3.270 \(\int \frac{1}{1+\sinh ^5(x)} \, dx\)
Optimal. Leaf size=242 \[ -\frac{1}{5} \sqrt{2} \tanh ^{-1}\left (\frac{1-\tanh \left (\frac{x}{2}\right )}{\sqrt{2}}\right )+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{7/10} \left (\sqrt [5]{-1} \tanh \left (\frac{x}{2}\right )+1\right )}{\sqrt{-(-1)^{2/5} \left (1+(-1)^{2/5}\right )}}\right )}{5 \sqrt{-(-1)^{2/5} \left (1+(-1)^{2/5}\right )}}-\frac{2 (-1)^{4/5} \tanh ^{-1}\left (\frac{1-(-1)^{4/5} \tanh \left (\frac{x}{2}\right )}{\sqrt{1-(-1)^{3/5}}}\right )}{5 \sqrt{1-(-1)^{3/5}}}-\frac{2 (-1)^{3/5} \tan ^{-1}\left (\frac{(-1)^{3/5} \tanh \left (\frac{x}{2}\right )+1}{\sqrt{\sqrt [5]{-1}-1}}\right )}{5 \sqrt{\sqrt [5]{-1}-1}}+\frac{2 (-1)^{9/10} \tan ^{-1}\left (\frac{-(-1)^{9/10} \tanh \left (\frac{x}{2}\right )+i}{\sqrt{1+(-1)^{4/5}}}\right )}{5 \sqrt{1+(-1)^{4/5}}} \]
[Out]
(-2*(-1)^(3/5)*ArcTan[(1 + (-1)^(3/5)*Tanh[x/2])/Sqrt[-1 + (-1)^(1/5)]])/(5*Sqrt[-1 + (-1)^(1/5)]) + (2*(-1)^(
9/10)*ArcTan[(I - (-1)^(9/10)*Tanh[x/2])/Sqrt[1 + (-1)^(4/5)]])/(5*Sqrt[1 + (-1)^(4/5)]) - (Sqrt[2]*ArcTanh[(1
- Tanh[x/2])/Sqrt[2]])/5 + (2*(-1)^(9/10)*ArcTanh[((-1)^(7/10)*(1 + (-1)^(1/5)*Tanh[x/2]))/Sqrt[-((-1)^(2/5)*
(1 + (-1)^(2/5)))]])/(5*Sqrt[-((-1)^(2/5)*(1 + (-1)^(2/5)))]) - (2*(-1)^(4/5)*ArcTanh[(1 - (-1)^(4/5)*Tanh[x/2
])/Sqrt[1 - (-1)^(3/5)]])/(5*Sqrt[1 - (-1)^(3/5)])
________________________________________________________________________________________
Rubi [A] time = 0.5185, antiderivative size = 242, normalized size of antiderivative = 1.,
number of steps used = 17, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used =
{3213, 2660, 618, 204, 206, 617} \[ -\frac{1}{5} \sqrt{2} \tanh ^{-1}\left (\frac{1-\tanh \left (\frac{x}{2}\right )}{\sqrt{2}}\right )+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{7/10} \left (\sqrt [5]{-1} \tanh \left (\frac{x}{2}\right )+1\right )}{\sqrt{-(-1)^{2/5} \left (1+(-1)^{2/5}\right )}}\right )}{5 \sqrt{-(-1)^{2/5} \left (1+(-1)^{2/5}\right )}}-\frac{2 (-1)^{4/5} \tanh ^{-1}\left (\frac{1-(-1)^{4/5} \tanh \left (\frac{x}{2}\right )}{\sqrt{1-(-1)^{3/5}}}\right )}{5 \sqrt{1-(-1)^{3/5}}}-\frac{2 (-1)^{3/5} \tan ^{-1}\left (\frac{(-1)^{3/5} \tanh \left (\frac{x}{2}\right )+1}{\sqrt{\sqrt [5]{-1}-1}}\right )}{5 \sqrt{\sqrt [5]{-1}-1}}+\frac{2 (-1)^{9/10} \tan ^{-1}\left (\frac{-(-1)^{9/10} \tanh \left (\frac{x}{2}\right )+i}{\sqrt{1+(-1)^{4/5}}}\right )}{5 \sqrt{1+(-1)^{4/5}}} \]
Antiderivative was successfully verified.
[In]
Int[(1 + Sinh[x]^5)^(-1),x]
[Out]
(-2*(-1)^(3/5)*ArcTan[(1 + (-1)^(3/5)*Tanh[x/2])/Sqrt[-1 + (-1)^(1/5)]])/(5*Sqrt[-1 + (-1)^(1/5)]) + (2*(-1)^(
9/10)*ArcTan[(I - (-1)^(9/10)*Tanh[x/2])/Sqrt[1 + (-1)^(4/5)]])/(5*Sqrt[1 + (-1)^(4/5)]) - (Sqrt[2]*ArcTanh[(1
- Tanh[x/2])/Sqrt[2]])/5 + (2*(-1)^(9/10)*ArcTanh[((-1)^(7/10)*(1 + (-1)^(1/5)*Tanh[x/2]))/Sqrt[-((-1)^(2/5)*
(1 + (-1)^(2/5)))]])/(5*Sqrt[-((-1)^(2/5)*(1 + (-1)^(2/5)))]) - (2*(-1)^(4/5)*ArcTanh[(1 - (-1)^(4/5)*Tanh[x/2
])/Sqrt[1 - (-1)^(3/5)]])/(5*Sqrt[1 - (-1)^(3/5)])
Rule 3213
Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Rule 2660
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*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 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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rubi steps
\begin{align*} \int \frac{1}{1+\sinh ^5(x)} \, dx &=\int \left (-\frac{(-1)^{9/10}}{5 \left (-(-1)^{9/10}-i \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 \left (-(-1)^{9/10}-\sqrt [10]{-1} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 \left (-(-1)^{9/10}+(-1)^{3/10} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 \left (-(-1)^{9/10}+(-1)^{7/10} \sinh (x)\right )}-\frac{(-1)^{9/10}}{5 \left (-(-1)^{9/10}-(-1)^{9/10} \sinh (x)\right )}\right ) \, dx\\ &=-\left (\frac{1}{5} (-1)^{9/10} \int \frac{1}{-(-1)^{9/10}-i \sinh (x)} \, dx\right )-\frac{1}{5} (-1)^{9/10} \int \frac{1}{-(-1)^{9/10}-\sqrt [10]{-1} \sinh (x)} \, dx-\frac{1}{5} (-1)^{9/10} \int \frac{1}{-(-1)^{9/10}+(-1)^{3/10} \sinh (x)} \, dx-\frac{1}{5} (-1)^{9/10} \int \frac{1}{-(-1)^{9/10}+(-1)^{7/10} \sinh (x)} \, dx-\frac{1}{5} (-1)^{9/10} \int \frac{1}{-(-1)^{9/10}-(-1)^{9/10} \sinh (x)} \, dx\\ &=-\left (\frac{1}{5} \left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10}-2 i x+(-1)^{9/10} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\right )-\frac{1}{5} \left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10}-2 \sqrt [10]{-1} x+(-1)^{9/10} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )-\frac{1}{5} \left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10}+2 (-1)^{3/10} x+(-1)^{9/10} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )-\frac{1}{5} \left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10}+2 (-1)^{7/10} x+(-1)^{9/10} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )-\frac{1}{5} \left (2 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{9/10}-2 (-1)^{9/10} x+(-1)^{9/10} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=-\left (\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,1-\tanh \left (\frac{x}{2}\right )\right )\right )+\frac{1}{5} \left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{4 (-1)^{3/5} \left (1-\sqrt [5]{-1}\right )-x^2} \, dx,x,2 (-1)^{3/10}+2 (-1)^{9/10} \tanh \left (\frac{x}{2}\right )\right )+\frac{1}{5} \left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{4 \sqrt [5]{-1} \left (1-(-1)^{3/5}\right )-x^2} \, dx,x,-2 \sqrt [10]{-1}+2 (-1)^{9/10} \tanh \left (\frac{x}{2}\right )\right )+\frac{1}{5} \left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (1+(-1)^{4/5}\right )-x^2} \, dx,x,-2 i+2 (-1)^{9/10} \tanh \left (\frac{x}{2}\right )\right )+\frac{1}{5} \left (4 (-1)^{9/10}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left ((-1)^{2/5}+(-1)^{4/5}\right )-x^2} \, dx,x,2 (-1)^{7/10}+2 (-1)^{9/10} \tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{2 (-1)^{9/10} \tan ^{-1}\left (\frac{i-(-1)^{9/10} \tanh \left (\frac{x}{2}\right )}{\sqrt{1+(-1)^{4/5}}}\right )}{5 \sqrt{1+(-1)^{4/5}}}-\frac{1}{5} \sqrt{2} \tanh ^{-1}\left (\frac{1-\tanh \left (\frac{x}{2}\right )}{\sqrt{2}}\right )-\frac{2 \sqrt [10]{-1} \tanh ^{-1}\left (\frac{i-\sqrt [10]{-1} \tanh \left (\frac{x}{2}\right )}{\sqrt{-1+\sqrt [5]{-1}}}\right )}{5 \sqrt{-1+\sqrt [5]{-1}}}-\frac{2 (-1)^{4/5} \tanh ^{-1}\left (\frac{1-(-1)^{4/5} \tanh \left (\frac{x}{2}\right )}{\sqrt{1-(-1)^{3/5}}}\right )}{5 \sqrt{1-(-1)^{3/5}}}+\frac{2 (-1)^{9/10} \tanh ^{-1}\left (\frac{(-1)^{7/10}+(-1)^{9/10} \tanh \left (\frac{x}{2}\right )}{\sqrt{-(-1)^{2/5} \left (1+(-1)^{2/5}\right )}}\right )}{5 \sqrt{-(-1)^{2/5} \left (1+(-1)^{2/5}\right )}}\\ \end{align*}
Mathematica [C] time = 0.915842, size = 439, normalized size = 1.81 \[ \frac{1}{10} \left (2 \sqrt{2} \tanh ^{-1}\left (\frac{\tanh \left (\frac{x}{2}\right )-1}{\sqrt{2}}\right )-\text{RootSum}\left [\text{$\#$1}^8-2 \text{$\#$1}^7-2 \text{$\#$1}^5+14 \text{$\#$1}^4+2 \text{$\#$1}^3+2 \text{$\#$1}+1\& ,\frac{\text{$\#$1}^6 x-4 \text{$\#$1}^5 x+9 \text{$\#$1}^4 x-24 \text{$\#$1}^3 x-9 \text{$\#$1}^2 x+2 \text{$\#$1}^6 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-8 \text{$\#$1}^5 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )+18 \text{$\#$1}^4 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-48 \text{$\#$1}^3 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-18 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-4 \text{$\#$1} x-8 \text{$\#$1} \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-x}{4 \text{$\#$1}^7-7 \text{$\#$1}^6-5 \text{$\#$1}^4+28 \text{$\#$1}^3+3 \text{$\#$1}^2+1}\& \right ]\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(1 + Sinh[x]^5)^(-1),x]
[Out]
(2*Sqrt[2]*ArcTanh[(-1 + Tanh[x/2])/Sqrt[2]] - RootSum[1 + 2*#1 + 2*#1^3 + 14*#1^4 - 2*#1^5 - 2*#1^7 + #1^8 &
, (-x - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] - 4*x*#1 - 8*Log[-Cosh[x/2] - Sinh[x/2] +
Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1 - 9*x*#1^2 - 18*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2
- 24*x*#1^3 - 48*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^3 + 9*x*#1^4 + 18*Log[-Cosh[x/2
] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^4 - 4*x*#1^5 - 8*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - S
inh[x/2]*#1]*#1^5 + x*#1^6 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(1 + 3*#1^2 + 2
8*#1^3 - 5*#1^4 - 7*#1^6 + 4*#1^7) & ])/10
________________________________________________________________________________________
Maple [C] time = 0.033, size = 124, normalized size = 0.5 \begin{align*}{\frac{2}{5}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+2\,{{\it \_Z}}^{7}+2\,{{\it \_Z}}^{5}+14\,{{\it \_Z}}^{4}-2\,{{\it \_Z}}^{3}-2\,{\it \_Z}+1 \right ) }{\frac{-2\,{{\it \_R}}^{6}-3\,{{\it \_R}}^{5}+2\,{{\it \_R}}^{4}+2\,{{\it \_R}}^{3}-2\,{{\it \_R}}^{2}-3\,{\it \_R}+2}{4\,{{\it \_R}}^{7}+7\,{{\it \_R}}^{6}+5\,{{\it \_R}}^{4}+28\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}-1}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }}+{\frac{\sqrt{2}}{5}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1+sinh(x)^5),x)
[Out]
2/5*sum((-2*_R^6-3*_R^5+2*_R^4+2*_R^3-2*_R^2-3*_R+2)/(4*_R^7+7*_R^6+5*_R^4+28*_R^3-3*_R^2-1)*ln(tanh(1/2*x)-_R
),_R=RootOf(_Z^8+2*_Z^7+2*_Z^5+14*_Z^4-2*_Z^3-2*_Z+1))+1/5*2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)-2)*2^(1/2))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{10} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{x} - 1}{\sqrt{2} + e^{x} + 1}\right ) - \int \frac{2 \,{\left (e^{\left (7 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 9 \, e^{\left (5 \, x\right )} - 24 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )} - e^{x}\right )}}{5 \,{\left (e^{\left (8 \, x\right )} - 2 \, e^{\left (7 \, x\right )} - 2 \, e^{\left (5 \, x\right )} + 14 \, e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + 2 \, e^{x} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+sinh(x)^5),x, algorithm="maxima")
[Out]
1/10*sqrt(2)*log(-(sqrt(2) - e^x - 1)/(sqrt(2) + e^x + 1)) - integrate(2/5*(e^(7*x) - 4*e^(6*x) + 9*e^(5*x) -
24*e^(4*x) - 9*e^(3*x) - 4*e^(2*x) - e^x)/(e^(8*x) - 2*e^(7*x) - 2*e^(5*x) + 14*e^(4*x) + 2*e^(3*x) + 2*e^x +
1), x)
________________________________________________________________________________________
Fricas [B] time = 3.48642, size = 11367, normalized size = 46.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+sinh(x)^5),x, algorithm="fricas")
[Out]
-1/200*sqrt(2)*sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4*sqrt(5) + 20)*(8*sqrt(5) + 24)^(1/4)*(3*sq
rt(5) - 5)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3)*arctan(1/40*sqrt(2)*((11*sqrt(5) - 25)*e^x - 7*sqrt(5) + 15)*
sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 1/80*sqrt(2)*(sqrt(2)*((11*sqrt(5) - 25)*e^x + 4*sqrt(5) - 10)*sqrt(2*
sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*((3*sqrt(5) - 5)*e^x + 7*sqrt(5) - 15)*sqrt(2*sqrt(5) + 5))*sqrt(sqrt(5) +
3) + 1/12800*(80*sqrt(2)*(5*sqrt(5) - 11)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 40*sqrt(2)*(sqrt(2)*(5*sqrt(
5) - 11)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*sqrt(2*sqrt(5) + 5)*(sqrt(5) - 3))*sqrt(sqrt(5) + 3) + sqrt
(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4*sqrt(5) + 20)*((sqrt(2)*(7*sqrt(5) - 15)*sqrt(2*sqrt(5) + 5)*
sqrt(sqrt(5) + 3) + 2*(11*sqrt(5) - 25)*sqrt(2*sqrt(5) + 5))*(8*sqrt(5) + 24)^(3/4) + 4*(sqrt(2)*(17*sqrt(5) -
35)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*(11*sqrt(5) - 25)*sqrt(2*sqrt(5) + 5))*(8*sqrt(5) + 24)^(1/4))
+ 320*sqrt(2*sqrt(5) + 5)*(sqrt(5) - 4))*sqrt(-20*sqrt(2)*sqrt(sqrt(5) + 3)*(sqrt(5) - 3) + 40*(sqrt(5) - 1)*e
^x - 2*(sqrt(2)*((2*sqrt(5) - 5)*e^x - 3*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*(sqrt(5) - 5)*e^x - 3*sqrt(5) + 5)
*sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4*sqrt(5) + 20)*(8*sqrt(5) + 24)^(1/4) + 80*e^(2*x) + 80)
+ 1/640*sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4*sqrt(5) + 20)*((sqrt(2)*((3*sqrt(5) - 7)*e^x + 8*
sqrt(5) - 18)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*((5*sqrt(5) - 11)*e^x + 2*sqrt(5) - 4)*sqrt(2*sqrt(5)
+ 5))*(8*sqrt(5) + 24)^(3/4) + 4*(sqrt(2)*((7*sqrt(5) - 17)*e^x + 8*sqrt(5) - 18)*sqrt(2*sqrt(5) + 5)*sqrt(sqr
t(5) + 3) + 2*((5*sqrt(5) - 11)*e^x + 5*sqrt(5) - 9)*sqrt(2*sqrt(5) + 5))*(8*sqrt(5) + 24)^(1/4)) + 1/20*(2*(4
*sqrt(5) - 5)*e^x - sqrt(5) + 5)*sqrt(2*sqrt(5) + 5)) - 1/200*sqrt(2)*sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt
(5) + 3) - 4*sqrt(5) + 20)*(8*sqrt(5) + 24)^(1/4)*(3*sqrt(5) - 5)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3)*arctan
(-1/40*sqrt(2)*((11*sqrt(5) - 25)*e^x - 7*sqrt(5) + 15)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) - 1/80*sqrt(2)*(
sqrt(2)*((11*sqrt(5) - 25)*e^x + 4*sqrt(5) - 10)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*((3*sqrt(5) - 5)*e^
x + 7*sqrt(5) - 15)*sqrt(2*sqrt(5) + 5))*sqrt(sqrt(5) + 3) - 1/12800*(80*sqrt(2)*(5*sqrt(5) - 11)*sqrt(2*sqrt(
5) + 5)*sqrt(sqrt(5) + 3) + 40*sqrt(2)*(sqrt(2)*(5*sqrt(5) - 11)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*sqr
t(2*sqrt(5) + 5)*(sqrt(5) - 3))*sqrt(sqrt(5) + 3) - sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4*sqrt(
5) + 20)*((sqrt(2)*(7*sqrt(5) - 15)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*(11*sqrt(5) - 25)*sqrt(2*sqrt(5)
+ 5))*(8*sqrt(5) + 24)^(3/4) + 4*(sqrt(2)*(17*sqrt(5) - 35)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*(11*sqr
t(5) - 25)*sqrt(2*sqrt(5) + 5))*(8*sqrt(5) + 24)^(1/4)) + 320*sqrt(2*sqrt(5) + 5)*(sqrt(5) - 4))*sqrt(-20*sqrt
(2)*sqrt(sqrt(5) + 3)*(sqrt(5) - 3) + 40*(sqrt(5) - 1)*e^x + 2*(sqrt(2)*((2*sqrt(5) - 5)*e^x - 3*sqrt(5) + 5)*
sqrt(sqrt(5) + 3) + 2*(sqrt(5) - 5)*e^x - 3*sqrt(5) + 5)*sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4*
sqrt(5) + 20)*(8*sqrt(5) + 24)^(1/4) + 80*e^(2*x) + 80) + 1/640*sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) +
3) - 4*sqrt(5) + 20)*((sqrt(2)*((3*sqrt(5) - 7)*e^x + 8*sqrt(5) - 18)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) +
2*((5*sqrt(5) - 11)*e^x + 2*sqrt(5) - 4)*sqrt(2*sqrt(5) + 5))*(8*sqrt(5) + 24)^(3/4) + 4*(sqrt(2)*((7*sqrt(5)
- 17)*e^x + 8*sqrt(5) - 18)*sqrt(2*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*((5*sqrt(5) - 11)*e^x + 5*sqrt(5) - 9)*s
qrt(2*sqrt(5) + 5))*(8*sqrt(5) + 24)^(1/4)) - 1/20*(2*(4*sqrt(5) - 5)*e^x - sqrt(5) + 5)*sqrt(2*sqrt(5) + 5))
+ 1/400*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sqrt(5) + 20)*(3*sqrt(5) + 5)*sqrt(-2*sqrt(5) + 5)*(-8
*sqrt(5) + 24)^(3/4)*arctan(-1/1280*((4*(5*sqrt(5) + 11)*e^x + ((3*sqrt(5) + 7)*e^x + 8*sqrt(5) + 18)*sqrt(-8*
sqrt(5) + 24) + 8*sqrt(5) + 16)*(-8*sqrt(5) + 24)^(3/4) + 4*(4*(5*sqrt(5) + 11)*e^x + ((7*sqrt(5) + 17)*e^x +
8*sqrt(5) + 18)*sqrt(-8*sqrt(5) + 24) + 20*sqrt(5) + 36)*(-8*sqrt(5) + 24)^(1/4))*sqrt(-(2*sqrt(5) + 5)*sqrt(-
8*sqrt(5) + 24) + 4*sqrt(5) + 20)*sqrt(-2*sqrt(5) + 5) + 1/25600*((((7*sqrt(5) + 15)*sqrt(-8*sqrt(5) + 24) + 4
4*sqrt(5) + 100)*(-8*sqrt(5) + 24)^(3/4) + 4*((17*sqrt(5) + 35)*sqrt(-8*sqrt(5) + 24) + 44*sqrt(5) + 100)*(-8*
sqrt(5) + 24)^(1/4))*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sqrt(5) + 20)*sqrt(-2*sqrt(5) + 5) + 20*(
((5*sqrt(5) + 11)*sqrt(-8*sqrt(5) + 24) + 4*sqrt(5) + 12)*sqrt(-8*sqrt(5) + 24) + 4*(5*sqrt(5) + 11)*sqrt(-8*s
qrt(5) + 24) + 32*sqrt(5) + 128)*sqrt(-2*sqrt(5) + 5))*sqrt(-40*(sqrt(5) + 1)*e^x + (4*(sqrt(5) + 5)*e^x + ((2
*sqrt(5) + 5)*e^x - 3*sqrt(5) - 5)*sqrt(-8*sqrt(5) + 24) - 6*sqrt(5) - 10)*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(
5) + 24) + 4*sqrt(5) + 20)*(-8*sqrt(5) + 24)^(1/4) + 10*(sqrt(5) + 3)*sqrt(-8*sqrt(5) + 24) + 80*e^(2*x) + 80)
- 1/320*(32*(4*sqrt(5) + 5)*e^x + 4*((11*sqrt(5) + 25)*e^x - 7*sqrt(5) - 15)*sqrt(-8*sqrt(5) + 24) + (4*(3*sq
rt(5) + 5)*e^x + ((11*sqrt(5) + 25)*e^x + 4*sqrt(5) + 10)*sqrt(-8*sqrt(5) + 24) + 28*sqrt(5) + 60)*sqrt(-8*sqr
t(5) + 24) - 16*sqrt(5) - 80)*sqrt(-2*sqrt(5) + 5)) + 1/400*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sq
rt(5) + 20)*(3*sqrt(5) + 5)*sqrt(-2*sqrt(5) + 5)*(-8*sqrt(5) + 24)^(3/4)*arctan(-1/1280*((4*(5*sqrt(5) + 11)*e
^x + ((3*sqrt(5) + 7)*e^x + 8*sqrt(5) + 18)*sqrt(-8*sqrt(5) + 24) + 8*sqrt(5) + 16)*(-8*sqrt(5) + 24)^(3/4) +
4*(4*(5*sqrt(5) + 11)*e^x + ((7*sqrt(5) + 17)*e^x + 8*sqrt(5) + 18)*sqrt(-8*sqrt(5) + 24) + 20*sqrt(5) + 36)*(
-8*sqrt(5) + 24)^(1/4))*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sqrt(5) + 20)*sqrt(-2*sqrt(5) + 5) + 1
/25600*((((7*sqrt(5) + 15)*sqrt(-8*sqrt(5) + 24) + 44*sqrt(5) + 100)*(-8*sqrt(5) + 24)^(3/4) + 4*((17*sqrt(5)
+ 35)*sqrt(-8*sqrt(5) + 24) + 44*sqrt(5) + 100)*(-8*sqrt(5) + 24)^(1/4))*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5)
+ 24) + 4*sqrt(5) + 20)*sqrt(-2*sqrt(5) + 5) - 20*(((5*sqrt(5) + 11)*sqrt(-8*sqrt(5) + 24) + 4*sqrt(5) + 12)*
sqrt(-8*sqrt(5) + 24) + 4*(5*sqrt(5) + 11)*sqrt(-8*sqrt(5) + 24) + 32*sqrt(5) + 128)*sqrt(-2*sqrt(5) + 5))*sqr
t(-40*(sqrt(5) + 1)*e^x - (4*(sqrt(5) + 5)*e^x + ((2*sqrt(5) + 5)*e^x - 3*sqrt(5) - 5)*sqrt(-8*sqrt(5) + 24) -
6*sqrt(5) - 10)*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sqrt(5) + 20)*(-8*sqrt(5) + 24)^(1/4) + 10*(s
qrt(5) + 3)*sqrt(-8*sqrt(5) + 24) + 80*e^(2*x) + 80) + 1/320*(32*(4*sqrt(5) + 5)*e^x + 4*((11*sqrt(5) + 25)*e^
x - 7*sqrt(5) - 15)*sqrt(-8*sqrt(5) + 24) + (4*(3*sqrt(5) + 5)*e^x + ((11*sqrt(5) + 25)*e^x + 4*sqrt(5) + 10)*
sqrt(-8*sqrt(5) + 24) + 28*sqrt(5) + 60)*sqrt(-8*sqrt(5) + 24) - 16*sqrt(5) - 80)*sqrt(-2*sqrt(5) + 5)) + 1/80
0*(sqrt(2)*(3*sqrt(5) - 5)*sqrt(sqrt(5) + 3) + 8*sqrt(5))*sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4
*sqrt(5) + 20)*(8*sqrt(5) + 24)^(1/4)*log(-4*sqrt(2)*sqrt(sqrt(5) + 3)*(sqrt(5) - 3) + 8*(sqrt(5) - 1)*e^x + 2
/5*(sqrt(2)*((2*sqrt(5) - 5)*e^x - 3*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*(sqrt(5) - 5)*e^x - 3*sqrt(5) + 5)*sqr
t(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4*sqrt(5) + 20)*(8*sqrt(5) + 24)^(1/4) + 16*e^(2*x) + 16) - 1/
800*(sqrt(2)*(3*sqrt(5) - 5)*sqrt(sqrt(5) + 3) + 8*sqrt(5))*sqrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) -
4*sqrt(5) + 20)*(8*sqrt(5) + 24)^(1/4)*log(-4*sqrt(2)*sqrt(sqrt(5) + 3)*(sqrt(5) - 3) + 8*(sqrt(5) - 1)*e^x -
2/5*(sqrt(2)*((2*sqrt(5) - 5)*e^x - 3*sqrt(5) + 5)*sqrt(sqrt(5) + 3) + 2*(sqrt(5) - 5)*e^x - 3*sqrt(5) + 5)*s
qrt(2*sqrt(2)*(2*sqrt(5) - 5)*sqrt(sqrt(5) + 3) - 4*sqrt(5) + 20)*(8*sqrt(5) + 24)^(1/4) + 16*e^(2*x) + 16) +
1/1600*((3*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 16*sqrt(5))*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sq
rt(5) + 20)*(-8*sqrt(5) + 24)^(1/4)*log(-8*(sqrt(5) + 1)*e^x + 1/5*(4*(sqrt(5) + 5)*e^x + ((2*sqrt(5) + 5)*e^x
- 3*sqrt(5) - 5)*sqrt(-8*sqrt(5) + 24) - 6*sqrt(5) - 10)*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sqrt
(5) + 20)*(-8*sqrt(5) + 24)^(1/4) + 2*(sqrt(5) + 3)*sqrt(-8*sqrt(5) + 24) + 16*e^(2*x) + 16) - 1/1600*((3*sqrt
(5) + 5)*sqrt(-8*sqrt(5) + 24) + 16*sqrt(5))*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sqrt(5) + 20)*(-8
*sqrt(5) + 24)^(1/4)*log(-8*(sqrt(5) + 1)*e^x - 1/5*(4*(sqrt(5) + 5)*e^x + ((2*sqrt(5) + 5)*e^x - 3*sqrt(5) -
5)*sqrt(-8*sqrt(5) + 24) - 6*sqrt(5) - 10)*sqrt(-(2*sqrt(5) + 5)*sqrt(-8*sqrt(5) + 24) + 4*sqrt(5) + 20)*(-8*s
qrt(5) + 24)^(1/4) + 2*(sqrt(5) + 3)*sqrt(-8*sqrt(5) + 24) + 16*e^(2*x) + 16) + 1/10*sqrt(2)*log(-(2*(sqrt(2)
- 1)*e^x + 2*sqrt(2) - e^(2*x) - 3)/(e^(2*x) + 2*e^x - 1))
________________________________________________________________________________________
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/(1+sinh(x)**5),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sinh \left (x\right )^{5} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+sinh(x)^5),x, algorithm="giac")
[Out]
integrate(1/(sinh(x)^5 + 1), x)