3.209 \(\int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx\)

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

[Out]

(2*A*a^3+3*A*a*b^2-4*B*a^2*b-B*b^3)*arctanh((a-b)^(1/2)*tanh(1/2*e*x+1/2*d)/(a+b)^(1/2))/(a-b)^(7/2)/(a+b)^(7/
2)/e-1/3*C/b/e/(a+b*cosh(e*x+d))^3-1/3*(A*b-B*a)*sinh(e*x+d)/(a^2-b^2)/e/(a+b*cosh(e*x+d))^3-1/6*(5*A*a*b-2*B*
a^2-3*B*b^2)*sinh(e*x+d)/(a^2-b^2)^2/e/(a+b*cosh(e*x+d))^2-1/6*(11*A*a^2*b+4*A*b^3-2*B*a^3-13*B*a*b^2)*sinh(e*
x+d)/(a^2-b^2)^3/e/(a+b*cosh(e*x+d))

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Rubi [A]  time = 0.45, antiderivative size = 260, normalized size of antiderivative = 1.00, 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} \[ \frac {\left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{e (a-b)^{7/2} (a+b)^{7/2}}-\frac {\left (11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right ) \sinh (d+e x)}{6 e \left (a^2-b^2\right )^3 (a+b \cosh (d+e x))}-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{6 e \left (a^2-b^2\right )^2 (a+b \cosh (d+e x))^2}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[(Sqrt[a - b]*Tanh[(d + e*x)/2])/Sqrt[a + b]])/((a - b)^(7/2
)*(a + b)^(7/2)*e) - C/(3*b*e*(a + b*Cosh[d + e*x])^3) - ((A*b - a*B)*Sinh[d + e*x])/(3*(a^2 - b^2)*e*(a + b*C
osh[d + e*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sinh[d + e*x])/(6*(a^2 - b^2)^2*e*(a + b*Cosh[d + e*x])^2) -
((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sinh[d + e*x])/(6*(a^2 - b^2)^3*e*(a + b*Cosh[d + e*x]))

Rule 12

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

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]

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 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 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 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])

Rubi steps

\begin {align*} \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx &=C \int \frac {\sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx+\int \frac {A+B \cosh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx\\ &=-\frac {(A b-a B) \sinh (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cosh (d+e x))^3}-\frac {\int \frac {-3 (a A-b B)+2 (A b-a B) \cosh (d+e x)}{(a+b \cosh (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 \cosh (d+e x)\right )}{b e}\\ &=-\frac {C}{3 b e (a+b \cosh (d+e x))^3}-\frac {(A b-a B) \sinh (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cosh (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cosh (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 ) \cosh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx}{6 \left (a^2-b^2\right )^2}\\ &=-\frac {C}{3 b e (a+b \cosh (d+e x))^3}-\frac {(A b-a B) \sinh (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cosh (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cosh (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 ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cosh (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 \cosh (d+e x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=-\frac {C}{3 b e (a+b \cosh (d+e x))^3}-\frac {(A b-a B) \sinh (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cosh (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cosh (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 ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cosh (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 \cosh (d+e x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=-\frac {C}{3 b e (a+b \cosh (d+e x))^3}-\frac {(A b-a B) \sinh (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cosh (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cosh (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 ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cosh (d+e x))}-\frac {\left (i \left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (i d+i 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 ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \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 \cosh (d+e x))^3}-\frac {(A b-a B) \sinh (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cosh (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cosh (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 ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cosh (d+e x))}\\ \end {align*}

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Mathematica [A]  time = 2.57, size = 245, normalized size = 0.94 \[ \frac {\frac {2 C \left (b^2-a^2\right )-2 b (A b-a B) \sinh (d+e x)}{b (a-b) (a+b) (a+b \cosh (d+e x))^3}+\frac {\left (2 a^2 B-5 a A b+3 b^2 B\right ) \sinh (d+e x)}{(a-b)^2 (a+b)^2 (a+b \cosh (d+e x))^2}+\frac {6 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}+\frac {\left (2 a^3 B-11 a^2 A b+13 a b^2 B-4 A b^3\right ) \sinh (d+e x)}{(a-b)^3 (a+b)^3 (a+b \cosh (d+e x))}}{6 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[((a - b)*Tanh[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^
2)^(7/2) + ((-5*a*A*b + 2*a^2*B + 3*b^2*B)*Sinh[d + e*x])/((a - b)^2*(a + b)^2*(a + b*Cosh[d + e*x])^2) + ((-1
1*a^2*A*b - 4*A*b^3 + 2*a^3*B + 13*a*b^2*B)*Sinh[d + e*x])/((a - b)^3*(a + b)^3*(a + b*Cosh[d + e*x])) + (2*(-
a^2 + b^2)*C - 2*b*(A*b - a*B)*Sinh[d + e*x])/((a - b)*b*(a + b)*(a + b*Cosh[d + e*x])^3))/(6*e)

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fricas [B]  time = 0.69, size = 8531, normalized size = 32.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(4*B*a^5*b^3 - 22*A*a^4*b^4 + 22*B*a^3*b^5 + 14*A*a^2*b^6 - 26*B*a*b^7 + 8*A*b^8 - 6*(2*A*a^5*b^3 - 4*B*
a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(e*x + d)^5 - 6*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*
b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*sinh(e*x + d)^5 - 30*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b
^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(e*x + d)^4 - 30*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^
2*b^6 + B*a*b^7 + (2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(e*x + d))*sin
h(e*x + d)^4 + 4*(4*(B + C)*a^8 - 22*A*a^7*b + 4*(7*B - 4*C)*a^6*b^2 - 19*A*a^5*b^3 + (7*B + 24*C)*a^4*b^4 + 2
9*A*a^3*b^5 - (39*B + 16*C)*a^2*b^6 + 12*A*a*b^7 + 4*C*b^8)*cosh(e*x + d)^3 + 4*(4*(B + C)*a^8 - 22*A*a^7*b +
4*(7*B - 4*C)*a^6*b^2 - 19*A*a^5*b^3 + (7*B + 24*C)*a^4*b^4 + 29*A*a^3*b^5 - (39*B + 16*C)*a^2*b^6 + 12*A*a*b^
7 + 4*C*b^8 - 15*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(e*x + d)^2 - 3
0*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(e*x + d))*sinh(e*x + d)^3
+ 12*(4*B*a^7*b - 17*A*a^6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b
^8)*cosh(e*x + d)^2 + 12*(4*B*a^7*b - 17*A*a^6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6
- 4*B*a*b^7 + 2*A*b^8 - 5*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(e*x +
d)^3 - 15*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(e*x + d)^2 + (4*
(B + C)*a^8 - 22*A*a^7*b + 4*(7*B - 4*C)*a^6*b^2 - 19*A*a^5*b^3 + (7*B + 24*C)*a^4*b^4 + 29*A*a^3*b^5 - (39*B
+ 16*C)*a^2*b^6 + 12*A*a*b^7 + 4*C*b^8)*cosh(e*x + d))*sinh(e*x + d)^2 - 3*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*
b^6 - B*b^7 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3
*A*a*b^6 - B*b^7)*sinh(e*x + d)^6 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(e*x + d)^5 + 6*
(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x
+ d))*sinh(e*x + d)^5 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e
*x + d)^4 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7 + 5*(2*A*a^3*b^4 -
4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^2 + 10*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh
(e*x + d))*sinh(e*x + d)^4 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^
6)*cosh(e*x + d)^3 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6 + 5*(2
*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^3 + 15*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 -
B*a*b^6)*cosh(e*x + d)^2 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cos
h(e*x + d))*sinh(e*x + d)^3 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*
cosh(e*x + d)^2 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7 + 5*(2*A*a^3*
b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^4 + 20*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6
)*cosh(e*x + d)^3 + 6*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x +
d)^2 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6)*cosh(e*x + d))*sin
h(e*x + d)^2 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(e*x + d) + 6*(2*A*a^4*b^3 - 4*B*a^3*
b^4 + 3*A*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^5 + 5*(2*A*a^4*b^3
- 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(e*x + d)^4 + 2*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a
^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^3 + 2*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a
^2*b^5 - 3*B*a*b^6)*cosh(e*x + d)^2 + (8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B
*b^7)*cosh(e*x + d))*sinh(e*x + d))*sqrt(a^2 - b^2)*log((b^2*cosh(e*x + d)^2 + b^2*sinh(e*x + d)^2 + 2*a*b*cos
h(e*x + d) + 2*a^2 - b^2 + 2*(b^2*cosh(e*x + d) + a*b)*sinh(e*x + d) - 2*sqrt(a^2 - b^2)*(b*cosh(e*x + d) + b*
sinh(e*x + d) + a))/(b*cosh(e*x + d)^2 + b*sinh(e*x + d)^2 + 2*a*cosh(e*x + d) + 2*(b*cosh(e*x + d) + a)*sinh(
e*x + d) + b)) + 6*(4*B*a^6*b^2 - 20*A*a^5*b^3 + 18*B*a^4*b^4 + 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 + B*b^
8)*cosh(e*x + d) + 6*(4*B*a^6*b^2 - 20*A*a^5*b^3 + 18*B*a^4*b^4 + 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 + B*
b^8 - 5*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(e*x + d)^4 - 20*(2*A*a^
6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(e*x + d)^3 + 2*(4*(B + C)*a^8 - 22
*A*a^7*b + 4*(7*B - 4*C)*a^6*b^2 - 19*A*a^5*b^3 + (7*B + 24*C)*a^4*b^4 + 29*A*a^3*b^5 - (39*B + 16*C)*a^2*b^6
+ 12*A*a*b^7 + 4*C*b^8)*cosh(e*x + d)^2 + 4*(4*B*a^7*b - 17*A*a^6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3
*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8)*cosh(e*x + d))*sinh(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*cosh(e*x + d)^6 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*sinh(e*x + d)^6 +
6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d)^5 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*
a^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d)^4 + 6*((a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*co
sh(e*x + d) + (a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e)*sinh(e*x + d)^5 + 4*(2*a^11*b - 5*a^9*
b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*e*cosh(e*x + d)^3 + 3*(5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^
10 + b^12)*e*cosh(e*x + d)^2 + 10*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d) + (4*
a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e)*sinh(e*x + d)^4 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a
^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d)^2 + 4*(5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*c
osh(e*x + d)^3 + 15*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d)^2 + 3*(4*a^10*b^2 -
15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d) + (2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^
9 + 3*a*b^11)*e)*sinh(e*x + d)^3 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d) +
3*(5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cosh(e*x + d)^4 + 20*(a^9*b^3 - 4*a^7*b^5 + 6*a^5
*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d)^3 + 6*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*
cosh(e*x + d)^2 + 4*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*e*cosh(e*x + d) + (4*a^10*b^2
- 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e)*sinh(e*x + d)^2 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b
^10 + b^12)*e + 6*((a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cosh(e*x + d)^5 + 5*(a^9*b^3 - 4*a^
7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d)^4 + 2*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b
^8 + b^12)*e*cosh(e*x + d)^3 + 2*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*e*cosh(e*x + d)^2
+ (4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d) + (a^9*b^3 - 4*a^7*b^5 + 6*a^5*b
^7 - 4*a^3*b^9 + a*b^11)*e)*sinh(e*x + d)), -1/3*(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 - 3*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(e*x + d)
^5 - 3*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 + B*b^8)*sinh(e*x + d)^5 - 15*(2*A*a^6
*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7)*cosh(e*x + d)^4 - 15*(2*A*a^6*b^2 - 4*B*
a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*a*b^7 + (2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b
^6 - 3*A*a*b^7 + B*b^8)*cosh(e*x + d))*sinh(e*x + d)^4 + 2*(4*(B + C)*a^8 - 22*A*a^7*b + 4*(7*B - 4*C)*a^6*b^2
- 19*A*a^5*b^3 + (7*B + 24*C)*a^4*b^4 + 29*A*a^3*b^5 - (39*B + 16*C)*a^2*b^6 + 12*A*a*b^7 + 4*C*b^8)*cosh(e*x
+ d)^3 + 2*(4*(B + C)*a^8 - 22*A*a^7*b + 4*(7*B - 4*C)*a^6*b^2 - 19*A*a^5*b^3 + (7*B + 24*C)*a^4*b^4 + 29*A*a
^3*b^5 - (39*B + 16*C)*a^2*b^6 + 12*A*a*b^7 + 4*C*b^8 - 15*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^
6 - 3*A*a*b^7 + B*b^8)*cosh(e*x + d)^2 - 30*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6
+ B*a*b^7)*cosh(e*x + d))*sinh(e*x + d)^3 + 6*(4*B*a^7*b - 17*A*a^6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*
a^3*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8)*cosh(e*x + d)^2 + 6*(4*B*a^7*b - 17*A*a^6*b^2 + 13*B*a^5*b^3 + 11
*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8 - 5*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B
*a^2*b^6 - 3*A*a*b^7 + B*b^8)*cosh(e*x + d)^3 - 15*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*
a^2*b^6 + B*a*b^7)*cosh(e*x + d)^2 + (4*(B + C)*a^8 - 22*A*a^7*b + 4*(7*B - 4*C)*a^6*b^2 - 19*A*a^5*b^3 + (7*B
+ 24*C)*a^4*b^4 + 29*A*a^3*b^5 - (39*B + 16*C)*a^2*b^6 + 12*A*a*b^7 + 4*C*b^8)*cosh(e*x + d))*sinh(e*x + d)^2
+ 3*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x
+ d)^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*sinh(e*x + d)^6 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A
*a^2*b^5 - B*a*b^6)*cosh(e*x + d)^5 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 - 4*
B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d))*sinh(e*x + d)^5 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 -
8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^4 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5
+ 3*A*a*b^6 - B*b^7 + 5*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^2 + 10*(2*A*a^4*b^3 - 4
*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(e*x + d))*sinh(e*x + d)^4 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3
- 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6)*cosh(e*x + d)^3 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*
a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6 + 5*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^3 + 15*(2*
A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(e*x + d)^2 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^
4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d))*sinh(e*x + d)^3 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3
*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^2 + 3*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a
^2*b^5 + 3*A*a*b^6 - B*b^7 + 5*(2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^4 + 20*(2*A*a^4*b
^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(e*x + d)^3 + 6*(8*A*a^5*b^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B
*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^2 + 4*(4*A*a^6*b - 8*B*a^5*b^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A
*a^2*b^5 - 3*B*a*b^6)*cosh(e*x + d))*sinh(e*x + d)^2 + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*c
osh(e*x + d) + 6*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 -
B*b^7)*cosh(e*x + d)^5 + 5*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cosh(e*x + d)^4 + 2*(8*A*a^5*b
^2 - 16*B*a^4*b^3 + 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d)^3 + 2*(4*A*a^6*b - 8*B*a^5*b
^2 + 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 - 3*B*a*b^6)*cosh(e*x + d)^2 + (8*A*a^5*b^2 - 16*B*a^4*b^3 + 14
*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cosh(e*x + d))*sinh(e*x + d))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2
+ b^2)*(b*cosh(e*x + d) + b*sinh(e*x + d) + a)/(a^2 - b^2)) + 3*(4*B*a^6*b^2 - 20*A*a^5*b^3 + 18*B*a^4*b^4 +
15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 + B*b^8)*cosh(e*x + d) + 3*(4*B*a^6*b^2 - 20*A*a^5*b^3 + 18*B*a^4*b^4
+ 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 + B*b^8 - 5*(2*A*a^5*b^3 - 4*B*a^4*b^4 + A*a^3*b^5 + 3*B*a^2*b^6 - 3
*A*a*b^7 + B*b^8)*cosh(e*x + d)^4 - 20*(2*A*a^6*b^2 - 4*B*a^5*b^3 + A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 + B*
a*b^7)*cosh(e*x + d)^3 + 2*(4*(B + C)*a^8 - 22*A*a^7*b + 4*(7*B - 4*C)*a^6*b^2 - 19*A*a^5*b^3 + (7*B + 24*C)*a
^4*b^4 + 29*A*a^3*b^5 - (39*B + 16*C)*a^2*b^6 + 12*A*a*b^7 + 4*C*b^8)*cosh(e*x + d)^2 + 4*(4*B*a^7*b - 17*A*a^
6*b^2 + 13*B*a^5*b^3 + 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 - 4*B*a*b^7 + 2*A*b^8)*cosh(e*x + d))*sinh(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*cosh(e*x + d)^6 + (a^8*b^4 - 4*a^6*b^6 + 6*a^
4*b^8 - 4*a^2*b^10 + b^12)*e*sinh(e*x + d)^6 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh
(e*x + d)^5 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d)^4 + 6*((a^8*b^4 - 4
*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cosh(e*x + d) + (a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b
^11)*e)*sinh(e*x + d)^5 + 4*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*e*cosh(e*x + d)^3 + 3*
(5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cosh(e*x + d)^2 + 10*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b
^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d) + (4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e)*sinh(
e*x + d)^4 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d)^2 + 4*(5*(a^8*b^4 -
4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cosh(e*x + d)^3 + 15*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9
+ a*b^11)*e*cosh(e*x + d)^2 + 3*(4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d) +
(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*b^9 + 3*a*b^11)*e)*sinh(e*x + d)^3 + 6*(a^9*b^3 - 4*a^7*b^5 + 6*a^
5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d) + 3*(5*(a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cos
h(e*x + d)^4 + 20*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d)^3 + 6*(4*a^10*b^2 - 1
5*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d)^2 + 4*(2*a^11*b - 5*a^9*b^3 + 10*a^5*b^7 - 10*a^3*
b^9 + 3*a*b^11)*e*cosh(e*x + d) + (4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e)*sinh(e*x + d)^
2 + (a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e + 6*((a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10
+ b^12)*e*cosh(e*x + d)^5 + 5*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cosh(e*x + d)^4 + 2*(4
*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*cosh(e*x + d)^3 + 2*(2*a^11*b - 5*a^9*b^3 + 10*a^5*
b^7 - 10*a^3*b^9 + 3*a*b^11)*e*cosh(e*x + d)^2 + (4*a^10*b^2 - 15*a^8*b^4 + 20*a^6*b^6 - 10*a^4*b^8 + b^12)*e*
cosh(e*x + d) + (a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e)*sinh(e*x + d))]

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giac [B]  time = 0.25, size = 688, normalized size = 2.65 \[ \frac {1}{3} \, {\left (\frac {3 \, {\left (2 \, A a^{3} - 4 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \arctan \left (\frac {b e^{\left (x e + d\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {6 \, A a^{3} b^{3} e^{\left (5 \, x e + 5 \, d\right )} - 12 \, B a^{2} b^{4} e^{\left (5 \, x e + 5 \, d\right )} + 9 \, A a b^{5} e^{\left (5 \, x e + 5 \, d\right )} - 3 \, B b^{6} e^{\left (5 \, x e + 5 \, d\right )} + 30 \, A a^{4} b^{2} e^{\left (4 \, x e + 4 \, d\right )} - 60 \, B a^{3} b^{3} e^{\left (4 \, x e + 4 \, d\right )} + 45 \, A a^{2} b^{4} e^{\left (4 \, x e + 4 \, d\right )} - 15 \, B a b^{5} e^{\left (4 \, x e + 4 \, d\right )} - 8 \, B a^{6} e^{\left (3 \, x e + 3 \, d\right )} - 8 \, C a^{6} e^{\left (3 \, x e + 3 \, d\right )} + 44 \, A a^{5} b e^{\left (3 \, x e + 3 \, d\right )} - 64 \, B a^{4} b^{2} e^{\left (3 \, x e + 3 \, d\right )} + 24 \, C a^{4} b^{2} e^{\left (3 \, x e + 3 \, d\right )} + 82 \, A a^{3} b^{3} e^{\left (3 \, x e + 3 \, d\right )} - 78 \, B a^{2} b^{4} e^{\left (3 \, x e + 3 \, d\right )} - 24 \, C a^{2} b^{4} e^{\left (3 \, x e + 3 \, d\right )} + 24 \, A a b^{5} e^{\left (3 \, x e + 3 \, d\right )} + 8 \, C b^{6} e^{\left (3 \, x e + 3 \, d\right )} - 24 \, B a^{5} b e^{\left (2 \, x e + 2 \, d\right )} + 102 \, A a^{4} b^{2} e^{\left (2 \, x e + 2 \, d\right )} - 102 \, B a^{3} b^{3} e^{\left (2 \, x e + 2 \, d\right )} + 36 \, A a^{2} b^{4} e^{\left (2 \, x e + 2 \, d\right )} - 24 \, B a b^{5} e^{\left (2 \, x e + 2 \, d\right )} + 12 \, A b^{6} e^{\left (2 \, x e + 2 \, d\right )} - 12 \, B a^{4} b^{2} e^{\left (x e + d\right )} + 60 \, A a^{3} b^{3} e^{\left (x e + d\right )} - 66 \, B a^{2} b^{4} e^{\left (x e + d\right )} + 15 \, A a b^{5} e^{\left (x e + d\right )} + 3 \, B b^{6} e^{\left (x e + d\right )} - 2 \, B a^{3} b^{3} + 11 \, A a^{2} b^{4} - 13 \, B a b^{5} + 4 \, A b^{6}}{{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} {\left (b e^{\left (2 \, x e + 2 \, d\right )} + 2 \, a e^{\left (x e + d\right )} + b\right )}^{3}}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(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)*arctan((b*e^(x*e + d) + a)/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*A*a^3*b^3*e^(5*x*e + 5*d) - 12*B*a^2*b^4*e^(5*x*e + 5*d) + 9*A*a*b
^5*e^(5*x*e + 5*d) - 3*B*b^6*e^(5*x*e + 5*d) + 30*A*a^4*b^2*e^(4*x*e + 4*d) - 60*B*a^3*b^3*e^(4*x*e + 4*d) + 4
5*A*a^2*b^4*e^(4*x*e + 4*d) - 15*B*a*b^5*e^(4*x*e + 4*d) - 8*B*a^6*e^(3*x*e + 3*d) - 8*C*a^6*e^(3*x*e + 3*d) +
44*A*a^5*b*e^(3*x*e + 3*d) - 64*B*a^4*b^2*e^(3*x*e + 3*d) + 24*C*a^4*b^2*e^(3*x*e + 3*d) + 82*A*a^3*b^3*e^(3*
x*e + 3*d) - 78*B*a^2*b^4*e^(3*x*e + 3*d) - 24*C*a^2*b^4*e^(3*x*e + 3*d) + 24*A*a*b^5*e^(3*x*e + 3*d) + 8*C*b^
6*e^(3*x*e + 3*d) - 24*B*a^5*b*e^(2*x*e + 2*d) + 102*A*a^4*b^2*e^(2*x*e + 2*d) - 102*B*a^3*b^3*e^(2*x*e + 2*d)
+ 36*A*a^2*b^4*e^(2*x*e + 2*d) - 24*B*a*b^5*e^(2*x*e + 2*d) + 12*A*b^6*e^(2*x*e + 2*d) - 12*B*a^4*b^2*e^(x*e
+ d) + 60*A*a^3*b^3*e^(x*e + d) - 66*B*a^2*b^4*e^(x*e + d) + 15*A*a*b^5*e^(x*e + d) + 3*B*b^6*e^(x*e + d) - 2*
B*a^3*b^3 + 11*A*a^2*b^4 - 13*B*a*b^5 + 4*A*b^6)/((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*(b*e^(2*x*e + 2*d) + 2
*a*e^(x*e + d) + b)^3))*e^(-1)

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maple [A]  time = 0.18, size = 459, normalized size = 1.77 \[ \frac {-\frac {2 \left (-\frac {\left (6 A \,a^{2} b +3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -2 B \,a^{2} b -6 B a \,b^{2}-B \,b^{3}\right ) \left (\tanh ^{5}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {C \left (\tanh ^{4}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a -b}+\frac {2 \left (9 A \,a^{2} b +A \,b^{3}-3 a^{3} B -7 B a \,b^{2}\right ) \left (\tanh ^{3}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 a C \left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a^{2}-2 a b +b^{2}}-\frac {\left (6 A \,a^{2} b -3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B +2 B \,a^{2} b -6 B a \,b^{2}+B \,b^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {C \left (3 a^{2}+b^{2}\right )}{3 a^{3}-9 a^{2} b +9 a \,b^{2}-3 b^{3}}\right )}{\left (a \left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right ) b -a -b \right )^{3}}+\frac {\left (2 a^{3} A +3 A a \,b^{2}-4 B \,a^{2} b -B \,b^{3}\right ) \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(-2*(-1/2*(6*A*a^2*b+3*A*a*b^2+2*A*b^3-2*B*a^3-2*B*a^2*b-6*B*a*b^2-B*b^3)/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*
tanh(1/2*e*x+1/2*d)^5+C/(a-b)*tanh(1/2*e*x+1/2*d)^4+2/3*(9*A*a^2*b+A*b^3-3*B*a^3-7*B*a*b^2)/(a^2+2*a*b+b^2)/(a
^2-2*a*b+b^2)*tanh(1/2*e*x+1/2*d)^3-2*a*C/(a^2-2*a*b+b^2)*tanh(1/2*e*x+1/2*d)^2-1/2*(6*A*a^2*b-3*A*a*b^2+2*A*b
^3-2*B*a^3+2*B*a^2*b-6*B*a*b^2+B*b^3)/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*e*x+1/2*d)+1/3*C*(3*a^2+b^2)/(a
^3-3*a^2*b+3*a*b^2-b^3))/(a*tanh(1/2*e*x+1/2*d)^2-tanh(1/2*e*x+1/2*d)^2*b-a-b)^3+(2*A*a^3+3*A*a*b^2-4*B*a^2*b-
B*b^3)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*e*x+1/2*d)/((a+b)*(a-b))^(1/2)
))

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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((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^4,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*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,\mathrm {cosh}\left (d+e\,x\right )+C\,\mathrm {sinh}\left (d+e\,x\right )}{{\left (a+b\,\mathrm {cosh}\left (d+e\,x\right )\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))**4,x)

[Out]

Timed out

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