3.21 \(\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^4} \, dx\)

Optimal. Leaf size=260 \[ \frac{\left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a+b}}\right )}{e (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right ) \sin (d+e x)}{6 e \left (a^2-b^2\right )^3 (a+b \cos (d+e x))}-\frac{\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sin (d+e x)}{6 e \left (a^2-b^2\right )^2 (a+b \cos (d+e x))^2}-\frac{(A b-a B) \sin (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cos (d+e x))^3}+\frac{C}{3 b e (a+b \cos (d+e x))^3} \]

[Out]

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

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Rubi [A]  time = 0.499284, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4377, 2754, 12, 2659, 205, 2668, 32} \[ \frac{\left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a+b}}\right )}{e (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right ) \sin (d+e x)}{6 e \left (a^2-b^2\right )^3 (a+b \cos (d+e x))}-\frac{\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sin (d+e x)}{6 e \left (a^2-b^2\right )^2 (a+b \cos (d+e x))^2}-\frac{(A b-a B) \sin (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cos (d+e x))^3}+\frac{C}{3 b e (a+b \cos (d+e x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^4,x]

[Out]

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

Rule 4377

Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] :> With[{e = FreeFactors[Cos[c*(a +
b*x)], x]}, Int[ActivateTrig[u*v], x] + Dist[d, Int[ActivateTrig[u]*Sin[c*(a + b*x)]^n, x], x] /; FunctionOfQ[
Cos[c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] &&  !FreeQ[v, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Sin] || EqQ[F, sin])

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]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 1.12727, size = 302, normalized size = 1.16 \[ \frac{\frac{-3 b \left (3 a^2 A b^3-24 a^4 A b+14 a^3 b^2 B+8 a^5 B+3 a b^4 B-4 A b^5\right ) \sin (d+e x)+6 b^2 \left (9 a^3 A b-9 a^2 b^2 B-2 a^4 B+a A b^3+b^4 B\right ) \sin (2 (d+e x))+11 a^2 A b^4 \sin (3 (d+e x))-2 a^3 b^3 B \sin (3 (d+e x))+24 a^4 b^2 C-24 a^2 b^4 C-8 a^6 C-13 a b^5 B \sin (3 (d+e x))+4 A b^6 \sin (3 (d+e x))+8 b^6 C}{b \left (b^2-a^2\right )^3 (a+b \cos (d+e x))^3}+\frac{24 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}}{24 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^4,x]

[Out]

((24*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[((a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b
^2)^(7/2) + (-8*a^6*C + 24*a^4*b^2*C - 24*a^2*b^4*C + 8*b^6*C - 3*b*(-24*a^4*A*b + 3*a^2*A*b^3 - 4*A*b^5 + 8*a
^5*B + 14*a^3*b^2*B + 3*a*b^4*B)*Sin[d + e*x] + 6*b^2*(9*a^3*A*b + a*A*b^3 - 2*a^4*B - 9*a^2*b^2*B + b^4*B)*Si
n[2*(d + e*x)] + 11*a^2*A*b^4*Sin[3*(d + e*x)] + 4*A*b^6*Sin[3*(d + e*x)] - 2*a^3*b^3*B*Sin[3*(d + e*x)] - 13*
a*b^5*B*Sin[3*(d + e*x)])/(b*(-a^2 + b^2)^3*(a + b*Cos[d + e*x])^3))/(24*e)

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Maple [B]  time = 0.052, size = 1974, normalized size = 7.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,x)

[Out]

-4/3/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*e*x+1/2*d
)^3*A*b^3+4/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*e*
x+1/2*d)^3*B*a^3-2/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan
(1/2*e*x+1/2*d)*A*b^3+2/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3
)*tan(1/2*e*x+1/2*d)*B*a^3-1/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^
2-b^3)*tan(1/2*e*x+1/2*d)*B*b^3-4/e/(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*e*x
+1/2*d)/((a+b)*(a-b))^(1/2))*B*a^2*b+3/e/(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*e*x+1/2*d)/((a+b)*(a-b))^(1/2))*A*a*b^2-2/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a-b)/(a^3
+3*a^2*b+3*a*b^2+b^3)*tan(1/2*e*x+1/2*d)^5*A*b^3+2/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a-
b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*e*x+1/2*d)^5*B*a^3+1/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b
)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*e*x+1/2*d)^5*B*b^3+2/e/(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*e*x+1/2*d)/((a+b)*(a-b))^(1/2))*A*a^3-1/e/(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*e*x+1/2*d)/((a+b)*(a-b))^(1/2))*B*b^3-4/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1
/2*d)^2*b+a+b)^3*a*C/(a^2-2*a*b+b^2)*tan(1/2*e*x+1/2*d)^2-6/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a
+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*e*x+1/2*d)*A*a^2*b+28/3/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/
2*d)^2*b+a+b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*e*x+1/2*d)^3*B*a*b^2+6/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1
/2*e*x+1/2*d)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*e*x+1/2*d)^5*B*a*b^2+2/e/(tan(1/2*e*x+1/2*d)^
2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*e*x+1/2*d)^5*B*a^2*b+3/e/(tan(1/2*e*
x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*e*x+1/2*d)*A*a*b^2-2/e/(tan
(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*e*x+1/2*d)*B*a^2*b-1
2/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*e*x+1/2*d)^3
*A*a^2*b+6/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*e*x
+1/2*d)*B*a*b^2-6/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(
1/2*e*x+1/2*d)^5*A*a^2*b-3/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+
b^3)*tan(1/2*e*x+1/2*d)^5*A*a*b^2-2/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3*C/(a^3-3*a^2*b+3*a
*b^2-b^3)*a^2-2/3/e/(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3*C/(a^3-3*a^2*b+3*a*b^2-b^3)*b^2-2/e/
(tan(1/2*e*x+1/2*d)^2*a-tan(1/2*e*x+1/2*d)^2*b+a+b)^3*C/(a-b)*tan(1/2*e*x+1/2*d)^4

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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((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.15324, size = 2912, normalized size = 11.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,x, algorithm="fricas")

[Out]

[1/12*(4*C*a^8 - 16*C*a^6*b^2 + 24*C*a^4*b^4 - 16*C*a^2*b^6 + 4*C*b^8 - 3*(2*A*a^6*b - 4*B*a^5*b^2 + 3*A*a^4*b
^3 - B*a^3*b^4 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cos(e*x + d)^3 + 3*(2*A*a^4*b^3 - 4*B*a^3*b^4
+ 3*A*a^2*b^5 - B*a*b^6)*cos(e*x + d)^2 + 3*(2*A*a^5*b^2 - 4*B*a^4*b^3 + 3*A*a^3*b^4 - B*a^2*b^5)*cos(e*x + d
))*sqrt(-a^2 + b^2)*log((2*a*b*cos(e*x + d) + (2*a^2 - b^2)*cos(e*x + d)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(e*x + d
) + b)*sin(e*x + d) - a^2 + 2*b^2)/(b^2*cos(e*x + d)^2 + 2*a*b*cos(e*x + d) + a^2)) + 2*(6*B*a^7*b - 18*A*a^6*
b^2 + 4*B*a^5*b^3 + 23*A*a^4*b^4 - 11*B*a^3*b^5 - 7*A*a^2*b^6 + B*a*b^7 + 2*A*b^8 + (2*B*a^5*b^3 - 11*A*a^4*b^
4 + 11*B*a^3*b^5 + 7*A*a^2*b^6 - 13*B*a*b^7 + 4*A*b^8)*cos(e*x + d)^2 + 3*(2*B*a^6*b^2 - 9*A*a^5*b^3 + 7*B*a^4
*b^4 + 8*A*a^3*b^5 - 10*B*a^2*b^6 + A*a*b^7 + B*b^8)*cos(e*x + d))*sin(e*x + d))/((a^8*b^4 - 4*a^6*b^6 + 6*a^4
*b^8 - 4*a^2*b^10 + b^12)*e*cos(e*x + d)^3 + 3*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cos(e*
x + d)^2 + 3*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*e*cos(e*x + d) + (a^11*b - 4*a^9*b^3 +
6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9)*e), 1/6*(2*C*a^8 - 8*C*a^6*b^2 + 12*C*a^4*b^4 - 8*C*a^2*b^6 + 2*C*b^8 + 3*(2*
A*a^6*b - 4*B*a^5*b^2 + 3*A*a^4*b^3 - B*a^3*b^4 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cos(e*x + d)
^3 + 3*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cos(e*x + d)^2 + 3*(2*A*a^5*b^2 - 4*B*a^4*b^3 + 3*A
*a^3*b^4 - B*a^2*b^5)*cos(e*x + d))*sqrt(a^2 - b^2)*arctan(-(a*cos(e*x + d) + b)/(sqrt(a^2 - b^2)*sin(e*x + d)
)) + (6*B*a^7*b - 18*A*a^6*b^2 + 4*B*a^5*b^3 + 23*A*a^4*b^4 - 11*B*a^3*b^5 - 7*A*a^2*b^6 + B*a*b^7 + 2*A*b^8 +
(2*B*a^5*b^3 - 11*A*a^4*b^4 + 11*B*a^3*b^5 + 7*A*a^2*b^6 - 13*B*a*b^7 + 4*A*b^8)*cos(e*x + d)^2 + 3*(2*B*a^6*
b^2 - 9*A*a^5*b^3 + 7*B*a^4*b^4 + 8*A*a^3*b^5 - 10*B*a^2*b^6 + A*a*b^7 + B*b^8)*cos(e*x + d))*sin(e*x + d))/((
a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cos(e*x + d)^3 + 3*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4
*a^3*b^9 + a*b^11)*e*cos(e*x + d)^2 + 3*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*e*cos(e*x +
d) + (a^11*b - 4*a^9*b^3 + 6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9)*e)]

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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((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.30711, size = 1296, normalized size = 4.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,x, algorithm="giac")

[Out]

-1/3*(3*(2*A*a^3 - 4*B*a^2*b + 3*A*a*b^2 - B*b^3)*(pi*floor(1/2*(x*e + d)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-
(a*tan(1/2*x*e + 1/2*d) - b*tan(1/2*x*e + 1/2*d))/sqrt(a^2 - b^2)))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(
a^2 - b^2)) - (6*B*a^5*tan(1/2*x*e + 1/2*d)^5 - 18*A*a^4*b*tan(1/2*x*e + 1/2*d)^5 - 6*B*a^4*b*tan(1/2*x*e + 1/
2*d)^5 + 27*A*a^3*b^2*tan(1/2*x*e + 1/2*d)^5 + 12*B*a^3*b^2*tan(1/2*x*e + 1/2*d)^5 - 6*A*a^2*b^3*tan(1/2*x*e +
1/2*d)^5 - 27*B*a^2*b^3*tan(1/2*x*e + 1/2*d)^5 + 3*A*a*b^4*tan(1/2*x*e + 1/2*d)^5 + 12*B*a*b^4*tan(1/2*x*e +
1/2*d)^5 - 6*A*b^5*tan(1/2*x*e + 1/2*d)^5 + 3*B*b^5*tan(1/2*x*e + 1/2*d)^5 - 6*C*a^5*tan(1/2*x*e + 1/2*d)^4 -
6*C*a^4*b*tan(1/2*x*e + 1/2*d)^4 + 12*C*a^3*b^2*tan(1/2*x*e + 1/2*d)^4 + 12*C*a^2*b^3*tan(1/2*x*e + 1/2*d)^4 -
6*C*a*b^4*tan(1/2*x*e + 1/2*d)^4 - 6*C*b^5*tan(1/2*x*e + 1/2*d)^4 + 12*B*a^5*tan(1/2*x*e + 1/2*d)^3 - 36*A*a^
4*b*tan(1/2*x*e + 1/2*d)^3 + 16*B*a^3*b^2*tan(1/2*x*e + 1/2*d)^3 + 32*A*a^2*b^3*tan(1/2*x*e + 1/2*d)^3 - 28*B*
a*b^4*tan(1/2*x*e + 1/2*d)^3 + 4*A*b^5*tan(1/2*x*e + 1/2*d)^3 - 12*C*a^5*tan(1/2*x*e + 1/2*d)^2 - 24*C*a^4*b*t
an(1/2*x*e + 1/2*d)^2 + 24*C*a^2*b^3*tan(1/2*x*e + 1/2*d)^2 + 12*C*a*b^4*tan(1/2*x*e + 1/2*d)^2 + 6*B*a^5*tan(
1/2*x*e + 1/2*d) - 18*A*a^4*b*tan(1/2*x*e + 1/2*d) + 6*B*a^4*b*tan(1/2*x*e + 1/2*d) - 27*A*a^3*b^2*tan(1/2*x*e
+ 1/2*d) + 12*B*a^3*b^2*tan(1/2*x*e + 1/2*d) - 6*A*a^2*b^3*tan(1/2*x*e + 1/2*d) + 27*B*a^2*b^3*tan(1/2*x*e +
1/2*d) - 3*A*a*b^4*tan(1/2*x*e + 1/2*d) + 12*B*a*b^4*tan(1/2*x*e + 1/2*d) - 6*A*b^5*tan(1/2*x*e + 1/2*d) - 3*B
*b^5*tan(1/2*x*e + 1/2*d) - 6*C*a^5 - 18*C*a^4*b - 20*C*a^3*b^2 - 12*C*a^2*b^3 - 6*C*a*b^4 - 2*C*b^5)/((a^6 -
3*a^4*b^2 + 3*a^2*b^4 - b^6)*(a*tan(1/2*x*e + 1/2*d)^2 - b*tan(1/2*x*e + 1/2*d)^2 + a + b)^3))*e^(-1)