3.8.45 \(\int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx\) [745]

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

[Out]

-a*(2*a^2+3*b^2-3*c^2)*arctanh((c-(a-b)*tanh(1/2*x))/(a^2-b^2+c^2)^(1/2))/(a^2-b^2+c^2)^(7/2)+1/3*(-c*cosh(x)-
b*sinh(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c*sinh(x))^3-5/6*(a*c*cosh(x)+a*b*sinh(x))/(a^2-b^2+c^2)^2/(a+b*cosh(x)+
c*sinh(x))^2+1/6*(-c*(11*a^2+4*b^2-4*c^2)*cosh(x)-b*(11*a^2+4*b^2-4*c^2)*sinh(x))/(a^2-b^2+c^2)^3/(a+b*cosh(x)
+c*sinh(x))

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Rubi [A]
time = 0.23, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3208, 3235, 3232, 3203, 632, 212} \begin {gather*} -\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} \end {gather*}

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 212

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

Rule 632

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 3203

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 3208

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 3232

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 3235

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]

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 ) \text {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 ) \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(488\) vs. \(2(220)=440\).
time = 0.67, size = 488, normalized size = 2.22 \begin {gather*} -\frac {a \left (2 a^2+3 b^2-3 c^2\right ) \text {ArcTan}\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 {-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 \left (2 a^2+3 b^2-3 c^2\right ) \cosh (x)-6 a c \left (a^2 \left (-7 b^2+11 c^2\right )+2 \left (b^4+b^2 c^2-2 c^4\right )\right ) \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 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))^3} \end {gather*}

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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1597\) vs. \(2(212)=424\).
time = 1.94, size = 1598, normalized size = 7.26

method	result	size
risch	\(\text {Expression too large to display}\)	\(1588\)
default	\(-\frac {2 \left (-\frac {\left (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}\right ) \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{2 \left (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}\right ) \left (a -b \right )}-\frac {c \left (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}\right ) \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{2 \left (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}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (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 b^{4} a^{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 c^{6} b^{2}-4 c^{8}\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3 \left (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}\right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {c \left (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 c^{6} b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{\left (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}\right ) \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )}-\frac {\left (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 b^{4} a^{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 c^{6} b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{2 \left (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}\right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {a c \left (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}\right )}{6 \left (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}\right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 c \tanh \left (\frac {x}{2}\right )-a -b \right )^{3}}-\frac {\left (2 a^{2}+3 b^{2}-3 c^{2}\right ) a \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\left (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}\right ) \sqrt {-a^{2}+b^{2}-c^{2}}}\)	\(1598\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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-b*tanh(1/2*x)^2-2*c*tanh(1/2*x)-a-b)^3-(2*a^2+3*b^2-3*c^2)*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))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2-b^2+a^2>0)', see `assume?`
for more de

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11492 vs. \(2 (210) = 420\).
time = 0.57, size = 23093, normalized size = 104.97 \begin {gather*} \text {Too large to display} \end {gather*}

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]

[1/6*(22*a^4*b^3 - 14*a^2*b^5 - 8*b^7 - 24*b*c^6 + 8*c^7 - 2*(7*a^2 - 4*b^2)*c^5 + 6*(2*a^5*b^2 + a^3*b^4 - 3*
a*b^6 - 6*a*b*c^5 - 3*a*c^6 - (a^3 - 3*a*b^2)*c^4 - 2*(a^3*b - 6*a*b^3)*c^3 + (2*a^5 + 3*a*b^4)*c^2 + 2*(2*a^5
*b + a^3*b^3 - 3*a*b^5)*c)*cosh(x)^5 + 6*(2*a^5*b^2 + a^3*b^4 - 3*a*b^6 - 6*a*b*c^5 - 3*a*c^6 - (a^3 - 3*a*b^2
)*c^4 - 2*(a^3*b - 6*a*b^3)*c^3 + (2*a^5 + 3*a*b^4)*c^2 + 2*(2*a^5*b + a^3*b^3 - 3*a*b^5)*c)*sinh(x)^5 + 2*(21
*a^2*b + 20*b^3)*c^4 + 30*(2*a^6*b + a^4*b^3 - 3*a^2*b^5 - 3*a^2*b*c^4 - 3*a^2*c^5 - (a^4 - 6*a^2*b^2)*c^3 - (
a^4*b - 6*a^2*b^3)*c^2 + (2*a^6 + a^4*b^2 - 3*a^2*b^4)*c)*cosh(x)^4 + 30*(2*a^6*b + a^4*b^3 - 3*a^2*b^5 - 3*a^
2*b*c^4 - 3*a^2*c^5 - (a^4 - 6*a^2*b^2)*c^3 - (a^4*b - 6*a^2*b^3)*c^2 + (2*a^6 + a^4*b^2 - 3*a^2*b^4)*c + (2*a
^5*b^2 + a^3*b^4 - 3*a*b^6 - 6*a*b*c^5 - 3*a*c^6 - (a^3 - 3*a*b^2)*c^4 - 2*(a^3*b - 6*a*b^3)*c^3 + (2*a^5 + 3*
a*b^4)*c^2 + 2*(2*a^5*b + a^3*b^3 - 3*a*b^5)*c)*cosh(x))*sinh(x)^4 - 2*(11*a^4 + 14*a^2*b^2 + 20*b^4)*c^3 + 4*
(22*a^7 + 19*a^5*b^2 - 29*a^3*b^4 - 12*a*b^6 + 12*a*c^6 - (29*a^3 + 36*a*b^2)*c^4 - (19*a^5 - 58*a^3*b^2 - 36*
a*b^4)*c^2)*cosh(x)^3 + 4*(22*a^7 + 19*a^5*b^2 - 29*a^3*b^4 - 12*a*b^6 + 12*a*c^6 - (29*a^3 + 36*a*b^2)*c^4 -
(19*a^5 - 58*a^3*b^2 - 36*a*b^4)*c^2 + 15*(2*a^5*b^2 + a^3*b^4 - 3*a*b^6 - 6*a*b*c^5 - 3*a*c^6 - (a^3 - 3*a*b^
2)*c^4 - 2*(a^3*b - 6*a*b^3)*c^3 + (2*a^5 + 3*a*b^4)*c^2 + 2*(2*a^5*b + a^3*b^3 - 3*a*b^5)*c)*cosh(x)^2 + 30*(
2*a^6*b + a^4*b^3 - 3*a^2*b^5 - 3*a^2*b*c^4 - 3*a^2*c^5 - (a^4 - 6*a^2*b^2)*c^3 - (a^4*b - 6*a^2*b^3)*c^2 + (2
*a^6 + a^4*b^2 - 3*a^2*b^4)*c)*cosh(x))*sinh(x)^3 + 2*(33*a^4*b - 14*a^2*b^3 - 4*b^5)*c^2 + 12*(17*a^6*b - 11*
a^4*b^3 - 4*a^2*b^5 - 2*b^7 + 2*b*c^6 - 2*c^7 + 2*(2*a^2 + 3*b^2)*c^5 - 2*(2*a^2*b + 3*b^3)*c^4 - (11*a^4 + 8*
a^2*b^2 + 6*b^4)*c^3 + (11*a^4*b + 8*a^2*b^3 + 6*b^5)*c^2 - (17*a^6 - 11*a^4*b^2 - 4*a^2*b^4 - 2*b^6)*c)*cosh(
x)^2 + 12*(17*a^6*b - 11*a^4*b^3 - 4*a^2*b^5 - 2*b^7 + 2*b*c^6 - 2*c^7 + 2*(2*a^2 + 3*b^2)*c^5 - 2*(2*a^2*b +
3*b^3)*c^4 - (11*a^4 + 8*a^2*b^2 + 6*b^4)*c^3 + 5*(2*a^5*b^2 + a^3*b^4 - 3*a*b^6 - 6*a*b*c^5 - 3*a*c^6 - (a^3
- 3*a*b^2)*c^4 - 2*(a^3*b - 6*a*b^3)*c^3 + (2*a^5 + 3*a*b^4)*c^2 + 2*(2*a^5*b + a^3*b^3 - 3*a*b^5)*c)*cosh(x)^
3 + (11*a^4*b + 8*a^2*b^3 + 6*b^5)*c^2 + 15*(2*a^6*b + a^4*b^3 - 3*a^2*b^5 - 3*a^2*b*c^4 - 3*a^2*c^5 - (a^4 -
6*a^2*b^2)*c^3 - (a^4*b - 6*a^2*b^3)*c^2 + (2*a^6 + a^4*b^2 - 3*a^2*b^4)*c)*cosh(x)^2 - (17*a^6 - 11*a^4*b^2 -
4*a^2*b^4 - 2*b^6)*c + (22*a^7 + 19*a^5*b^2 - 29*a^3*b^4 - 12*a*b^6 + 12*a*c^6 - (29*a^3 + 36*a*b^2)*c^4 - (1
9*a^5 - 58*a^3*b^2 - 36*a*b^4)*c^2)*cosh(x))*sinh(x)^2 - 3*((2*a^3*b^3 + 3*a*b^5 - 9*a*b*c^4 - 3*a*c^5 + 2*(a^
3 - 3*a*b^2)*c^3 + 6*(a^3*b + a*b^3)*c^2 + 3*(2*a^3*b^2 + 3*a*b^4)*c)*cosh(x)^6 + (2*a^3*b^3 + 3*a*b^5 - 9*a*b
*c^4 - 3*a*c^5 + 2*(a^3 - 3*a*b^2)*c^3 + 6*(a^3*b + a*b^3)*c^2 + 3*(2*a^3*b^2 + 3*a*b^4)*c)*sinh(x)^6 + 2*a^3*
b^3 + 3*a*b^5 - 9*a*b*c^4 + 3*a*c^5 + 6*(2*a^4*b^2 + 3*a^2*b^4 + 2*a^4*c^2 - 6*a^2*b*c^3 - 3*a^2*c^4 + 2*(2*a^
4*b + 3*a^2*b^3)*c)*cosh(x)^5 + 6*(2*a^4*b^2 + 3*a^2*b^4 + 2*a^4*c^2 - 6*a^2*b*c^3 - 3*a^2*c^4 + 2*(2*a^4*b +
3*a^2*b^3)*c + (2*a^3*b^3 + 3*a*b^5 - 9*a*b*c^4 - 3*a*c^5 + 2*(a^3 - 3*a*b^2)*c^3 + 6*(a^3*b + a*b^3)*c^2 + 3*
(2*a^3*b^2 + 3*a*b^4)*c)*cosh(x))*sinh(x)^5 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 + 3*a*b*c^4 + 3*a*c^5 - 2*(7*a
^3 + 3*a*b^2)*c^3 - 2*(7*a^3*b + 3*a*b^3)*c^2 + (8*a^5 + 14*a^3*b^2 + 3*a*b^4)*c)*cosh(x)^4 + 3*(8*a^5*b + 14*
a^3*b^3 + 3*a*b^5 + 3*a*b*c^4 + 3*a*c^5 - 2*(7*a^3 + 3*a*b^2)*c^3 - 2*(7*a^3*b + 3*a*b^3)*c^2 + 5*(2*a^3*b^3 +
3*a*b^5 - 9*a*b*c^4 - 3*a*c^5 + 2*(a^3 - 3*a*b^2)*c^3 + 6*(a^3*b + a*b^3)*c^2 + 3*(2*a^3*b^2 + 3*a*b^4)*c)*co
sh(x)^2 + (8*a^5 + 14*a^3*b^2 + 3*a*b^4)*c + 10*(2*a^4*b^2 + 3*a^2*b^4 + 2*a^4*c^2 - 6*a^2*b*c^3 - 3*a^2*c^4 +
2*(2*a^4*b + 3*a^2*b^3)*c)*cosh(x))*sinh(x)^4 - 2*(a^3 - 3*a*b^2)*c^3 + 4*(4*a^6 + 12*a^4*b^2 + 9*a^2*b^4 + 9
*a^2*c^4 - 6*(2*a^4 + 3*a^2*b^2)*c^2)*cosh(x)^3 + 4*(4*a^6 + 12*a^4*b^2 + 9*a^2*b^4 + 9*a^2*c^4 + 5*(2*a^3*b^3
+ 3*a*b^5 - 9*a*b*c^4 - 3*a*c^5 + 2*(a^3 - 3*a*b^2)*c^3 + 6*(a^3*b + a*b^3)*c^2 + 3*(2*a^3*b^2 + 3*a*b^4)*c)*
cosh(x)^3 - 6*(2*a^4 + 3*a^2*b^2)*c^2 + 15*(2*a^4*b^2 + 3*a^2*b^4 + 2*a^4*c^2 - 6*a^2*b*c^3 - 3*a^2*c^4 + 2*(2
*a^4*b + 3*a^2*b^3)*c)*cosh(x)^2 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 + 3*a*b*c^4 + 3*a*c^5 - 2*(7*a^3 + 3*a*b^
2)*c^3 - 2*(7*a^3*b + 3*a*b^3)*c^2 + (8*a^5 + 14*a^3*b^2 + 3*a*b^4)*c)*cosh(x))*sinh(x)^3 + 6*(a^3*b + a*b^3)*
c^2 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 + 3*a*b*c^4 - 3*a*c^5 + 2*(7*a^3 + 3*a*b^2)*c^3 - 2*(7*a^3*b + 3*a*b^3
)*c^2 - (8*a^5 + 14*a^3*b^2 + 3*a*b^4)*c)*cosh(x)^2 + 3*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 + 3*a*b*c^4 - 3*a*c^5
+ 5*(2*a^3*b^3 + 3*a*b^5 - 9*a*b*c^4 - 3*a*c^5 + 2*(a^3 - 3*a*b^2)*c^3 + 6*(a^3*b + a*b^3)*c^2 + 3*(2*a^3*b^2
+ 3*a*b^4)*c)*cosh(x)^4 + 2*(7*a^3 + 3*a*b^2)*c^3 + 20*(2*a^4*b^2 + 3*a^2*b^4 + 2*a^4*c^2 - 6*a^2*b*c^3 - 3*a^
2*c^4 + 2*(2*a^4*b + 3*a^2*b^3)*c)*cosh(x)^3 - 2*(7*a^3*b + 3*a*b^3)*c^2 + 6*(8*a^5*b + 14*a^3*b^3 + 3*a*b^5 +
3*a*b*c^4 + 3*a*c^5 - 2*(7*a^3 + 3*a*b^2)*c^3 ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (210) = 420\).
time = 0.43, size = 717, normalized size = 3.26 \begin {gather*} \frac {{\left (2 \, a^{3} + 3 \, a b^{2} - 3 \, a c^{2}\right )} \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{{\left (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}\right )} \sqrt {-a^{2} + b^{2} - c^{2}}} + \frac {6 \, a^{3} b^{2} e^{\left (5 \, x\right )} + 9 \, a b^{4} e^{\left (5 \, x\right )} + 12 \, a^{3} b c e^{\left (5 \, x\right )} + 18 \, a b^{3} c e^{\left (5 \, x\right )} + 6 \, a^{3} c^{2} e^{\left (5 \, x\right )} - 18 \, a b c^{3} e^{\left (5 \, x\right )} - 9 \, a c^{4} e^{\left (5 \, x\right )} + 30 \, a^{4} b e^{\left (4 \, x\right )} + 45 \, a^{2} b^{3} e^{\left (4 \, x\right )} + 30 \, a^{4} c e^{\left (4 \, x\right )} + 45 \, a^{2} b^{2} c e^{\left (4 \, x\right )} - 45 \, a^{2} b c^{2} e^{\left (4 \, x\right )} - 45 \, a^{2} c^{3} e^{\left (4 \, x\right )} + 44 \, a^{5} e^{\left (3 \, x\right )} + 82 \, a^{3} b^{2} e^{\left (3 \, x\right )} + 24 \, a b^{4} e^{\left (3 \, x\right )} - 82 \, a^{3} c^{2} e^{\left (3 \, x\right )} - 48 \, a b^{2} c^{2} e^{\left (3 \, x\right )} + 24 \, a c^{4} e^{\left (3 \, x\right )} + 102 \, a^{4} b e^{\left (2 \, x\right )} + 36 \, a^{2} b^{3} e^{\left (2 \, x\right )} + 12 \, b^{5} e^{\left (2 \, x\right )} - 102 \, a^{4} c e^{\left (2 \, x\right )} - 36 \, a^{2} b^{2} c e^{\left (2 \, x\right )} - 12 \, b^{4} c e^{\left (2 \, x\right )} - 36 \, a^{2} b c^{2} e^{\left (2 \, x\right )} - 24 \, b^{3} c^{2} e^{\left (2 \, x\right )} + 36 \, a^{2} c^{3} e^{\left (2 \, x\right )} + 24 \, b^{2} c^{3} e^{\left (2 \, x\right )} + 12 \, b c^{4} e^{\left (2 \, x\right )} - 12 \, c^{5} e^{\left (2 \, x\right )} + 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}}{3 \, {\left (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}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}^{3}} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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