\(\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx\) [832]
Optimal result
Integrand size = 14, antiderivative size = 223 \[
\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\frac {4 c \text {arctanh}\left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {4 c \text {arctanh}\left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}
\]
[Out]
4*c*arctanh((b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)*tanh(1/2*x)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2))/(-4*a*c+b^2)^(1/2)/
(b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)-4*c*arctanh((b-2*c+(-4*a*c+b^2)^(1/2))^(1/2)
*tanh(1/2*x)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2))/(-4*a*c+b^2)^(1/2)/(b-2*c+(-4*a*c+b^2)^(1/2))^(1/2)/(b+2*c+(-4*
a*c+b^2)^(1/2))^(1/2)
Rubi [A] (verified)
Time = 0.37 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3330, 2738, 214}
\[
\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\frac {4 c \text {arctanh}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {-\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c}+b-2 c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}-\frac {4 c \text {arctanh}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b-2 c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}
\]
[In]
Int[(a + b*Cosh[x] + c*Cosh[x]^2)^(-1),x]
[Out]
(4*c*ArcTanh[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a
*c]*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (4*c*ArcTanh[(Sqrt[b - 2*c + Sqrt[b
^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c
]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 2738
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 3330
Int[((a_.) + cos[(d_.) + (e_.)*(x_)]^(n_.)*(b_.) + cos[(d_.) + (e_.)*(x_)]^(n2_.)*(c_.))^(-1), x_Symbol] :> Mo
dule[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[1/(b - q + 2*c*Cos[d + e*x]^n), x], x] - Dist[2*(c/q), Int[1/
(b + q + 2*c*Cos[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {(2 c) \int \frac {1}{b-\sqrt {b^2-4 a c}+2 c \cosh (x)} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {1}{b+\sqrt {b^2-4 a c}+2 c \cosh (x)} \, dx}{\sqrt {b^2-4 a c}} \\ & = \frac {(4 c) \text {Subst}\left (\int \frac {1}{b+2 c-\sqrt {b^2-4 a c}-\left (b-2 c-\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {b^2-4 a c}}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{b+2 c+\sqrt {b^2-4 a c}-\left (b-2 c+\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {b^2-4 a c}} \\ & = \frac {4 c \text {arctanh}\left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {4 c \text {arctanh}\left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tanh \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.79 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.89
\[
\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\frac {2 \sqrt {2} c \left (\frac {\arctan \left (\frac {\left (b-2 c+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)-2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\left (-b+2 c+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}}
\]
[In]
Integrate[(a + b*Cosh[x] + c*Cosh[x]^2)^(-1),x]
[Out]
(2*Sqrt[2]*c*(ArcTan[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tanh[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*
c]]]/Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]] + ArcTan[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tanh[x/2])/Sqrt[-
2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2 - 4*a*c]]]/Sqrt[-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*
c]
Maple [A] (verified)
Time = 0.54 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.93
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method | result | size |
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default |
\(2 \left (a -b +c \right ) \left (\frac {\left (-b +2 c -\sqrt {-4 a c +b^{2}}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b -c \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}+\frac {\left (b -2 c -\sqrt {-4 a c +b^{2}}\right ) \arctan \left (\frac {\left (a -b +c \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )\) |
\(208\) |
risch |
\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{4} c^{2}-8 a^{3} b^{2} c +32 a^{3} c^{3}+a^{2} b^{4}-32 a^{2} b^{2} c^{2}+16 a^{2} c^{4}+10 a \,b^{4} c -8 a \,b^{2} c^{3}-b^{6}+b^{4} c^{2}\right ) \textit {\_Z}^{4}+\left (-8 a^{2} c^{2}+6 a \,b^{2} c -8 a \,c^{3}-b^{4}+2 b^{2} c^{2}\right ) \textit {\_Z}^{2}+c^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}+\left (\frac {6 b \,a^{3}}{c}-\frac {24 c \,a^{3}}{b}-\frac {b^{3} a^{2}}{c^{2}}-\frac {24 c^{2} a^{2}}{b}-\frac {8 b^{3} a}{c}+18 b c a -\frac {8 c^{3} a}{b}-3 b^{3}-\frac {8 a^{4}}{b}+22 a^{2} b +\frac {b^{5}}{c^{2}}+2 b \,c^{2}\right ) \textit {\_R}^{3}+\left (-\frac {b \,a^{2}}{c}+\frac {8 c \,a^{2}}{b}+\frac {4 c^{2} a}{b}+\frac {4 a^{3}}{b}+\frac {b^{3}}{c}-b c -6 a b \right ) \textit {\_R}^{2}+\left (-\frac {4 b a}{c}+\frac {4 c a}{b}-2 b +\frac {b^{3}}{c^{2}}+\frac {2 c^{2}}{b}+\frac {2 a^{2}}{b}\right ) \textit {\_R} +\frac {b}{c}-\frac {c}{b}-\frac {a}{b}\right )\) |
\(353\) |
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[In]
int(1/(a+b*cosh(x)+c*cosh(x)^2),x,method=_RETURNVERBOSE)
[Out]
2*(a-b+c)*(1/2*(-b+2*c-(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)/(a-b+c)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)
*arctanh((-a+b-c)*tanh(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))+1/2*(b-2*c-(-4*a*c+b^2)^(1/2))/(-4*a*c
+b^2)^(1/2)/(a-b+c)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)*arctan((a-b+c)*tanh(1/2*x)/(((-4*a*c+b^2)^(1/2)-a
+c)*(a-b+c))^(1/2)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3485 vs. \(2 (183) = 366\).
Time = 0.38 (sec) , antiderivative size = 3485, normalized size of antiderivative = 15.63
\[
\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+b*cosh(x)+c*cosh(x)^2),x, algorithm="fricas")
[Out]
1/2*sqrt(2)*sqrt((b^2 - 2*a*c - 2*c^2 + (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*
sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^
2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3
- 3*a*b^2)*c))*log(4*b*c^2*cosh(x) + 4*b*c^2*sinh(x) + 2*b^2*c + sqrt(2)*(b^4 - 4*a*b^2*c - (a^2*b^4 - b^6 + 8
*a*c^5 + 2*(12*a^2 - b^2)*c^4 + 6*(4*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2 - 2*(3*a^3*b^2 - 4*
a*b^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a
^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))*sqrt((b^2 - 2*a*c - 2*c^2 + (a^2*b^2 - b^4 - 4
*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 -
b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2
*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)) + 2*(4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3
- 3*a*b^2)*c^2 - (a^2*b^2 - b^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(
2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))) - 1/2*sqrt(2)*sqrt(
(b^2 - 2*a*c - 2*c^2 + (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2
- 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2
- 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))*lo
g(4*b*c^2*cosh(x) + 4*b*c^2*sinh(x) + 2*b^2*c - sqrt(2)*(b^4 - 4*a*b^2*c - (a^2*b^4 - b^6 + 8*a*c^5 + 2*(12*a^
2 - b^2)*c^4 + 6*(4*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2 - 2*(3*a^3*b^2 - 4*a*b^4)*c)*sqrt(b^
2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 +
b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))*sqrt((b^2 - 2*a*c - 2*c^2 + (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 -
b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*
a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*
c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)) + 2*(4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3 - 3*a*b^2)*c^2 -
(a^2*b^2 - b^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^
3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))) + 1/2*sqrt(2)*sqrt((b^2 - 2*a*c - 2*
c^2 - (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^
6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*
b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))*log(4*b*c^2*cosh(x)
+ 4*b*c^2*sinh(x) + 2*b^2*c + sqrt(2)*(b^4 - 4*a*b^2*c + (a^2*b^4 - b^6 + 8*a*c^5 + 2*(12*a^2 - b^2)*c^4 + 6*
(4*a^3 - 3*a*b^2)*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2 - 2*(3*a^3*b^2 - 4*a*b^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^
2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^
5 - 3*a^3*b^2 + 2*a*b^4)*c)))*sqrt((b^2 - 2*a*c - 2*c^2 - (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*
a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3
- 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^
2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)) - 2*(4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3 - 3*a*b^2)*c^2 - (a^2*b^2 - b^4)*
c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11
*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c))) - 1/2*sqrt(2)*sqrt((b^2 - 2*a*c - 2*c^2 - (a^2*b^2 -
b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16
*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)
))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c))*log(4*b*c^2*cosh(x) + 4*b*c^2*sinh(x
) + 2*b^2*c - sqrt(2)*(b^4 - 4*a*b^2*c + (a^2*b^4 - b^6 + 8*a*c^5 + 2*(12*a^2 - b^2)*c^4 + 6*(4*a^3 - 3*a*b^2)
*c^3 + (8*a^4 - 22*a^2*b^2 + 3*b^4)*c^2 - 2*(3*a^3*b^2 - 4*a*b^4)*c)*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a
*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2
*a*b^4)*c)))*sqrt((b^2 - 2*a*c - 2*c^2 - (a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3 - 3*a*b^2)*c)
*sqrt(b^2/(a^4*b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a
^2*b^2 + b^4)*c^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))/(a^2*b^2 - b^4 - 4*a*c^3 - (8*a^2 - b^2)*c^2 - 2*(2*a^3
- 3*a*b^2)*c)) - 2*(4*a*c^4 + (8*a^2 - b^2)*c^3 + 2*(2*a^3 - 3*a*b^2)*c^2 - (a^2*b^2 - b^4)*c)*sqrt(b^2/(a^4*
b^2 - 2*a^2*b^4 + b^6 - 4*a*c^5 - (16*a^2 - b^2)*c^4 - 12*(2*a^3 - a*b^2)*c^3 - 2*(8*a^4 - 11*a^2*b^2 + b^4)*c
^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c)))
Sympy [F(-1)]
Timed out. \[
\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\text {Timed out}
\]
[In]
integrate(1/(a+b*cosh(x)+c*cosh(x)**2),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\int { \frac {1}{c \cosh \left (x\right )^{2} + b \cosh \left (x\right ) + a} \,d x }
\]
[In]
integrate(1/(a+b*cosh(x)+c*cosh(x)^2),x, algorithm="maxima")
[Out]
integrate(1/(c*cosh(x)^2 + b*cosh(x) + a), x)
Giac [A] (verification not implemented)
none
Time = 61.08 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.00
\[
\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=0
\]
[In]
integrate(1/(a+b*cosh(x)+c*cosh(x)^2),x, algorithm="giac")
[Out]
0
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{a+b \cosh (x)+c \cosh ^2(x)} \, dx=\text {Hanged}
\]
[In]
int(1/(a + b*cosh(x) + c*cosh(x)^2),x)
[Out]
\text{Hanged}