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

Optimal. Leaf size=361 \[ \frac{\sqrt{\sqrt{a}-\sqrt{a+b}} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a+b}+\sqrt{a}}-\sqrt{2} \sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}-\sqrt{a+b}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{a+b}}-\frac{\sqrt{\sqrt{a}-\sqrt{a+b}} \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a+b}+\sqrt{a}}+\sqrt{2} \sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}-\sqrt{a+b}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{a+b}}-\frac{\sqrt{\sqrt{a+b}+\sqrt{a}} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt{\sqrt{a+b}+\sqrt{a}} \tanh (x)+\sqrt{a+b}+\sqrt{a} \tanh ^2(x)\right )}{4 \sqrt{2} a^{3/4} \sqrt{a+b}}+\frac{\sqrt{\sqrt{a+b}+\sqrt{a}} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt{\sqrt{a+b}+\sqrt{a}} \tanh (x)+\sqrt{a+b}+\sqrt{a} \tanh ^2(x)\right )}{4 \sqrt{2} a^{3/4} \sqrt{a+b}} \]

[Out]

(Sqrt[Sqrt[a] - Sqrt[a + b]]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[a + b]] - Sqrt[2]*a^(1/4)*Tanh[x])/Sqrt[Sqrt[a] - Sq
rt[a + b]]])/(2*Sqrt[2]*a^(3/4)*Sqrt[a + b]) - (Sqrt[Sqrt[a] - Sqrt[a + b]]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[a + b
]] + Sqrt[2]*a^(1/4)*Tanh[x])/Sqrt[Sqrt[a] - Sqrt[a + b]]])/(2*Sqrt[2]*a^(3/4)*Sqrt[a + b]) - (Sqrt[Sqrt[a] +
Sqrt[a + b]]*Log[Sqrt[a + b] - Sqrt[2]*a^(1/4)*Sqrt[Sqrt[a] + Sqrt[a + b]]*Tanh[x] + Sqrt[a]*Tanh[x]^2])/(4*Sq
rt[2]*a^(3/4)*Sqrt[a + b]) + (Sqrt[Sqrt[a] + Sqrt[a + b]]*Log[Sqrt[a + b] + Sqrt[2]*a^(1/4)*Sqrt[Sqrt[a] + Sqr
t[a + b]]*Tanh[x] + Sqrt[a]*Tanh[x]^2])/(4*Sqrt[2]*a^(3/4)*Sqrt[a + b])

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Rubi [A]  time = 1.03716, antiderivative size = 485, normalized size of antiderivative = 1.34, number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3209, 1169, 634, 618, 204, 628} \[ -\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \log \left ((a+b)^{3/4} \coth ^2(x)-\sqrt{2} \sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b} \coth (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \log \left ((a+b)^{3/4} \coth ^2(x)+\sqrt{2} \sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b} \coth (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}-\sqrt{2} (a+b)^{3/4} \coth (x)}{\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}-\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \tan ^{-1}\left (\frac{\sqrt{2} (a+b)^{3/4} \coth (x)+\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}{\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^4)^(-1),x]

[Out]

((Sqrt[a] - Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]] - Sqrt[2]*(a + b)^(3/4)*Coth[x])/(a
^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]
) - ((Sqrt[a] - Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]] + Sqrt[2]*(a + b)^(3/4)*Coth[x]
)/(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a +
b]]) - ((Sqrt[a] + Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]
*Coth[x] + (a + b)^(3/4)*Coth[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) + ((S
qrt[a] + Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]*Coth[x] +
(a + b)^(3/4)*Coth[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])

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]

Rule 1169

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

Rule 634

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

Rule 618

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 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

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

Mathematica [C]  time = 0.229228, size = 121, normalized size = 0.34 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{a} \sqrt{a+i \sqrt{a} \sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{-a+i \sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{a} \sqrt{-a+i \sqrt{a} \sqrt{b}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^4)^(-1),x]

[Out]

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

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Maple [C]  time = 0.023, size = 121, normalized size = 0.3 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a+b \right ){{\it \_Z}}^{8}+ \left ( -4\,a+4\,b \right ){{\it \_Z}}^{6}+ \left ( 6\,a+6\,b \right ){{\it \_Z}}^{4}+ \left ( -4\,a+4\,b \right ){{\it \_Z}}^{2}+a+b \right ) }{\frac{-{{\it \_R}}^{6}+3\,{{\it \_R}}^{4}-3\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{7}a+{{\it \_R}}^{7}b-3\,{{\it \_R}}^{5}a+3\,{{\it \_R}}^{5}b+3\,{{\it \_R}}^{3}a+3\,{{\it \_R}}^{3}b-{\it \_R}\,a+{\it \_R}\,b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \cosh \left (x\right )^{4} + a}\,{d x} \end{align*}

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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Fricas [B]  time = 2.67857, size = 1728, normalized size = 4.79 \begin{align*} \text{result too large to display} \end{align*}

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*sin
h(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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \cosh \left (x\right )^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*cosh(x)^4 + a), x)