### 3.745 $$\int \frac{1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx$$

Optimal. Leaf size=220 $-\frac{a \left (2 a^2+3 b^2-3 c^2\right ) \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{7/2}}-\frac{b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}-\frac{5 (a b \sinh (x)+a c \cosh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac{b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}$

[Out]

-((a*(2*a^2 + 3*b^2 - 3*c^2)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(7/2))
- (c*Cosh[x] + b*Sinh[x])/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^3) - (5*(a*c*Cosh[x] + a*b*Sinh[x])
)/(6*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x])^2) - (c*(11*a^2 + 4*b^2 - 4*c^2)*Cosh[x] + b*(11*a^2 + 4*
b^2 - 4*c^2)*Sinh[x])/(6*(a^2 - b^2 + c^2)^3*(a + b*Cosh[x] + c*Sinh[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.301706, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {3129, 3156, 3153, 3124, 618, 206} $-\frac{a \left (2 a^2+3 b^2-3 c^2\right ) \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{7/2}}-\frac{b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}-\frac{5 (a b \sinh (x)+a c \cosh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac{b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*Cosh[x] + c*Sinh[x])^(-4),x]

[Out]

-((a*(2*a^2 + 3*b^2 - 3*c^2)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(7/2))
- (c*Cosh[x] + b*Sinh[x])/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^3) - (5*(a*c*Cosh[x] + a*b*Sinh[x])
)/(6*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x])^2) - (c*(11*a^2 + 4*b^2 - 4*c^2)*Cosh[x] + b*(11*a^2 + 4*
b^2 - 4*c^2)*Sinh[x])/(6*(a^2 - b^2 + c^2)^3*(a + b*Cosh[x] + c*Sinh[x]))

Rule 3129

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[((-(c*Cos[d
+ e*x]) + b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] +
Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*C
os[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n
, -1] && NeQ[n, -3/2]

Rule 3156

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_)*((A_.) + cos[(d_.) + (e_.)*(x
_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> -Simp[((c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B -
b*A)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] + Dist[1/
((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C)
+ (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A,
B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]

Rule 3153

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^2, x_Symbol] :> Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)
*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - b*B - c*C)/(a^2 -
b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[
a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]

Rule 3124

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[(2*f)/e, Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx &=-\frac{c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac{\int \frac{-3 a+2 b \cosh (x)+2 c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx}{3 \left (a^2-b^2+c^2\right )}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac{5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}+\frac{\int \frac{2 \left (3 a^2+2 b^2-2 c^2\right )-5 a b \cosh (x)-5 a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx}{6 \left (a^2-b^2+c^2\right )^2}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac{5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac{c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}+\frac{\left (a \left (2 a^2+3 b^2-3 c^2\right )\right ) \int \frac{1}{a+b \cosh (x)+c \sinh (x)} \, dx}{2 \left (a^2-b^2+c^2\right )^3}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac{5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac{c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}+\frac{\left (a \left (2 a^2+3 b^2-3 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+2 c x-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2+c^2\right )^3}\\ &=-\frac{c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac{5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac{c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}-\frac{\left (2 a \left (2 a^2+3 b^2-3 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2-b^2+c^2\right )-x^2} \, dx,x,2 c+2 (-a+b) \tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2+c^2\right )^3}\\ &=-\frac{a \left (2 a^2+3 b^2-3 c^2\right ) \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{7/2}}-\frac{c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac{5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac{c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))}\\ \end{align*}

Mathematica [B]  time = 0.996838, size = 488, normalized size = 2.22 $-\frac{a \left (2 a^2+3 b^2-3 c^2\right ) \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )+c}{\sqrt{-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{7/2}}-\frac{-72 a^2 b^2 c^2 \sinh (x)-30 a^2 b c \cosh (x) \left (2 a^2+3 b^2-3 c^2\right )-6 a c \cosh (2 x) \left (a^2 \left (11 c^2-7 b^2\right )+2 \left (b^2 c^2+b^4-2 c^4\right )\right )+22 a^2 b^3 c \cosh (3 x)-82 a^3 b^2 c-9 a^2 b^4 \sinh (x)+11 a^2 b^4 \sinh (3 x)+54 a^3 b^3 \sinh (2 x)+72 a^4 b^2 \sinh (x)-78 a^3 b c^2 \sinh (2 x)-22 a^2 b c^3 \cosh (3 x)+81 a^2 c^4 \sinh (x)-132 a^4 c^2 \sinh (x)-11 a^2 c^4 \sinh (3 x)+82 a^3 c^3-44 a^5 c-48 a b^3 c^2 \sinh (2 x)+48 a b^2 c^3-24 a b^4 c+6 a b^5 \sinh (2 x)+42 a b c^4 \sinh (2 x)-24 a c^5-36 b^4 c^2 \sinh (x)-4 b^4 c^2 \sinh (3 x)+36 b^2 c^4 \sinh (x)-4 b^2 c^4 \sinh (3 x)-16 b^3 c^3 \cosh (3 x)+8 b^5 c \cosh (3 x)+12 b^6 \sinh (x)+4 b^6 \sinh (3 x)+8 b c^5 \cosh (3 x)-12 c^6 \sinh (x)+4 c^6 \sinh (3 x)}{24 b \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*Cosh[x] + c*Sinh[x])^(-4),x]

[Out]

-((a*(2*a^2 + 3*b^2 - 3*c^2)*ArcTan[(c + (-a + b)*Tanh[x/2])/Sqrt[-a^2 + b^2 - c^2]])/(-a^2 + b^2 - c^2)^(7/2)
) - (-44*a^5*c - 82*a^3*b^2*c - 24*a*b^4*c + 82*a^3*c^3 + 48*a*b^2*c^3 - 24*a*c^5 - 30*a^2*b*c*(2*a^2 + 3*b^2
- 3*c^2)*Cosh[x] - 6*a*c*(a^2*(-7*b^2 + 11*c^2) + 2*(b^4 + b^2*c^2 - 2*c^4))*Cosh[2*x] + 22*a^2*b^3*c*Cosh[3*x
] + 8*b^5*c*Cosh[3*x] - 22*a^2*b*c^3*Cosh[3*x] - 16*b^3*c^3*Cosh[3*x] + 8*b*c^5*Cosh[3*x] + 72*a^4*b^2*Sinh[x]
- 9*a^2*b^4*Sinh[x] + 12*b^6*Sinh[x] - 132*a^4*c^2*Sinh[x] - 72*a^2*b^2*c^2*Sinh[x] - 36*b^4*c^2*Sinh[x] + 81
*a^2*c^4*Sinh[x] + 36*b^2*c^4*Sinh[x] - 12*c^6*Sinh[x] + 54*a^3*b^3*Sinh[2*x] + 6*a*b^5*Sinh[2*x] - 78*a^3*b*c
^2*Sinh[2*x] - 48*a*b^3*c^2*Sinh[2*x] + 42*a*b*c^4*Sinh[2*x] + 11*a^2*b^4*Sinh[3*x] + 4*b^6*Sinh[3*x] - 4*b^4*
c^2*Sinh[3*x] - 11*a^2*c^4*Sinh[3*x] - 4*b^2*c^4*Sinh[3*x] + 4*c^6*Sinh[3*x])/(24*b*(a^2 - b^2 + c^2)^3*(a + b
*Cosh[x] + c*Sinh[x])^3)

________________________________________________________________________________________

Maple [B]  time = 0.118, size = 1842, normalized size = 8.4 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x)+c*sinh(x))^4,x)

[Out]

-2*(-1/2*(6*a^5*b-15*a^4*b^2+9*a^4*c^2+11*a^3*b^3-9*a^3*b*c^2-3*a^2*b^4-3*a^2*b^2*c^2+6*a^2*c^4+3*a*b^5-3*a*b^
3*c^2-2*b^6+6*b^4*c^2-6*b^2*c^4+2*c^6)/(a^6-3*a^4*b^2+3*a^4*c^2+3*a^2*b^4-6*a^2*b^2*c^2+3*a^2*c^4-b^6+3*b^4*c^
2-3*b^2*c^4+c^6)/(a-b)*tanh(1/2*x)^5-1/2*c*(6*a^6-30*a^5*b+57*a^4*b^2-27*a^4*c^2-55*a^3*b^3+45*a^3*b*c^2+33*a^
2*b^4-21*a^2*b^2*c^2-12*a^2*c^4-15*a*b^5+15*a*b^3*c^2+4*b^6-12*b^4*c^2+12*b^2*c^4-4*c^6)/(a^6-3*a^4*b^2+3*a^4*
c^2+3*a^2*b^4-6*a^2*b^2*c^2+3*a^2*c^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)/(a^2-2*a*b+b^2)*tanh(1/2*x)^4+1/3*(18*a^7*b
-54*a^6*b^2+54*a^6*c^2+38*a^5*b^3-120*a^5*b*c^2+30*a^4*b^4+81*a^4*b^2*c^2-21*a^4*c^4-50*a^3*b^5-61*a^3*b^3*c^2
+81*a^3*b*c^4+22*a^2*b^6+87*a^2*b^4*c^2-105*a^2*b^2*c^4-4*a^2*c^6-6*a*b^7-39*a*b^5*c^2+51*a*b^3*c^4-6*a*b*c^6+
2*b^8-2*b^6*c^2-6*b^4*c^4+10*b^2*c^6-4*c^8)/(a^6-3*a^4*b^2+3*a^4*c^2+3*a^2*b^4-6*a^2*b^2*c^2+3*a^2*c^4-b^6+3*b
^4*c^2-3*b^2*c^4+c^6)/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*x)^3+c*(6*a^7-18*a^6*b+18*a^5*b^2-20*a^5*c^2-2*a^4*b^
3+22*a^4*b*c^2-14*a^3*b^4+7*a^3*b^2*c^2-3*a^3*c^4+18*a^2*b^5-6*a^2*b^3*c^2-12*a^2*b*c^4-10*a*b^6+3*a*b^4*c^2+9
*a*b^2*c^4-2*a*c^6+2*b^7-6*b^5*c^2+6*b^3*c^4-2*b*c^6)/(a^6-3*a^4*b^2+3*a^4*c^2+3*a^2*b^4-6*a^2*b^2*c^2+3*a^2*c
^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)/(a^2-2*a*b+b^2)/(a-b)*tanh(1/2*x)^2-1/2*(6*a^7*b-9*a^6*b^2+27*a^6*c^2-7*a^5*b^
3-9*a^5*b*c^2+16*a^4*b^4-30*a^4*b^2*c^2+4*a^4*c^4-4*a^3*b^5+14*a^3*b*c^4-5*a^2*b^6-3*a^2*b^4*c^2+6*a^2*b^2*c^4
+2*a^2*c^6+5*a*b^7+9*a*b^5*c^2-18*a*b^3*c^4+4*a*b*c^6-2*b^8+6*b^6*c^2-6*b^4*c^4+2*b^2*c^6)/(a^6-3*a^4*b^2+3*a^
4*c^2+3*a^2*b^4-6*a^2*b^2*c^2+3*a^2*c^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)/(a^3-3*a^2*b+3*a*b^2-b^3)*tanh(1/2*x)-1/6
*a*c*(18*a^6-21*a^4*b^2+5*a^4*c^2-12*a^2*b^4+16*a^2*b^2*c^2+2*a^2*c^4+15*b^6-21*b^4*c^2+6*b^2*c^4)/(a^6-3*a^4*
b^2+3*a^4*c^2+3*a^2*b^4-6*a^2*b^2*c^2+3*a^2*c^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)/(a^3-3*a^2*b+3*a*b^2-b^3))/(a*tan
h(1/2*x)^2-tanh(1/2*x)^2*b-2*c*tanh(1/2*x)-a-b)^3-2*a^3/(a^6-3*a^4*b^2+3*a^4*c^2+3*a^2*b^4-6*a^2*b^2*c^2+3*a^2
*c^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)/(-a^2+b^2-c^2)^(1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/(-a^2+b^2-c^2)^(1/
2))-3*a/(a^6-3*a^4*b^2+3*a^4*c^2+3*a^2*b^4-6*a^2*b^2*c^2+3*a^2*c^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)/(-a^2+b^2-c^2)
^(1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/(-a^2+b^2-c^2)^(1/2))*b^2+3*a/(a^6-3*a^4*b^2+3*a^4*c^2+3*a^2*b^4-6
*a^2*b^2*c^2+3*a^2*c^4-b^6+3*b^4*c^2-3*b^2*c^4+c^6)/(-a^2+b^2-c^2)^(1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/
(-a^2+b^2-c^2)^(1/2))*c^2

________________________________________________________________________________________

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(1/(a+b*cosh(x)+c*sinh(x))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [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(1/(a+b*cosh(x)+c*sinh(x))^4,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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(1/(a+b*cosh(x)+c*sinh(x))**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.20635, size = 968, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^4,x, algorithm="giac")

[Out]

(2*a^3 + 3*a*b^2 - 3*a*c^2)*arctan((b*e^x + c*e^x + a)/sqrt(-a^2 + b^2 - c^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 -
b^6 + 3*a^4*c^2 - 6*a^2*b^2*c^2 + 3*b^4*c^2 + 3*a^2*c^4 - 3*b^2*c^4 + c^6)*sqrt(-a^2 + b^2 - c^2)) + 1/3*(6*a
^3*b^2*e^(5*x) + 9*a*b^4*e^(5*x) + 12*a^3*b*c*e^(5*x) + 18*a*b^3*c*e^(5*x) + 6*a^3*c^2*e^(5*x) - 18*a*b*c^3*e^
(5*x) - 9*a*c^4*e^(5*x) + 30*a^4*b*e^(4*x) + 45*a^2*b^3*e^(4*x) + 30*a^4*c*e^(4*x) + 45*a^2*b^2*c*e^(4*x) - 45
*a^2*b*c^2*e^(4*x) - 45*a^2*c^3*e^(4*x) + 44*a^5*e^(3*x) + 82*a^3*b^2*e^(3*x) + 24*a*b^4*e^(3*x) - 82*a^3*c^2*
e^(3*x) - 48*a*b^2*c^2*e^(3*x) + 24*a*c^4*e^(3*x) + 102*a^4*b*e^(2*x) + 36*a^2*b^3*e^(2*x) + 12*b^5*e^(2*x) -
102*a^4*c*e^(2*x) - 36*a^2*b^2*c*e^(2*x) - 12*b^4*c*e^(2*x) - 36*a^2*b*c^2*e^(2*x) - 24*b^3*c^2*e^(2*x) + 36*a
^2*c^3*e^(2*x) + 24*b^2*c^3*e^(2*x) + 12*b*c^4*e^(2*x) - 12*c^5*e^(2*x) + 60*a^3*b^2*e^x + 15*a*b^4*e^x - 120*
a^3*b*c*e^x - 30*a*b^3*c*e^x + 60*a^3*c^2*e^x + 30*a*b*c^3*e^x - 15*a*c^4*e^x + 11*a^2*b^3 + 4*b^5 - 33*a^2*b^
2*c - 12*b^4*c + 33*a^2*b*c^2 + 8*b^3*c^2 - 11*a^2*c^3 + 8*b^2*c^3 - 12*b*c^4 + 4*c^5)/((a^6 - 3*a^4*b^2 + 3*a
^2*b^4 - b^6 + 3*a^4*c^2 - 6*a^2*b^2*c^2 + 3*b^4*c^2 + 3*a^2*c^4 - 3*b^2*c^4 + c^6)*(b*e^(2*x) + c*e^(2*x) + 2
*a*e^x + b - c)^3)