B + 4*b^
3*B - 13*a^3*C - 17*a*b^2*C)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rule 2754

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^5} \, dx &=\frac{\int \frac{b^2 (b B-a C)+b^3 C \cos (c+d x)}{(a+b \cos (c+d x))^4} \, dx}{b^2}\\ &=-\frac{b (b B-2 a C) \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\int \frac{-3 b^2 \left (a b B-a^2 C-b^2 C\right )+2 b^3 (b B-2 a C) \cos (c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=-\frac{b (b B-2 a C) \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{b \left (5 a b B-7 a^2 C-3 b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\int \frac{2 b^2 \left (3 a^2 b B+2 b^3 B-3 a^3 C-7 a b^2 C\right )-b^3 \left (5 a b B-7 a^2 C-3 b^2 C\right ) \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{b (b B-2 a C) \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{b \left (5 a b B-7 a^2 C-3 b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{b \left (11 a^2 b B+4 b^3 B-13 a^3 C-17 a b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\int -\frac{3 b^2 \left (2 a^3 b B+3 a b^3 B-2 a^4 C-7 a^2 b^2 C-b^4 C\right )}{a+b \cos (c+d x)} \, dx}{6 b^2 \left (a^2-b^2\right )^3}\\ &=-\frac{b (b B-2 a C) \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{b \left (5 a b B-7 a^2 C-3 b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{b \left (11 a^2 b B+4 b^3 B-13 a^3 C-17 a b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (2 a^3 b B+3 a b^3 B-2 a^4 C-7 a^2 b^2 C-b^4 C\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=-\frac{b (b B-2 a C) \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{b \left (5 a b B-7 a^2 C-3 b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{b \left (11 a^2 b B+4 b^3 B-13 a^3 C-17 a b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (2 a^3 b B+3 a b^3 B-2 a^4 C-7 a^2 b^2 C-b^4 C\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 )}{\left (a^2-b^2\right )^3 d}\\ &=\frac{\left (2 a^3 b B+3 a b^3 B-2 a^4 C-7 a^2 b^2 C-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)^{7/2} (a+b)^{7/2} d}-\frac{b (b B-2 a C) \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{b \left (5 a b B-7 a^2 C-3 b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{b \left (11 a^2 b B+4 b^3 B-13 a^3 C-17 a b^2 C\right ) \sin (c+d x)}{6 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 1.00508, size = 246, normalized size = 0.99 \[ \frac{\frac{24 \left (7 a^2 b^2 C-2 a^3 b B+2 a^4 C-3 a b^3 B+b^4 C\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}-\frac{2 b \sin (c+d x) \left (6 b \left (-10 a^2 b^2 C+9 a^3 b B-11 a^4 C+a b^3 B+b^4 C\right ) \cos (c+d x)+b^2 \left (11 a^2 b B-13 a^3 C-17 a b^2 C+4 b^3 B\right ) \cos (2 (c+d x))+a^2 b^3 B-23 a^3 b^2 C+36 a^4 b B-48 a^5 C-19 a b^4 C+8 b^5 B\right )}{(a+b \cos (c+d x))^3}}{24 d \left (a^2-b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^5,x]

[Out]

((24*(-2*a^3*b*B - 3*a*b^3*B + 2*a^4*C + 7*a^2*b^2*C + b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b
^2]])/Sqrt[-a^2 + b^2] - (2*b*(36*a^4*b*B + a^2*b^3*B + 8*b^5*B - 48*a^5*C - 23*a^3*b^2*C - 19*a*b^4*C + 6*b*(
9*a^3*b*B + a*b^3*B - 11*a^4*C - 10*a^2*b^2*C + b^4*C)*Cos[c + d*x] + b^2*(11*a^2*b*B + 4*b^3*B - 13*a^3*C - 1
7*a*b^2*C)*Cos[2*(c + d*x)])*Sin[c + d*x])/(a + b*Cos[c + d*x])^3)/(24*(a^2 - b^2)^3*d)

________________________________________________________________________________________

Maple [B]  time = 0.049, size = 1817, normalized size = 7.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*b*B-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^5,x)

[Out]

-6/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^2/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2
*c)^5*a^2*B-3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(
1/2*d*x+1/2*c)^5*a*B-2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+
b^3)*tan(1/2*d*x+1/2*c)^5*B+8/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b/(a-b)/(a^3+3*a^2*b+3*a
*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*a^3*C+5/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^2/(a-b)/(a^3+
3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*a^2*C+8/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^3/
(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*a+1/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+
b)^3*b^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C-12/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c
)^2*b+a+b)^3*b^2/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*a^2*B-4/3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(
1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+16/d/(a*tan(1/2*d*x+1/2*c
)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*a^3*C+32/3/d/(a*tan(1
/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*a-6/d
/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^2/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*
a^2*B+3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*
x+1/2*c)*a*B-2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan
(1/2*d*x+1/2*c)*B+8/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*
tan(1/2*d*x+1/2*c)*a^3*C-5/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^2/(a+b)/(a^3-3*a^2*b+3*a*
b^2-b^3)*tan(1/2*d*x+1/2*c)*a^2*C+8/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^3/(a+b)/(a^3-3*a
^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C*a-1/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a+b)/(
a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+2/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-
b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a^3*b*B+3/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arcta
n((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a*b^3*B-2/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*
arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a^4*C-7/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/
2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a^2*b^2*C-1/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-
b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C*b^4

________________________________________________________________________________________

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((a*b*B-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.19836, size = 2859, normalized size = 11.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

[1/12*(3*(2*C*a^7 - 2*B*a^6*b + 7*C*a^5*b^2 - 3*B*a^4*b^3 + C*a^3*b^4 + (2*C*a^4*b^3 - 2*B*a^3*b^4 + 7*C*a^2*b
^5 - 3*B*a*b^6 + C*b^7)*cos(d*x + c)^3 + 3*(2*C*a^5*b^2 - 2*B*a^4*b^3 + 7*C*a^3*b^4 - 3*B*a^2*b^5 + C*a*b^6)*c
os(d*x + c)^2 + 3*(2*C*a^6*b - 2*B*a^5*b^2 + 7*C*a^4*b^3 - 3*B*a^3*b^4 + 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*cos(d*x + c) + a^2)) + 2*(24*C*a^7*b - 18*B*a^6*b^2 - 19*C*a^5
*b^3 + 23*B*a^4*b^4 - 4*C*a^3*b^5 - 7*B*a^2*b^6 - C*a*b^7 + 2*B*b^8 + (13*C*a^5*b^3 - 11*B*a^4*b^4 + 4*C*a^3*b
^5 + 7*B*a^2*b^6 - 17*C*a*b^7 + 4*B*b^8)*cos(d*x + c)^2 + 3*(11*C*a^6*b^2 - 9*B*a^5*b^3 - C*a^4*b^4 + 8*B*a^3*
b^5 - 11*C*a^2*b^6 + B*a*b^7 + C*b^8)*cos(d*x + c))*sin(d*x + c))/((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^
9 + b^11)*d*cos(d*x + c)^3 + 3*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cos(d*x + c)^2 + 3*(a^
10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c) + (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6
+ a^3*b^8)*d), -1/6*(3*(2*C*a^7 - 2*B*a^6*b + 7*C*a^5*b^2 - 3*B*a^4*b^3 + C*a^3*b^4 + (2*C*a^4*b^3 - 2*B*a^3*
b^4 + 7*C*a^2*b^5 - 3*B*a*b^6 + C*b^7)*cos(d*x + c)^3 + 3*(2*C*a^5*b^2 - 2*B*a^4*b^3 + 7*C*a^3*b^4 - 3*B*a^2*b
^5 + C*a*b^6)*cos(d*x + c)^2 + 3*(2*C*a^6*b - 2*B*a^5*b^2 + 7*C*a^4*b^3 - 3*B*a^3*b^4 + 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))) - (24*C*a^7*b - 18*B*a^6*b^2 -
19*C*a^5*b^3 + 23*B*a^4*b^4 - 4*C*a^3*b^5 - 7*B*a^2*b^6 - C*a*b^7 + 2*B*b^8 + (13*C*a^5*b^3 - 11*B*a^4*b^4 +
4*C*a^3*b^5 + 7*B*a^2*b^6 - 17*C*a*b^7 + 4*B*b^8)*cos(d*x + c)^2 + 3*(11*C*a^6*b^2 - 9*B*a^5*b^3 - C*a^4*b^4 +
8*B*a^3*b^5 - 11*C*a^2*b^6 + B*a*b^7 + C*b^8)*cos(d*x + c))*sin(d*x + c))/((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 -
4*a^2*b^9 + b^11)*d*cos(d*x + c)^3 + 3*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cos(d*x + c)^
2 + 3*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c) + (a^11 - 4*a^9*b^2 + 6*a^7*b^4 -
4*a^5*b^6 + a^3*b^8)*d)]

________________________________________________________________________________________

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((a*b*B-a**2*C+b**2*B*cos(d*x+c)+b**2*C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.44506, size = 960, normalized size = 3.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^5,x, algorithm="giac")

[Out]

-1/3*(3*(2*C*a^4 - 2*B*a^3*b + 7*C*a^2*b^2 - 3*B*a*b^3 + C*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^6 - 3*a^4*b^2 + 3*a^2*b^4
- b^6)*sqrt(a^2 - b^2)) - (24*C*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 18*B*a^4*b^2*tan(1/2*d*x + 1/2*c)^5 - 33*C*a^4*
b^2*tan(1/2*d*x + 1/2*c)^5 + 27*B*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 + 18*C*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*B*a
^2*b^4*tan(1/2*d*x + 1/2*c)^5 - 30*C*a^2*b^4*tan(1/2*d*x + 1/2*c)^5 + 3*B*a*b^5*tan(1/2*d*x + 1/2*c)^5 + 18*C*
a*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*B*b^6*tan(1/2*d*x + 1/2*c)^5 + 3*C*b^6*tan(1/2*d*x + 1/2*c)^5 + 48*C*a^5*b*ta
n(1/2*d*x + 1/2*c)^3 - 36*B*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 - 16*C*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 32*B*a^2*b^
4*tan(1/2*d*x + 1/2*c)^3 - 32*C*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 4*B*b^6*tan(1/2*d*x + 1/2*c)^3 + 24*C*a^5*b*tan
(1/2*d*x + 1/2*c) - 18*B*a^4*b^2*tan(1/2*d*x + 1/2*c) + 33*C*a^4*b^2*tan(1/2*d*x + 1/2*c) - 27*B*a^3*b^3*tan(1
/2*d*x + 1/2*c) + 18*C*a^3*b^3*tan(1/2*d*x + 1/2*c) - 6*B*a^2*b^4*tan(1/2*d*x + 1/2*c) + 30*C*a^2*b^4*tan(1/2*
d*x + 1/2*c) - 3*B*a*b^5*tan(1/2*d*x + 1/2*c) + 18*C*a*b^5*tan(1/2*d*x + 1/2*c) - 6*B*b^6*tan(1/2*d*x + 1/2*c)
- 3*C*b^6*tan(1/2*d*x + 1/2*c))/((a^6 - 3*a^4*b^2 + 3*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)^3))/d