Integrand size = 12, antiderivative size = 140 \[ \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx=\frac {a \left (5 a^2-3 c^2\right ) \log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{2 c^7}-\frac {c \cosh (x)+a \sinh (x)}{3 c^2 (a+a \cosh (x)+c \sinh (x))^3}-\frac {5 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{6 c^4 (a+a \cosh (x)+c \sinh (x))^2}-\frac {c \left (15 a^2-4 c^2\right ) \cosh (x)+a \left (15 a^2-4 c^2\right ) \sinh (x)}{6 c^6 (a+a \cosh (x)+c \sinh (x))} \] Output:
1/2*a*(5*a^2-3*c^2)*ln(a+c*tanh(1/2*x))/c^7-1/3*(c*cosh(x)+a*sinh(x))/c^2/ (a+a*cosh(x)+c*sinh(x))^3-5/6*(a*c*cosh(x)+a^2*sinh(x))/c^4/(a+a*cosh(x)+c *sinh(x))^2-1/6*(c*(15*a^2-4*c^2)*cosh(x)+a*(15*a^2-4*c^2)*sinh(x))/c^6/(a +a*cosh(x)+c*sinh(x))
Leaf count is larger than twice the leaf count of optimal. \(300\) vs. \(2(140)=280\).
Time = 0.40 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.14 \[ \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx=\frac {192 \left (-5 a^3+3 a c^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )+192 a \left (5 a^2-3 c^2\right ) \log \left (a \cosh \left (\frac {x}{2}\right )+c \sinh \left (\frac {x}{2}\right )\right )-\frac {c \text {sech}^6\left (\frac {x}{2}\right ) \left (-150 a^5 c+130 a^3 c^3-24 a c^5+\left (-75 a^5 c+75 a^3 c^3+12 a c^5\right ) \cosh (x)+6 a c \left (25 a^4-15 a^2 c^2+4 c^4\right ) \cosh (2 x)+75 a^5 c \cosh (3 x)-35 a^3 c^3 \cosh (3 x)+4 a c^5 \cosh (3 x)+150 a^6 \sinh (x)-255 a^4 c^2 \sinh (x)+129 a^2 c^4 \sinh (x)-12 c^6 \sinh (x)+120 a^6 \sinh (2 x)-72 a^4 c^2 \sinh (2 x)+36 a^2 c^4 \sinh (2 x)+30 a^6 \sinh (3 x)+37 a^4 c^2 \sinh (3 x)-27 a^2 c^4 \sinh (3 x)+4 c^6 \sinh (3 x)\right )}{a \left (a+c \tanh \left (\frac {x}{2}\right )\right )^3}}{384 c^7} \] Input:
Integrate[(a + a*Cosh[x] + c*Sinh[x])^(-4),x]
Output:
(192*(-5*a^3 + 3*a*c^2)*Log[Cosh[x/2]] + 192*a*(5*a^2 - 3*c^2)*Log[a*Cosh[ x/2] + c*Sinh[x/2]] - (c*Sech[x/2]^6*(-150*a^5*c + 130*a^3*c^3 - 24*a*c^5 + (-75*a^5*c + 75*a^3*c^3 + 12*a*c^5)*Cosh[x] + 6*a*c*(25*a^4 - 15*a^2*c^2 + 4*c^4)*Cosh[2*x] + 75*a^5*c*Cosh[3*x] - 35*a^3*c^3*Cosh[3*x] + 4*a*c^5* Cosh[3*x] + 150*a^6*Sinh[x] - 255*a^4*c^2*Sinh[x] + 129*a^2*c^4*Sinh[x] - 12*c^6*Sinh[x] + 120*a^6*Sinh[2*x] - 72*a^4*c^2*Sinh[2*x] + 36*a^2*c^4*Sin h[2*x] + 30*a^6*Sinh[3*x] + 37*a^4*c^2*Sinh[3*x] - 27*a^2*c^4*Sinh[3*x] + 4*c^6*Sinh[3*x]))/(a*(a + c*Tanh[x/2])^3))/(384*c^7)
Time = 0.71 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3608, 25, 3042, 3635, 25, 3042, 3632, 3042, 3603, 16}
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.
| \(\displaystyle \int \frac {1}{(a \cosh (x)+a+c \sinh (x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \cos (i x)+a-i c \sin (i x))^4}dx\) |
| \(\Big \downarrow \) 3608 |
| \(\displaystyle -\frac {\int -\frac {-2 \cosh (x) a+3 a-2 c \sinh (x)}{(\cosh (x) a+a+c \sinh (x))^3}dx}{3 c^2}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-2 \cosh (x) a+3 a-2 c \sinh (x)}{(\cosh (x) a+a+c \sinh (x))^3}dx}{3 c^2}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}+\frac {\int \frac {-2 \cos (i x) a+3 a+2 i c \sin (i x)}{(\cos (i x) a+a-i c \sin (i x))^3}dx}{3 c^2}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {-\frac {\int -\frac {-5 \cosh (x) a^2-5 c \sinh (x) a+2 \left (5 a^2-2 c^2\right )}{(\cosh (x) a+a+c \sinh (x))^2}dx}{2 c^2}-\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}}{3 c^2}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-5 \cosh (x) a^2-5 c \sinh (x) a+2 \left (5 a^2-2 c^2\right )}{(\cosh (x) a+a+c \sinh (x))^2}dx}{2 c^2}-\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}}{3 c^2}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}+\frac {-\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}+\frac {\int \frac {-5 \cos (i x) a^2+5 i c \sin (i x) a+2 \left (5 a^2-2 c^2\right )}{(\cos (i x) a+a-i c \sin (i x))^2}dx}{2 c^2}}{3 c^2}\) |
| \(\Big \downarrow \) 3632 |
| \(\displaystyle \frac {\frac {-3 a \left (3-\frac {5 a^2}{c^2}\right ) \int \frac {1}{\cosh (x) a+a+c \sinh (x)}dx-\frac {a \left (15 a^2-4 c^2\right ) \sinh (x)+c \left (15 a^2-4 c^2\right ) \cosh (x)}{c^2 (a \cosh (x)+a+c \sinh (x))}}{2 c^2}-\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}}{3 c^2}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}+\frac {-\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}+\frac {-\frac {a \left (15 a^2-4 c^2\right ) \sinh (x)+c \left (15 a^2-4 c^2\right ) \cosh (x)}{c^2 (a \cosh (x)+a+c \sinh (x))}-3 a \left (3-\frac {5 a^2}{c^2}\right ) \int \frac {1}{\cos (i x) a+a-i c \sin (i x)}dx}{2 c^2}}{3 c^2}\) |
| \(\Big \downarrow \) 3603 |
| \(\displaystyle \frac {\frac {-6 a \left (3-\frac {5 a^2}{c^2}\right ) \int \frac {1}{2 a+2 c \tanh \left (\frac {x}{2}\right )}d\tanh \left (\frac {x}{2}\right )-\frac {a \left (15 a^2-4 c^2\right ) \sinh (x)+c \left (15 a^2-4 c^2\right ) \cosh (x)}{c^2 (a \cosh (x)+a+c \sinh (x))}}{2 c^2}-\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}}{3 c^2}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {-\frac {3 a \left (3-\frac {5 a^2}{c^2}\right ) \log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{c}-\frac {a \left (15 a^2-4 c^2\right ) \sinh (x)+c \left (15 a^2-4 c^2\right ) \cosh (x)}{c^2 (a \cosh (x)+a+c \sinh (x))}}{2 c^2}-\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}}{3 c^2}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3}\) |
Input:
Int[(a + a*Cosh[x] + c*Sinh[x])^(-4),x]
Output:
-1/3*(c*Cosh[x] + a*Sinh[x])/(c^2*(a + a*Cosh[x] + c*Sinh[x])^3) + ((-5*(a *c*Cosh[x] + a^2*Sinh[x]))/(2*c^2*(a + a*Cosh[x] + c*Sinh[x])^2) + ((-3*a* (3 - (5*a^2)/c^2)*Log[a + c*Tanh[x/2]])/c - (c*(15*a^2 - 4*c^2)*Cosh[x] + a*(15*a^2 - 4*c^2)*Sinh[x])/(c^2*(a + a*Cosh[x] + c*Sinh[x])))/(2*c^2))/(3 *c^2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), 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]
Time = 21.93 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3} c^{2}}{3}-2 a \tanh \left (\frac {x}{2}\right )^{2} c +10 a^{2} \tanh \left (\frac {x}{2}\right )-3 c^{2} \tanh \left (\frac {x}{2}\right )}{8 c^{6}}-\frac {-15 a^{4}+18 a^{2} c^{2}-3 c^{4}}{8 c^{7} \left (a +c \tanh \left (\frac {x}{2}\right )\right )}+\frac {a \left (5 a^{2}-3 c^{2}\right ) \ln \left (a +c \tanh \left (\frac {x}{2}\right )\right )}{2 c^{7}}-\frac {3 a \left (a^{4}-2 a^{2} c^{2}+c^{4}\right )}{8 c^{7} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {-a^{6}+3 a^{4} c^{2}-3 a^{2} c^{4}+c^{6}}{24 c^{7} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{3}}\) | \(178\) |
| risch | \(\frac {-150 \,{\mathrm e}^{x} c \,a^{4}+30 a^{2} c^{3} {\mathrm e}^{x}-45 a^{4} c +4 c^{5}+150 \,{\mathrm e}^{3 x} a^{5}-12 c^{5} {\mathrm e}^{2 x}+15 a^{5} {\mathrm e}^{5 x}+75 a^{5} {\mathrm e}^{4 x}+150 a^{5} {\mathrm e}^{2 x}+75 \,{\mathrm e}^{x} a^{5}-12 a \,c^{4}+41 a^{3} c^{2}+60 a^{3} c^{2} {\mathrm e}^{x}-15 a \,c^{4} {\mathrm e}^{x}+30 a^{4} c \,{\mathrm e}^{5 x}-18 a^{2} c^{3} {\mathrm e}^{5 x}-45 a^{3} c^{2} {\mathrm e}^{4 x}-60 a^{3} c^{2} {\mathrm e}^{2 x}+12 a \,c^{4} {\mathrm e}^{2 x}+24 a \,c^{4} {\mathrm e}^{3 x}-150 a^{4} c \,{\mathrm e}^{2 x}-130 a^{3} c^{2} {\mathrm e}^{3 x}-45 a^{2} c^{3} {\mathrm e}^{4 x}+6 a^{3} c^{2} {\mathrm e}^{5 x}-9 a \,c^{4} {\mathrm e}^{5 x}+75 a^{4} c \,{\mathrm e}^{4 x}+60 a^{2} c^{3} {\mathrm e}^{2 x}-3 a^{2} c^{3}+15 a^{5}}{3 \left ({\mathrm e}^{2 x} a +c \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+a -c \right )^{3} c^{6}}-\frac {5 a^{3} \ln \left (1+{\mathrm e}^{x}\right )}{2 c^{7}}+\frac {3 a \ln \left (1+{\mathrm e}^{x}\right )}{2 c^{5}}+\frac {5 a^{3} \ln \left ({\mathrm e}^{x}+\frac {a -c}{a +c}\right )}{2 c^{7}}-\frac {3 a \ln \left ({\mathrm e}^{x}+\frac {a -c}{a +c}\right )}{2 c^{5}}\) | \(370\) |
Input:
int(1/(a+a*cosh(x)+c*sinh(x))^4,x,method=_RETURNVERBOSE)
Output:
-1/8/c^6*(1/3*tanh(1/2*x)^3*c^2-2*a*tanh(1/2*x)^2*c+10*a^2*tanh(1/2*x)-3*c ^2*tanh(1/2*x))-1/8*(-15*a^4+18*a^2*c^2-3*c^4)/c^7/(a+c*tanh(1/2*x))+1/2*a *(5*a^2-3*c^2)*ln(a+c*tanh(1/2*x))/c^7-3/8*a/c^7*(a^4-2*a^2*c^2+c^4)/(a+c* tanh(1/2*x))^2-1/24/c^7*(-a^6+3*a^4*c^2-3*a^2*c^4+c^6)/(a+c*tanh(1/2*x))^3
Leaf count of result is larger than twice the leaf count of optimal. 4015 vs. \(2 (130) = 260\).
Time = 0.13 (sec) , antiderivative size = 4015, normalized size of antiderivative = 28.68 \[ \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*cosh(x)+c*sinh(x))^4,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx=\text {Timed out} \] Input:
integrate(1/(a+a*cosh(x)+c*sinh(x))**4,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (130) = 260\).
Time = 0.09 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.48 \[ \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx=-\frac {15 \, a^{5} + 45 \, a^{4} c + 41 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - 12 \, a c^{4} - 4 \, c^{5} + 15 \, {\left (5 \, a^{5} + 10 \, a^{4} c + 4 \, a^{3} c^{2} - 2 \, a^{2} c^{3} - a c^{4}\right )} e^{\left (-x\right )} + 6 \, {\left (25 \, a^{5} + 25 \, a^{4} c - 10 \, a^{3} c^{2} - 10 \, a^{2} c^{3} + 2 \, a c^{4} + 2 \, c^{5}\right )} e^{\left (-2 \, x\right )} + 2 \, {\left (75 \, a^{5} - 65 \, a^{3} c^{2} + 12 \, a c^{4}\right )} e^{\left (-3 \, x\right )} + 15 \, {\left (5 \, a^{5} - 5 \, a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (5 \, a^{5} - 10 \, a^{4} c + 2 \, a^{3} c^{2} + 6 \, a^{2} c^{3} - 3 \, a c^{4}\right )} e^{\left (-5 \, x\right )}}{3 \, {\left (a^{3} c^{6} + 3 \, a^{2} c^{7} + 3 \, a c^{8} + c^{9} + 6 \, {\left (a^{3} c^{6} + 2 \, a^{2} c^{7} + a c^{8}\right )} e^{\left (-x\right )} + 3 \, {\left (5 \, a^{3} c^{6} + 5 \, a^{2} c^{7} - a c^{8} - c^{9}\right )} e^{\left (-2 \, x\right )} + 4 \, {\left (5 \, a^{3} c^{6} - 3 \, a c^{8}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (5 \, a^{3} c^{6} - 5 \, a^{2} c^{7} - a c^{8} + c^{9}\right )} e^{\left (-4 \, x\right )} + 6 \, {\left (a^{3} c^{6} - 2 \, a^{2} c^{7} + a c^{8}\right )} e^{\left (-5 \, x\right )} + {\left (a^{3} c^{6} - 3 \, a^{2} c^{7} + 3 \, a c^{8} - c^{9}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac {{\left (5 \, a^{3} - 3 \, a c^{2}\right )} \log \left (-{\left (a - c\right )} e^{\left (-x\right )} - a - c\right )}{2 \, c^{7}} - \frac {{\left (5 \, a^{3} - 3 \, a c^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, c^{7}} \] Input:
integrate(1/(a+a*cosh(x)+c*sinh(x))^4,x, algorithm="maxima")
Output:
-1/3*(15*a^5 + 45*a^4*c + 41*a^3*c^2 + 3*a^2*c^3 - 12*a*c^4 - 4*c^5 + 15*( 5*a^5 + 10*a^4*c + 4*a^3*c^2 - 2*a^2*c^3 - a*c^4)*e^(-x) + 6*(25*a^5 + 25* a^4*c - 10*a^3*c^2 - 10*a^2*c^3 + 2*a*c^4 + 2*c^5)*e^(-2*x) + 2*(75*a^5 - 65*a^3*c^2 + 12*a*c^4)*e^(-3*x) + 15*(5*a^5 - 5*a^4*c - 3*a^3*c^2 + 3*a^2* c^3)*e^(-4*x) + 3*(5*a^5 - 10*a^4*c + 2*a^3*c^2 + 6*a^2*c^3 - 3*a*c^4)*e^( -5*x))/(a^3*c^6 + 3*a^2*c^7 + 3*a*c^8 + c^9 + 6*(a^3*c^6 + 2*a^2*c^7 + a*c ^8)*e^(-x) + 3*(5*a^3*c^6 + 5*a^2*c^7 - a*c^8 - c^9)*e^(-2*x) + 4*(5*a^3*c ^6 - 3*a*c^8)*e^(-3*x) + 3*(5*a^3*c^6 - 5*a^2*c^7 - a*c^8 + c^9)*e^(-4*x) + 6*(a^3*c^6 - 2*a^2*c^7 + a*c^8)*e^(-5*x) + (a^3*c^6 - 3*a^2*c^7 + 3*a*c^ 8 - c^9)*e^(-6*x)) + 1/2*(5*a^3 - 3*a*c^2)*log(-(a - c)*e^(-x) - a - c)/c^ 7 - 1/2*(5*a^3 - 3*a*c^2)*log(e^(-x) + 1)/c^7
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (130) = 260\).
Time = 0.14 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.69 \[ \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx=\frac {{\left (5 \, a^{4} + 5 \, a^{3} c - 3 \, a^{2} c^{2} - 3 \, a c^{3}\right )} \log \left ({\left | a e^{x} + c e^{x} + a - c \right |}\right )}{2 \, {\left (a c^{7} + c^{8}\right )}} - \frac {{\left (5 \, a^{3} - 3 \, a c^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, c^{7}} + \frac {15 \, a^{5} e^{\left (5 \, x\right )} + 30 \, a^{4} c e^{\left (5 \, x\right )} + 6 \, a^{3} c^{2} e^{\left (5 \, x\right )} - 18 \, a^{2} c^{3} e^{\left (5 \, x\right )} - 9 \, a c^{4} e^{\left (5 \, x\right )} + 75 \, a^{5} e^{\left (4 \, x\right )} + 75 \, a^{4} c e^{\left (4 \, x\right )} - 45 \, a^{3} c^{2} e^{\left (4 \, x\right )} - 45 \, a^{2} c^{3} e^{\left (4 \, x\right )} + 150 \, a^{5} e^{\left (3 \, x\right )} - 130 \, a^{3} c^{2} e^{\left (3 \, x\right )} + 24 \, a c^{4} e^{\left (3 \, x\right )} + 150 \, a^{5} e^{\left (2 \, x\right )} - 150 \, a^{4} c e^{\left (2 \, x\right )} - 60 \, a^{3} c^{2} e^{\left (2 \, x\right )} + 60 \, a^{2} c^{3} e^{\left (2 \, x\right )} + 12 \, a c^{4} e^{\left (2 \, x\right )} - 12 \, c^{5} e^{\left (2 \, x\right )} + 75 \, a^{5} e^{x} - 150 \, a^{4} c e^{x} + 60 \, a^{3} c^{2} e^{x} + 30 \, a^{2} c^{3} e^{x} - 15 \, a c^{4} e^{x} + 15 \, a^{5} - 45 \, a^{4} c + 41 \, a^{3} c^{2} - 3 \, a^{2} c^{3} - 12 \, a c^{4} + 4 \, c^{5}}{3 \, {\left (a e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + a - c\right )}^{3} c^{6}} \] Input:
integrate(1/(a+a*cosh(x)+c*sinh(x))^4,x, algorithm="giac")
Output:
1/2*(5*a^4 + 5*a^3*c - 3*a^2*c^2 - 3*a*c^3)*log(abs(a*e^x + c*e^x + a - c) )/(a*c^7 + c^8) - 1/2*(5*a^3 - 3*a*c^2)*log(e^x + 1)/c^7 + 1/3*(15*a^5*e^( 5*x) + 30*a^4*c*e^(5*x) + 6*a^3*c^2*e^(5*x) - 18*a^2*c^3*e^(5*x) - 9*a*c^4 *e^(5*x) + 75*a^5*e^(4*x) + 75*a^4*c*e^(4*x) - 45*a^3*c^2*e^(4*x) - 45*a^2 *c^3*e^(4*x) + 150*a^5*e^(3*x) - 130*a^3*c^2*e^(3*x) + 24*a*c^4*e^(3*x) + 150*a^5*e^(2*x) - 150*a^4*c*e^(2*x) - 60*a^3*c^2*e^(2*x) + 60*a^2*c^3*e^(2 *x) + 12*a*c^4*e^(2*x) - 12*c^5*e^(2*x) + 75*a^5*e^x - 150*a^4*c*e^x + 60* a^3*c^2*e^x + 30*a^2*c^3*e^x - 15*a*c^4*e^x + 15*a^5 - 45*a^4*c + 41*a^3*c ^2 - 3*a^2*c^3 - 12*a*c^4 + 4*c^5)/((a*e^(2*x) + c*e^(2*x) + 2*a*e^x + a - c)^3*c^6)
Timed out. \[ \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^4} \,d x \] Input:
int(1/(a + a*cosh(x) + c*sinh(x))^4,x)
Output:
int(1/(a + a*cosh(x) + c*sinh(x))^4, x)
Time = 0.34 (sec) , antiderivative size = 2186, normalized size of antiderivative = 15.61 \[ \int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx =\text {Too large to display} \] Input:
int(1/(a+a*cosh(x)+c*sinh(x))^4,x)
Output:
( - 15*e**(6*x)*log(e**x + 1)*a**6 - 45*e**(6*x)*log(e**x + 1)*a**5*c - 36 *e**(6*x)*log(e**x + 1)*a**4*c**2 + 12*e**(6*x)*log(e**x + 1)*a**3*c**3 + 27*e**(6*x)*log(e**x + 1)*a**2*c**4 + 9*e**(6*x)*log(e**x + 1)*a*c**5 + 15 *e**(6*x)*log(e**x*a + e**x*c + a - c)*a**6 + 45*e**(6*x)*log(e**x*a + e** x*c + a - c)*a**5*c + 36*e**(6*x)*log(e**x*a + e**x*c + a - c)*a**4*c**2 - 12*e**(6*x)*log(e**x*a + e**x*c + a - c)*a**3*c**3 - 27*e**(6*x)*log(e**x *a + e**x*c + a - c)*a**2*c**4 - 9*e**(6*x)*log(e**x*a + e**x*c + a - c)*a *c**5 - 5*e**(6*x)*a**5*c - 15*e**(6*x)*a**4*c**2 - 12*e**(6*x)*a**3*c**3 + 4*e**(6*x)*a**2*c**4 + 9*e**(6*x)*a*c**5 + 3*e**(6*x)*c**6 - 90*e**(5*x) *log(e**x + 1)*a**6 - 180*e**(5*x)*log(e**x + 1)*a**5*c - 36*e**(5*x)*log( e**x + 1)*a**4*c**2 + 108*e**(5*x)*log(e**x + 1)*a**3*c**3 + 54*e**(5*x)*l og(e**x + 1)*a**2*c**4 + 90*e**(5*x)*log(e**x*a + e**x*c + a - c)*a**6 + 1 80*e**(5*x)*log(e**x*a + e**x*c + a - c)*a**5*c + 36*e**(5*x)*log(e**x*a + e**x*c + a - c)*a**4*c**2 - 108*e**(5*x)*log(e**x*a + e**x*c + a - c)*a** 3*c**3 - 54*e**(5*x)*log(e**x*a + e**x*c + a - c)*a**2*c**4 - 225*e**(4*x) *log(e**x + 1)*a**6 - 225*e**(4*x)*log(e**x + 1)*a**5*c + 180*e**(4*x)*log (e**x + 1)*a**4*c**2 + 180*e**(4*x)*log(e**x + 1)*a**3*c**3 - 27*e**(4*x)* log(e**x + 1)*a**2*c**4 - 27*e**(4*x)*log(e**x + 1)*a*c**5 + 225*e**(4*x)* log(e**x*a + e**x*c + a - c)*a**6 + 225*e**(4*x)*log(e**x*a + e**x*c + a - c)*a**5*c - 180*e**(4*x)*log(e**x*a + e**x*c + a - c)*a**4*c**2 - 180*...