3.183 \(\int \frac{1}{1-\sinh ^3(x)} \, dx\)

Optimal. Leaf size=133 \[ \frac{1}{3} \sqrt{2} \tanh ^{-1}\left (\frac{\tanh \left (\frac{x}{2}\right )+1}{\sqrt{2}}\right )-\frac{1}{3} (-1)^{5/6} \log \left ((-1)^{2/3} \tanh \left (\frac{x}{2}\right )+(-1)^{5/6}+1\right )+\frac{1}{3} (-1)^{5/6} \log \left ((-1)^{5/6} \tanh \left (\frac{x}{2}\right )+\sqrt [6]{-1}+1\right )+\frac{2 (-1)^{5/6} \tan ^{-1}\left (\frac{-(-1)^{5/6} \tanh \left (\frac{x}{2}\right )+i}{\sqrt{1+(-1)^{2/3}}}\right )}{3 \sqrt{1+(-1)^{2/3}}} \]

[Out]

(2*(-1)^(5/6)*ArcTan[(I - (-1)^(5/6)*Tanh[x/2])/Sqrt[1 + (-1)^(2/3)]])/(3*Sqrt[1 + (-1)^(2/3)]) + (Sqrt[2]*Arc
Tanh[(1 + Tanh[x/2])/Sqrt[2]])/3 - ((-1)^(5/6)*Log[1 + (-1)^(5/6) + (-1)^(2/3)*Tanh[x/2]])/3 + ((-1)^(5/6)*Log
[1 + (-1)^(1/6) + (-1)^(5/6)*Tanh[x/2]])/3

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Rubi [A]  time = 0.188952, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {3213, 2660, 618, 204, 616, 31, 617, 206} \[ \frac{1}{3} \sqrt{2} \tanh ^{-1}\left (\frac{\tanh \left (\frac{x}{2}\right )+1}{\sqrt{2}}\right )-\frac{1}{3} (-1)^{5/6} \log \left ((-1)^{2/3} \tanh \left (\frac{x}{2}\right )+(-1)^{5/6}+1\right )+\frac{1}{3} (-1)^{5/6} \log \left ((-1)^{5/6} \tanh \left (\frac{x}{2}\right )+\sqrt [6]{-1}+1\right )+\frac{2 (-1)^{5/6} \tan ^{-1}\left (\frac{-(-1)^{5/6} \tanh \left (\frac{x}{2}\right )+i}{\sqrt{1+(-1)^{2/3}}}\right )}{3 \sqrt{1+(-1)^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sinh[x]^3)^(-1),x]

[Out]

(2*(-1)^(5/6)*ArcTan[(I - (-1)^(5/6)*Tanh[x/2])/Sqrt[1 + (-1)^(2/3)]])/(3*Sqrt[1 + (-1)^(2/3)]) + (Sqrt[2]*Arc
Tanh[(1 + Tanh[x/2])/Sqrt[2]])/3 - ((-1)^(5/6)*Log[1 + (-1)^(5/6) + (-1)^(2/3)*Tanh[x/2]])/3 + ((-1)^(5/6)*Log
[1 + (-1)^(1/6) + (-1)^(5/6)*Tanh[x/2]])/3

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 616

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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]

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])

Rubi steps

\begin{align*} \int \frac{1}{1-\sinh ^3(x)} \, dx &=\int \left (-\frac{(-1)^{5/6}}{3 \left (-(-1)^{5/6}-i \sinh (x)\right )}-\frac{(-1)^{5/6}}{3 \left (-(-1)^{5/6}+\sqrt [6]{-1} \sinh (x)\right )}-\frac{(-1)^{5/6}}{3 \left (-(-1)^{5/6}+(-1)^{5/6} \sinh (x)\right )}\right ) \, dx\\ &=-\left (\frac{1}{3} (-1)^{5/6} \int \frac{1}{-(-1)^{5/6}-i \sinh (x)} \, dx\right )-\frac{1}{3} (-1)^{5/6} \int \frac{1}{-(-1)^{5/6}+\sqrt [6]{-1} \sinh (x)} \, dx-\frac{1}{3} (-1)^{5/6} \int \frac{1}{-(-1)^{5/6}+(-1)^{5/6} \sinh (x)} \, dx\\ &=-\left (\frac{1}{3} \left (2 (-1)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{5/6}-2 i x+(-1)^{5/6} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\right )-\frac{1}{3} \left (2 (-1)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{5/6}+2 \sqrt [6]{-1} x+(-1)^{5/6} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )-\frac{1}{3} \left (2 (-1)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{-(-1)^{5/6}+2 (-1)^{5/6} x+(-1)^{5/6} x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,1+\tanh \left (\frac{x}{2}\right )\right )+\frac{1}{3} (-1)^{2/3} \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt [6]{-1}+(-1)^{5/6} x} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )-\frac{1}{3} (-1)^{2/3} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [6]{-1}+(-1)^{5/6} x} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )+\frac{1}{3} \left (4 (-1)^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (1+(-1)^{2/3}\right )-x^2} \, dx,x,-2 i+2 (-1)^{5/6} \tanh \left (\frac{x}{2}\right )\right )\\ &=\frac{2 (-1)^{5/6} \tan ^{-1}\left (\frac{i-(-1)^{5/6} \tanh \left (\frac{x}{2}\right )}{\sqrt{1+(-1)^{2/3}}}\right )}{3 \sqrt{1+(-1)^{2/3}}}+\frac{1}{3} \sqrt{2} \tanh ^{-1}\left (\frac{1+\tanh \left (\frac{x}{2}\right )}{\sqrt{2}}\right )-\frac{1}{3} (-1)^{5/6} \log \left (1+(-1)^{5/6}+(-1)^{2/3} \tanh \left (\frac{x}{2}\right )\right )+\frac{1}{3} (-1)^{5/6} \log \left (1+\sqrt [6]{-1}+(-1)^{5/6} \tanh \left (\frac{x}{2}\right )\right )\\ \end{align*}

Mathematica [A]  time = 1.21926, size = 156, normalized size = 1.17 \[ \frac{2 \tanh ^{-1}\left (\frac{\tanh \left (\frac{x}{2}\right )+1}{\sqrt{2}}\right )+\sqrt{-1+i \sqrt{3}} \left (1+i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2+\left (-1-i \sqrt{3}\right ) \tanh \left (\frac{x}{2}\right )}{\sqrt{-2-2 i \sqrt{3}}}\right )+\sqrt{-1-i \sqrt{3}} \left (1-i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2+i \left (\sqrt{3}+i\right ) \tanh \left (\frac{x}{2}\right )}{\sqrt{-2+2 i \sqrt{3}}}\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sinh[x]^3)^(-1),x]

[Out]

(Sqrt[-1 + I*Sqrt[3]]*(1 + I*Sqrt[3])*ArcTan[(2 + (-1 - I*Sqrt[3])*Tanh[x/2])/Sqrt[-2 - (2*I)*Sqrt[3]]] + Sqrt
[-1 - I*Sqrt[3]]*(1 - I*Sqrt[3])*ArcTan[(2 + I*(I + Sqrt[3])*Tanh[x/2])/Sqrt[-2 + (2*I)*Sqrt[3]]] + 2*ArcTanh[
(1 + Tanh[x/2])/Sqrt[2]])/(3*Sqrt[2])

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Maple [C]  time = 0.029, size = 80, normalized size = 0.6 \begin{align*}{\frac{2}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-2\,{{\it \_Z}}^{3}+2\,{{\it \_Z}}^{2}+2\,{\it \_Z}+1 \right ) }{\frac{-{{\it \_R}}^{2}+{\it \_R}+1}{2\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+2\,{\it \_R}+1}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }}+{\frac{\sqrt{2}}{3}{\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)^3),x)

[Out]

2/3*sum((-_R^2+_R+1)/(2*_R^3-3*_R^2+2*_R+1)*ln(tanh(1/2*x)-_R),_R=RootOf(_Z^4-2*_Z^3+2*_Z^2+2*_Z+1))+1/3*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}{6} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{x} + 1}{\sqrt{2} + e^{x} - 1}\right ) + \int \frac{2 \,{\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )} - e^{x}\right )}}{3 \,{\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, 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)^3),x, algorithm="maxima")

[Out]

-1/6*sqrt(2)*log(-(sqrt(2) - e^x + 1)/(sqrt(2) + e^x - 1)) + integrate(2/3*(e^(3*x) + 4*e^(2*x) - e^x)/(e^(4*x
) + 2*e^(3*x) + 2*e^(2*x) - 2*e^x + 1), x)

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Fricas [B]  time = 1.85063, size = 590, normalized size = 4.44 \begin{align*} -\frac{1}{6} \, \sqrt{3} \log \left (4 \,{\left (\sqrt{3} + 1\right )} e^{x} + 4 \, \sqrt{3} + 4 \, e^{\left (2 \, x\right )} + 8\right ) + \frac{1}{6} \, \sqrt{3} \log \left (-4 \,{\left (\sqrt{3} - 1\right )} e^{x} - 4 \, \sqrt{3} + 4 \, e^{\left (2 \, x\right )} + 8\right ) + \frac{1}{6} \, \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2} - 1\right )} e^{x} - 2 \, \sqrt{2} + e^{\left (2 \, x\right )} + 3}{e^{\left (2 \, x\right )} - 2 \, e^{x} - 1}\right ) - \frac{2}{3} \, \arctan \left (-{\left (\sqrt{3} + 1\right )} e^{x} + \frac{1}{2} \, \sqrt{-4 \,{\left (\sqrt{3} - 1\right )} e^{x} - 4 \, \sqrt{3} + 4 \, e^{\left (2 \, x\right )} + 8}{\left (\sqrt{3} + 1\right )} + 1\right ) + \frac{2}{3} \, \arctan \left (-{\left (\sqrt{3} - 1\right )} e^{x} + \sqrt{{\left (\sqrt{3} + 1\right )} e^{x} + \sqrt{3} + e^{\left (2 \, x\right )} + 2}{\left (\sqrt{3} - 1\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^3),x, algorithm="fricas")

[Out]

-1/6*sqrt(3)*log(4*(sqrt(3) + 1)*e^x + 4*sqrt(3) + 4*e^(2*x) + 8) + 1/6*sqrt(3)*log(-4*(sqrt(3) - 1)*e^x - 4*s
qrt(3) + 4*e^(2*x) + 8) + 1/6*sqrt(2)*log((2*(sqrt(2) - 1)*e^x - 2*sqrt(2) + e^(2*x) + 3)/(e^(2*x) - 2*e^x - 1
)) - 2/3*arctan(-(sqrt(3) + 1)*e^x + 1/2*sqrt(-4*(sqrt(3) - 1)*e^x - 4*sqrt(3) + 4*e^(2*x) + 8)*(sqrt(3) + 1)
+ 1) + 2/3*arctan(-(sqrt(3) - 1)*e^x + sqrt((sqrt(3) + 1)*e^x + sqrt(3) + e^(2*x) + 2)*(sqrt(3) - 1) - 1)

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Sympy [B]  time = 72.4185, size = 3742, normalized size = 28.14 \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)**3),x)

[Out]

831721548108794611356*sqrt(6)*I*log(tanh(x/2) + 1 + sqrt(2))/(-10586087041080284410320*sqrt(2) + 1497098786595
8303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 122237604
05665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 3528695680360094803440*log(tanh(x/2)
+ 1 + sqrt(2))/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sq
rt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 499032928
8652767668136*sqrt(3)*I) + 2037293400944170367740*sqrt(2)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) + 1 + sqrt(2))/(-1
0586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I)
- 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt
(3)*I) - 2881167958148549067306*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) + 1 + sqrt(2))/(-10586087041080284410320*sqr
t(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*s
qrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 11762318934533649
34480*sqrt(3)*I*log(tanh(x/2) + 1 + sqrt(2))/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 864
3503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*s
qrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 2495164644326383834068*sqrt(2)*log(tanh(x/2) + 1 + sq
rt(2))/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + s
qrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 49903292886527676
68136*sqrt(3)*I) - 2495164644326383834068*sqrt(2)*log(tanh(x/2) - sqrt(2) + 1)/(-10586087041080284410320*sqrt(
2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqr
t(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 1176231893453364934
480*sqrt(3)*I*log(tanh(x/2) - sqrt(2) + 1)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 86435
03874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqr
t(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 2881167958148549067306*sqrt(1 + sqrt(3)*I)*log(tanh(x/2
) - sqrt(2) + 1)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*
sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329
288652767668136*sqrt(3)*I) - 2037293400944170367740*sqrt(2)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - sqrt(2) + 1)/(
-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I
) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sq
rt(3)*I) + 3528695680360094803440*log(tanh(x/2) - sqrt(2) + 1)/(-10586087041080284410320*sqrt(2) + 14970987865
958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 1222376
0405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 831721548108794611356*sqrt(6)*I*log
(tanh(x/2) - sqrt(2) + 1)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918
*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I)
+ 4990329288652767668136*sqrt(3)*I) + 1440583979074274533653*sqrt(3)*I*sqrt(1 - sqrt(3)*I)*sqrt(1 + sqrt(3)*I)
*log(tanh(x/2) - 1/2 - sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-10586087041080284410320*sqrt(2) + 149709
87865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12
223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 2352463786906729868960*sqrt(3)
*I*sqrt(1 - sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-105860870410802844
10320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 352869568036009
4803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 10186467
00472085183870*sqrt(2)*sqrt(1 - sqrt(3)*I)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(2)*sqrt(1 - sqrt(3)*
I)/2 + sqrt(3)*I/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(
2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990
329288652767668136*sqrt(3)*I) + 1440583979074274533653*sqrt(1 - sqrt(3)*I)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) -
1/2 - sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-10586087041080284410320*sqrt(2) + 1497098786595830300440
8 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 122237604056650222
06440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 1663443096217589222712*sqrt(6)*I*sqrt(1 - sqrt
(3)*I)*log(tanh(x/2) - 1/2 - sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-10586087041080284410320*sqrt(2) +
14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*
I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 1018646700472085183870*s
qrt(6)*I*sqrt(1 - sqrt(3)*I)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)
*I/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sq
rt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 499032928865276766
8136*sqrt(3)*I) + 1018646700472085183870*sqrt(6)*I*sqrt(1 - sqrt(3)*I)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2
+ sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 -
8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 1222376040566502220644
0*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 1663443096217589222712*sqrt(6)*I*sqrt(1 - sqrt(3)*
I)*log(tanh(x/2) - 1/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-10586087041080284410320*sqrt(2) + 1497
0987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I +
12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 1440583979074274533653*sqrt(
1 - sqrt(3)*I)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-105860
87041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 352
8695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I
) + 1018646700472085183870*sqrt(2)*sqrt(1 - sqrt(3)*I)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 + sqrt(2)*sqrt(
1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647
201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(
3)*I) + 4990329288652767668136*sqrt(3)*I) - 2352463786906729868960*sqrt(3)*I*sqrt(1 - sqrt(3)*I)*log(tanh(x/2)
- 1/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004
408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 1222376040566502
2206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 1440583979074274533653*sqrt(3)*I*sqrt(1 - sq
rt(3)*I)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2)/(-105860870410
80284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 352869568
0360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 83
1721548108794611356*sqrt(6)*I*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(3)*I/2 - sqrt(2)*sqrt(1 + sqrt(3)
*I)/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + s
qrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 49903292886527676
68136*sqrt(3)*I) - 4074586801888340735480*sqrt(2)*log(tanh(x/2) - 1/2 - sqrt(3)*I/2 - sqrt(2)*sqrt(1 + sqrt(3)
*I)/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + s
qrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 49903292886527676
68136*sqrt(3)*I) + 3528695680360094803440*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(3)*I/2 - sqrt(2)*sqrt
(1 + sqrt(3)*I)/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2
)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 49903
29288652767668136*sqrt(3)*I) - 2495164644326383834068*sqrt(2)*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(3
)*I/2 - sqrt(2)*sqrt(1 + sqrt(3)*I)/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 864350387
4445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1
+ sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 1176231893453364934480*sqrt(3)*I*sqrt(1 + sqrt(3)*I)*log(ta
nh(x/2) - 1/2 - sqrt(3)*I/2 - sqrt(2)*sqrt(1 + sqrt(3)*I)/2)/(-10586087041080284410320*sqrt(2) + 1497098786595
8303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 122237604
05665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 5762335916297098134612*log(tanh(x/2)
- 1/2 - sqrt(3)*I/2 - sqrt(2)*sqrt(1 + sqrt(3)*I)/2)/(-10586087041080284410320*sqrt(2) + 149709878659583030044
08 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022
206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 5762335916297098134612*log(tanh(x/2) - 1/2 -
sqrt(3)*I/2 + sqrt(2)*sqrt(1 + sqrt(3)*I)/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 864
3503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*s
qrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 1176231893453364934480*sqrt(3)*I*sqrt(1 + sqrt(3)*I)*
log(tanh(x/2) - 1/2 - sqrt(3)*I/2 + sqrt(2)*sqrt(1 + sqrt(3)*I)/2)/(-10586087041080284410320*sqrt(2) + 1497098
7865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 122
23760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) + 2495164644326383834068*sqrt(2)*
sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(3)*I/2 + sqrt(2)*sqrt(1 + sqrt(3)*I)/2)/(-105860870410802844103
20*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 352869568036009480
3440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) - 35286956803
60094803440*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(3)*I/2 + sqrt(2)*sqrt(1 + sqrt(3)*I)/2)/(-105860870
41080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 352869
5680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) +
4074586801888340735480*sqrt(2)*log(tanh(x/2) - 1/2 - sqrt(3)*I/2 + sqrt(2)*sqrt(1 + sqrt(3)*I)/2)/(-105860870
41080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1 + sqrt(3)*I) - 352869
5680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 4990329288652767668136*sqrt(3)*I) -
831721548108794611356*sqrt(6)*I*sqrt(1 + sqrt(3)*I)*log(tanh(x/2) - 1/2 - sqrt(3)*I/2 + sqrt(2)*sqrt(1 + sqrt
(3)*I)/2)/(-10586087041080284410320*sqrt(2) + 14970987865958303004408 - 8643503874445647201918*sqrt(2)*sqrt(1
+ sqrt(3)*I) - 3528695680360094803440*sqrt(6)*I + 12223760405665022206440*sqrt(1 + sqrt(3)*I) + 49903292886527
67668136*sqrt(3)*I)

________________________________________________________________________________________

Giac [A]  time = 1.18007, size = 146, normalized size = 1.1 \begin{align*} -\frac{1}{6} \,{\left (\sqrt{3} - i\right )} \log \left (\sqrt{3} + \left (i + 1\right ) \, e^{x} + 1\right ) - \frac{1}{6} \,{\left (\sqrt{3} + i\right )} \log \left (i \, \sqrt{3} + \left (i + 1\right ) \, e^{x} + i\right ) + \frac{1}{6} \,{\left (\sqrt{3} - i\right )} \log \left (-i \, \sqrt{3} + \left (i + 1\right ) \, e^{x} + i\right ) + \frac{1}{6} \,{\left (\sqrt{3} + i\right )} \log \left (-\sqrt{3} + \left (i + 1\right ) \, e^{x} + 1\right ) - \frac{1}{6} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, e^{x} - 2 \right |}}{{\left | 2 \, \sqrt{2} + 2 \, e^{x} - 2 \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^3),x, algorithm="giac")

[Out]

-1/6*(sqrt(3) - I)*log(sqrt(3) + (I + 1)*e^x + 1) - 1/6*(sqrt(3) + I)*log(I*sqrt(3) + (I + 1)*e^x + I) + 1/6*(
sqrt(3) - I)*log(-I*sqrt(3) + (I + 1)*e^x + I) + 1/6*(sqrt(3) + I)*log(-sqrt(3) + (I + 1)*e^x + 1) - 1/6*sqrt(
2)*log(abs(-2*sqrt(2) + 2*e^x - 2)/abs(2*sqrt(2) + 2*e^x - 2))