3.61 \(\int \frac {1}{a-b \cosh ^4(x)} \, dx\)

Optimal. Leaf size=101 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}} \]

[Out]

1/2*arctanh(a^(1/4)*tanh(x)/(a^(1/2)-b^(1/2))^(1/2))/a^(3/4)/(a^(1/2)-b^(1/2))^(1/2)+1/2*arctanh(a^(1/4)*tanh(
x)/(a^(1/2)+b^(1/2))^(1/2))/a^(3/4)/(a^(1/2)+b^(1/2))^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3209, 1166, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Cosh[x]^4)^(-1),x]

[Out]

ArcTanh[(a^(1/4)*Tanh[x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[(a^(1/4)*Tanh
[x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]])

Rule 208

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

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 3209

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{a-b \cosh ^4(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1-x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \left (-1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\coth (x)\right )+\frac {1}{2} \left (-1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\coth (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 109, normalized size = 1.08 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}+a}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*Cosh[x]^4)^(-1),x]

[Out]

-1/2*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ArcTanh[(Sqrt
[a]*Tanh[x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]])

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fricas [B]  time = 0.51, size = 779, normalized size = 7.71 \[ -\frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} - a b}} \log \left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} + 2 \, {\left (a b - {\left (a^{4} - a^{3} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} - a b}} - 2 \, {\left (a^{3} - a^{2} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} - a b}} \log \left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} - 2 \, {\left (a b - {\left (a^{4} - a^{3} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} - a b}} - 2 \, {\left (a^{3} - a^{2} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} - a b}} \log \left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} + 2 \, {\left (a b + {\left (a^{4} - a^{3} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} - a b}} + 2 \, {\left (a^{3} - a^{2} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} - a b}} \log \left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} - 2 \, {\left (a b + {\left (a^{4} - a^{3} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} - a b}} + 2 \, {\left (a^{3} - a^{2} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cosh(x)^4),x, algorithm="fricas")

[Out]

-1/4*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(
x) + b*sinh(x)^2 + 2*(a*b - (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2
*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b)) - 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) + 1/4*sqrt(((a^2
- a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^
2 - 2*(a*b - (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^
2)) + 1)/(a^2 - a*b)) - 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) - 1/4*sqrt(-((a^2 - a*b)*sqrt(b
/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 2*(a*b + (
a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2
- a*b)) + 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) + 1/4*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4
*b + a^3*b^2)) - 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 - 2*(a*b + (a^4 - a^3*b)*
sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b)) + 2*
(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[34,61]Warning,
need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done
assuming [a,b]=[95,-54]Precision problem choosing root in common_EXT, current precision 14Warning, need to cho
ose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a
,b]=[-81,-38]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wron
g.The choice was done assuming [a,b]=[-70,-54]8*(sqrt(1/1024)*sqrt(1/1048576*(1048576*a^2+1048576*a*sqrt(a*b))
/(a^4-a^3*b))*ln(abs(60*a^4*b*exp(x)^2+6*a^4*b-24*a^4*sqrt(a*b)*exp(x)^2-68*a^3*b^2*exp(x)^2-2*a^3*b^2-16*a^3*
b*sqrt(a*b)*exp(x)^2-12*a^3*b*sqrt(a*b)+48*a^3*b*sqrt(a^2-a*sqrt(a*b))*exp(x)^2+6*a^3*b*sqrt(a^2-a*sqrt(a*b))-
24*a^3*sqrt(a*b)*sqrt(a^2-a*sqrt(a*b))*exp(x)^2-16*a^2*b^3*exp(x)^2-8*a^2*b^3+64*a^2*b^2*sqrt(a*b)*exp(x)^2+16
*a^2*b^2*sqrt(a*b)-61*a^2*b^2*sqrt(a^2-a*sqrt(a*b))*exp(x)^2-5*a^2*b^2*sqrt(a^2-a*sqrt(a*b))+5*a^2*b*sqrt(a*b)
*sqrt(a^2-a*sqrt(a*b))*exp(x)^2-9*a^2*b*sqrt(a*b)*sqrt(a^2-a*sqrt(a*b))-4*a*b^3*sqrt(a^2-a*sqrt(a*b))*exp(x)^2
-4*a*b^3*sqrt(a^2-a*sqrt(a*b))+36*a*b^2*sqrt(a*b)*sqrt(a^2-a*sqrt(a*b))*exp(x)^2+12*a*b^2*sqrt(a*b)*sqrt(a^2-a
*sqrt(a*b))))-sqrt(1/1024)*sqrt(1/1048576*(1048576*a^2+1048576*a*sqrt(a*b))/(a^4-a^3*b))*ln(abs(60*a^4*b*exp(x
)^2+6*a^4*b-24*a^4*sqrt(a*b)*exp(x)^2-68*a^3*b^2*exp(x)^2-2*a^3*b^2-16*a^3*b*sqrt(a*b)*exp(x)^2-12*a^3*b*sqrt(
a*b)-48*a^3*b*sqrt(a^2-a*sqrt(a*b))*exp(x)^2-6*a^3*b*sqrt(a^2-a*sqrt(a*b))+24*a^3*sqrt(a*b)*sqrt(a^2-a*sqrt(a*
b))*exp(x)^2-16*a^2*b^3*exp(x)^2-8*a^2*b^3+64*a^2*b^2*sqrt(a*b)*exp(x)^2+16*a^2*b^2*sqrt(a*b)+61*a^2*b^2*sqrt(
a^2-a*sqrt(a*b))*exp(x)^2+5*a^2*b^2*sqrt(a^2-a*sqrt(a*b))-5*a^2*b*sqrt(a*b)*sqrt(a^2-a*sqrt(a*b))*exp(x)^2+9*a
^2*b*sqrt(a*b)*sqrt(a^2-a*sqrt(a*b))+4*a*b^3*sqrt(a^2-a*sqrt(a*b))*exp(x)^2+4*a*b^3*sqrt(a^2-a*sqrt(a*b))-36*a
*b^2*sqrt(a*b)*sqrt(a^2-a*sqrt(a*b))*exp(x)^2-12*a*b^2*sqrt(a*b)*sqrt(a^2-a*sqrt(a*b))))+sqrt(1/1024)*sqrt(1/1
048576*(1048576*a^2-1048576*a*sqrt(a*b))/(a^4-a^3*b))*ln(abs(60*a^4*b*exp(x)^2+6*a^4*b+24*a^4*sqrt(a*b)*exp(x)
^2-68*a^3*b^2*exp(x)^2-2*a^3*b^2+16*a^3*b*sqrt(a*b)*exp(x)^2+12*a^3*b*sqrt(a*b)+48*a^3*b*sqrt(a^2+a*sqrt(a*b))
*exp(x)^2+6*a^3*b*sqrt(a^2+a*sqrt(a*b))+24*a^3*sqrt(a*b)*sqrt(a^2+a*sqrt(a*b))*exp(x)^2-16*a^2*b^3*exp(x)^2-8*
a^2*b^3-64*a^2*b^2*sqrt(a*b)*exp(x)^2-16*a^2*b^2*sqrt(a*b)-61*a^2*b^2*sqrt(a^2+a*sqrt(a*b))*exp(x)^2-5*a^2*b^2
*sqrt(a^2+a*sqrt(a*b))-5*a^2*b*sqrt(a*b)*sqrt(a^2+a*sqrt(a*b))*exp(x)^2+9*a^2*b*sqrt(a*b)*sqrt(a^2+a*sqrt(a*b)
)-4*a*b^3*sqrt(a^2+a*sqrt(a*b))*exp(x)^2-4*a*b^3*sqrt(a^2+a*sqrt(a*b))-36*a*b^2*sqrt(a*b)*sqrt(a^2+a*sqrt(a*b)
)*exp(x)^2-12*a*b^2*sqrt(a*b)*sqrt(a^2+a*sqrt(a*b))))-sqrt(1/1024)*sqrt(1/1048576*(1048576*a^2-1048576*a*sqrt(
a*b))/(a^4-a^3*b))*ln(abs(60*a^4*b*exp(x)^2+6*a^4*b+24*a^4*sqrt(a*b)*exp(x)^2-68*a^3*b^2*exp(x)^2-2*a^3*b^2+16
*a^3*b*sqrt(a*b)*exp(x)^2+12*a^3*b*sqrt(a*b)-48*a^3*b*sqrt(a^2+a*sqrt(a*b))*exp(x)^2-6*a^3*b*sqrt(a^2+a*sqrt(a
*b))-24*a^3*sqrt(a*b)*sqrt(a^2+a*sqrt(a*b))*exp(x)^2-16*a^2*b^3*exp(x)^2-8*a^2*b^3-64*a^2*b^2*sqrt(a*b)*exp(x)
^2-16*a^2*b^2*sqrt(a*b)+61*a^2*b^2*sqrt(a^2+a*sqrt(a*b))*exp(x)^2+5*a^2*b^2*sqrt(a^2+a*sqrt(a*b))+5*a^2*b*sqrt
(a*b)*sqrt(a^2+a*sqrt(a*b))*exp(x)^2-9*a^2*b*sqrt(a*b)*sqrt(a^2+a*sqrt(a*b))+4*a*b^3*sqrt(a^2+a*sqrt(a*b))*exp
(x)^2+4*a*b^3*sqrt(a^2+a*sqrt(a*b))+36*a*b^2*sqrt(a*b)*sqrt(a^2+a*sqrt(a*b))*exp(x)^2+12*a*b^2*sqrt(a*b)*sqrt(
a^2+a*sqrt(a*b)))))

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maple [C]  time = 0.09, size = 127, normalized size = 1.26 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a -b \right ) \textit {\_Z}^{8}+\left (-4 a -4 b \right ) \textit {\_Z}^{6}+\left (6 a -6 b \right ) \textit {\_Z}^{4}+\left (-4 a -4 b \right ) \textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 \textit {\_R}^{4}-3 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -\textit {\_R}^{7} b -3 \textit {\_R}^{5} a -3 \textit {\_R}^{5} b +3 \textit {\_R}^{3} a -3 \textit {\_R}^{3} b -\textit {\_R} a -\textit {\_R} b}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*cosh(x)^4),x)

[Out]

1/4*sum((-_R^6+3*_R^4-3*_R^2+1)/(_R^7*a-_R^7*b-3*_R^5*a-3*_R^5*b+3*_R^3*a-3*_R^3*b-_R*a-_R*b)*ln(tanh(1/2*x)-_
R),_R=RootOf((a-b)*_Z^8+(-4*a-4*b)*_Z^6+(6*a-6*b)*_Z^4+(-4*a-4*b)*_Z^2+a-b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{b \cosh \relax (x)^{4} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cosh(x)^4),x, algorithm="maxima")

[Out]

-integrate(1/(b*cosh(x)^4 - a), x)

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mupad [B]  time = 8.90, size = 1487, normalized size = 14.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*cosh(x)^4),x)

[Out]

log(((((1/(a^2 - (a^3*b)^(1/2)))^(1/2)*((4194304*(b^4*exp(2*x) + 253*a*b^3 + 1184*a^3*b - 512*a^4 + b^4 - 930*
a^2*b^2 - 1392*a^2*b^2*exp(2*x) + 627*a*b^3*exp(2*x) + 768*a^3*b*exp(2*x)))/(b^6*(a - b)^2) - (8388608*a*(1/(a
^2 - (a^3*b)^(1/2)))^(1/2)*(512*a^3*exp(2*x) + 6*b^3*exp(2*x) + 181*a*b^2 - 432*a^2*b + 256*a^3 + 5*b^3 + 622*
a*b^2*exp(2*x) - 1152*a^2*b*exp(2*x)))/(b^6*(a - b))))/4 - (2097152*(176*a*b - 1536*a^2*exp(2*x) + 134*b^2*exp
(2*x) - 256*a^2 + 75*b^2 + 1408*a*b*exp(2*x)))/(b^6*(a - b)))*(1/(a^2 - (a^3*b)^(1/2)))^(1/2))/4 + (524288*(10
24*a^3*exp(2*x) + 35*b^3*exp(2*x) + 185*a*b^2 - 464*a^2*b + 256*a^3 + 24*b^3 + 988*a*b^2*exp(2*x) - 2048*a^2*b
*exp(2*x)))/(a*b^6*(a - b)^2))*(-(a^2 + (a^3*b)^(1/2))/(16*(a^3*b - a^4)))^(1/2) - log(((((1/(a^2 - (a^3*b)^(1
/2)))^(1/2)*((4194304*(b^4*exp(2*x) + 253*a*b^3 + 1184*a^3*b - 512*a^4 + b^4 - 930*a^2*b^2 - 1392*a^2*b^2*exp(
2*x) + 627*a*b^3*exp(2*x) + 768*a^3*b*exp(2*x)))/(b^6*(a - b)^2) + (8388608*a*(1/(a^2 - (a^3*b)^(1/2)))^(1/2)*
(512*a^3*exp(2*x) + 6*b^3*exp(2*x) + 181*a*b^2 - 432*a^2*b + 256*a^3 + 5*b^3 + 622*a*b^2*exp(2*x) - 1152*a^2*b
*exp(2*x)))/(b^6*(a - b))))/4 + (2097152*(176*a*b - 1536*a^2*exp(2*x) + 134*b^2*exp(2*x) - 256*a^2 + 75*b^2 +
1408*a*b*exp(2*x)))/(b^6*(a - b)))*(1/(a^2 - (a^3*b)^(1/2)))^(1/2))/4 + (524288*(1024*a^3*exp(2*x) + 35*b^3*ex
p(2*x) + 185*a*b^2 - 464*a^2*b + 256*a^3 + 24*b^3 + 988*a*b^2*exp(2*x) - 2048*a^2*b*exp(2*x)))/(a*b^6*(a - b)^
2))*(-(a^2 + (a^3*b)^(1/2))/(16*(a^3*b - a^4)))^(1/2) + log(((1/(a^2 + (a^3*b)^(1/2)))^(1/2)*(((1/(a^2 + (a^3*
b)^(1/2)))^(1/2)*((4194304*(b^4*exp(2*x) + 253*a*b^3 + 1184*a^3*b - 512*a^4 + b^4 - 930*a^2*b^2 - 1392*a^2*b^2
*exp(2*x) + 627*a*b^3*exp(2*x) + 768*a^3*b*exp(2*x)))/(b^6*(a - b)^2) - (8388608*a*(1/(a^2 + (a^3*b)^(1/2)))^(
1/2)*(512*a^3*exp(2*x) + 6*b^3*exp(2*x) + 181*a*b^2 - 432*a^2*b + 256*a^3 + 5*b^3 + 622*a*b^2*exp(2*x) - 1152*
a^2*b*exp(2*x)))/(b^6*(a - b))))/4 - (2097152*(176*a*b - 1536*a^2*exp(2*x) + 134*b^2*exp(2*x) - 256*a^2 + 75*b
^2 + 1408*a*b*exp(2*x)))/(b^6*(a - b))))/4 + (524288*(1024*a^3*exp(2*x) + 35*b^3*exp(2*x) + 185*a*b^2 - 464*a^
2*b + 256*a^3 + 24*b^3 + 988*a*b^2*exp(2*x) - 2048*a^2*b*exp(2*x)))/(a*b^6*(a - b)^2))*(-(a^2 - (a^3*b)^(1/2))
/(16*(a^3*b - a^4)))^(1/2) - log(((1/(a^2 + (a^3*b)^(1/2)))^(1/2)*(((1/(a^2 + (a^3*b)^(1/2)))^(1/2)*((4194304*
(b^4*exp(2*x) + 253*a*b^3 + 1184*a^3*b - 512*a^4 + b^4 - 930*a^2*b^2 - 1392*a^2*b^2*exp(2*x) + 627*a*b^3*exp(2
*x) + 768*a^3*b*exp(2*x)))/(b^6*(a - b)^2) + (8388608*a*(1/(a^2 + (a^3*b)^(1/2)))^(1/2)*(512*a^3*exp(2*x) + 6*
b^3*exp(2*x) + 181*a*b^2 - 432*a^2*b + 256*a^3 + 5*b^3 + 622*a*b^2*exp(2*x) - 1152*a^2*b*exp(2*x)))/(b^6*(a -
b))))/4 + (2097152*(176*a*b - 1536*a^2*exp(2*x) + 134*b^2*exp(2*x) - 256*a^2 + 75*b^2 + 1408*a*b*exp(2*x)))/(b
^6*(a - b))))/4 + (524288*(1024*a^3*exp(2*x) + 35*b^3*exp(2*x) + 185*a*b^2 - 464*a^2*b + 256*a^3 + 24*b^3 + 98
8*a*b^2*exp(2*x) - 2048*a^2*b*exp(2*x)))/(a*b^6*(a - b)^2))*(-(a^2 - (a^3*b)^(1/2))/(16*(a^3*b - a^4)))^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cosh(x)**4),x)

[Out]

Timed out

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