Integrand size = 12, antiderivative size = 220 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=-\frac {a \left (2 a^2+3 b^2-3 c^2\right ) \text {arctanh}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{7/2}}-\frac {c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac {5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac {c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))} \]
-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)*sin h(x))/(a^2-b^2+c^2)^3/(a+b*cosh(x)+c*sinh(x))
Leaf count is larger than twice the leaf count of optimal. \(488\) vs. \(2(220)=440\).
Time = 0.63 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.22 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=-\frac {a \left (2 a^2+3 b^2-3 c^2\right ) \arctan \left (\frac {c+(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{7/2}}-\frac {-44 a^5 c-82 a^3 b^2 c-24 a b^4 c+82 a^3 c^3+48 a b^2 c^3-24 a c^5-30 a^2 b c \left (2 a^2+3 b^2-3 c^2\right ) \cosh (x)-6 a c \left (a^2 \left (-7 b^2+11 c^2\right )+2 \left (b^4+b^2 c^2-2 c^4\right )\right ) \cosh (2 x)+22 a^2 b^3 c \cosh (3 x)+8 b^5 c \cosh (3 x)-22 a^2 b c^3 \cosh (3 x)-16 b^3 c^3 \cosh (3 x)+8 b c^5 \cosh (3 x)+72 a^4 b^2 \sinh (x)-9 a^2 b^4 \sinh (x)+12 b^6 \sinh (x)-132 a^4 c^2 \sinh (x)-72 a^2 b^2 c^2 \sinh (x)-36 b^4 c^2 \sinh (x)+81 a^2 c^4 \sinh (x)+36 b^2 c^4 \sinh (x)-12 c^6 \sinh (x)+54 a^3 b^3 \sinh (2 x)+6 a b^5 \sinh (2 x)-78 a^3 b c^2 \sinh (2 x)-48 a b^3 c^2 \sinh (2 x)+42 a b c^4 \sinh (2 x)+11 a^2 b^4 \sinh (3 x)+4 b^6 \sinh (3 x)-4 b^4 c^2 \sinh (3 x)-11 a^2 c^4 \sinh (3 x)-4 b^2 c^4 \sinh (3 x)+4 c^6 \sinh (3 x)}{24 b \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))^3} \]
-((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*Cos h[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*S inh[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*Sin h[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)
Time = 0.90 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3608, 25, 3042, 3635, 25, 3042, 3632, 3042, 3603, 1083, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+b \cos (i x)-i c \sin (i x))^4}dx\) |
\(\Big \downarrow \) 3608 |
\(\displaystyle -\frac {\int -\frac {3 a-2 b \cosh (x)-2 c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {3 a-2 b \cosh (x)-2 c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}+\frac {\int \frac {3 a-2 b \cos (i x)+2 i c \sin (i x)}{(a+b \cos (i x)-i c \sin (i x))^3}dx}{3 \left (a^2-b^2+c^2\right )}\) |
\(\Big \downarrow \) 3635 |
\(\displaystyle \frac {-\frac {\int -\frac {2 \left (3 a^2+2 b^2-2 c^2\right )-5 a b \cosh (x)-5 a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int \frac {2 \left (3 a^2+2 b^2-2 c^2\right )-5 a b \cosh (x)-5 a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}+\frac {-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 \left (3 a^2+2 b^2-2 c^2\right )-5 a b \cos (i x)+5 i a c \sin (i x)}{(a+b \cos (i x)-i c \sin (i x))^2}dx}{2 \left (a^2-b^2+c^2\right )}}{3 \left (a^2-b^2+c^2\right )}\) |
\(\Big \downarrow \) 3632 |
\(\displaystyle \frac {\frac {\frac {3 a \left (2 a^2+3 b^2-3 c^2\right ) \int \frac {1}{a+b \cosh (x)+c \sinh (x)}dx}{a^2-b^2+c^2}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}+\frac {-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}+\frac {3 a \left (2 a^2+3 b^2-3 c^2\right ) \int \frac {1}{a+b \cos (i x)-i c \sin (i x)}dx}{a^2-b^2+c^2}}{2 \left (a^2-b^2+c^2\right )}}{3 \left (a^2-b^2+c^2\right )}\) |
\(\Big \downarrow \) 3603 |
\(\displaystyle \frac {\frac {\frac {6 a \left (2 a^2+3 b^2-3 c^2\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+2 c \tanh \left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2+c^2}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {\frac {-\frac {12 a \left (2 a^2+3 b^2-3 c^2\right ) \int \frac {1}{4 \left (a^2-b^2+c^2\right )-\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2+c^2}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {-\frac {6 a \left (2 a^2+3 b^2-3 c^2\right ) \text {arctanh}\left (\frac {2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\) |
-1/3*(c*Cosh[x] + b*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x] )^3) + ((-5*(a*c*Cosh[x] + a*b*Sinh[x]))/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[ x] + c*Sinh[x])^2) + ((-6*a*(2*a^2 + 3*b^2 - 3*c^2)*ArcTanh[(2*c - 2*(a - b)*Tanh[x/2])/(2*Sqrt[a^2 - b^2 + c^2])])/(a^2 - b^2 + c^2)^(3/2) - (c*(11 *a^2 + 4*b^2 - 4*c^2)*Cosh[x] + b*(11*a^2 + 4*b^2 - 4*c^2)*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])))/(2*(a^2 - b^2 + c^2)))/(3*(a^2 - b^2 + c^2))
3.8.45.3.1 Defintions of rubi rules used
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (-1), x_Symbol] :> Module[{f = FreeFactors[Tan[(d + e*x)/2], x]}, Simp[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]
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] + Simp[ 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*Cos[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]
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] + Simp[(a*A - b*B - c*C)/(a^2 - b^2 - c^2) Int[1/(a + b*Cos[d + e*x] + c*S in[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]
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] + Simp[1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a + b*Co s[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1587\) vs. \(2(212)=424\).
Time = 118.96 (sec) , antiderivative size = 1588, normalized size of antiderivative = 7.22
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1588\) |
default | \(\text {Expression too large to display}\) | \(1598\) |
1/3*(33*a^2*b*c^2+102*a^4*b*exp(x)^2+36*a^2*b^3*exp(x)^2+30*a^4*b*exp(x)^4 +45*a^2*b^3*exp(x)^4+8*b^2*c^3+15*exp(x)*a*b^4-12*b*c^4+8*b^3*c^2-11*c^3*a ^2+4*b^5+6*a^3*b^2*exp(x)^5+9*a*b^4*exp(x)^5+82*a^3*b^2*exp(x)^3+24*a*b^4* exp(x)^3+60*a^3*b^2*exp(x)+11*a^2*b^3+12*b^5*exp(x)^2+4*c^5+44*a^5*exp(x)^ 3-12*b^4*c-33*a^2*b^2*c-12*c^5*exp(x)^2-12*b^4*c*exp(x)^2-24*b^3*c^2*exp(x )^2-102*a^4*c*exp(x)^2-9*a*c^4*exp(x)^5+24*a*c^4*exp(x)^3+6*a^3*c^2*exp(x) ^5+24*b^2*c^3*exp(x)^2+36*a^2*c^3*exp(x)^2+12*b*c^4*exp(x)^2-82*a^3*c^2*ex p(x)^3-45*a^2*c^3*exp(x)^4+30*a^4*c*exp(x)^4+60*a^3*c^2*exp(x)-15*a*c^4*ex p(x)+12*a^3*b*c*exp(x)^5+18*a*b^3*c*exp(x)^5-18*a*b*c^3*exp(x)^5+45*a^2*b^ 2*c*exp(x)^4-45*a^2*b*c^2*exp(x)^4-48*a*b^2*c^2*exp(x)^3-36*a^2*b^2*c*exp( x)^2-36*a^2*b*c^2*exp(x)^2-120*a^3*b*c*exp(x)-30*a*b^3*c*exp(x)+30*a*b*c^3 *exp(x))/(a^2-b^2+c^2)^3/(b*exp(x)^2+exp(x)^2*c+2*a*exp(x)+b-c)^3+1/(a^2-b ^2+c^2)^(7/2)*a^3*ln(exp(x)+((a^2-b^2+c^2)^(7/2)*a-a^8+4*a^6*b^2-4*a^6*c^2 -6*a^4*b^4+12*a^4*b^2*c^2-6*a^4*c^4+4*a^2*b^6-12*a^2*b^4*c^2+12*a^2*b^2*c^ 4-4*a^2*c^6-b^8+4*c^2*b^6-6*b^4*c^4+4*c^6*b^2-c^8)/(a^2-b^2+c^2)^(7/2)/(b+ c))+3/2/(a^2-b^2+c^2)^(7/2)*a*ln(exp(x)+((a^2-b^2+c^2)^(7/2)*a-a^8+4*a^6*b ^2-4*a^6*c^2-6*a^4*b^4+12*a^4*b^2*c^2-6*a^4*c^4+4*a^2*b^6-12*a^2*b^4*c^2+1 2*a^2*b^2*c^4-4*a^2*c^6-b^8+4*c^2*b^6-6*b^4*c^4+4*c^6*b^2-c^8)/(a^2-b^2+c^ 2)^(7/2)/(b+c))*b^2-3/2/(a^2-b^2+c^2)^(7/2)*a*ln(exp(x)+((a^2-b^2+c^2)^(7/ 2)*a-a^8+4*a^6*b^2-4*a^6*c^2-6*a^4*b^4+12*a^4*b^2*c^2-6*a^4*c^4+4*a^2*b...
Leaf count of result is larger than twice the leaf count of optimal. 11492 vs. \(2 (210) = 420\).
Time = 0.49 (sec) , antiderivative size = 23093, normalized size of antiderivative = 104.97 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c^2-b^2+a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (210) = 420\).
Time = 0.30 (sec) , antiderivative size = 717, normalized size of antiderivative = 3.26 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\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}} \]
(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)
Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^4} \,d x \]