3.274 \(\int \frac {1}{1-\sinh ^6(x)} \, dx\)

Optimal. Leaf size=83 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{3 \sqrt {2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1+(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1+(-1)^{2/3}}} \]

[Out]

1/6*arctanh(2^(1/2)*tanh(x))*2^(1/2)+1/3*arctanh((1-(-1)^(1/3))^(1/2)*tanh(x))/(1-(-1)^(1/3))^(1/2)+1/3*arctan
h((1+(-1)^(2/3))^(1/2)*tanh(x))/(1+(-1)^(2/3))^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3211, 3181, 206} \[ \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{3 \sqrt {2}}+\frac {\tanh ^{-1}\left (\sqrt {1-\sqrt [3]{-1}} \tanh (x)\right )}{3 \sqrt {1-\sqrt [3]{-1}}}+\frac {\tanh ^{-1}\left (\sqrt {1+(-1)^{2/3}} \tanh (x)\right )}{3 \sqrt {1+(-1)^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sinh[x]^6)^(-1),x]

[Out]

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

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 3181

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

Rule 3211

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{k}, Dist[2/(a*n), Sum[Int[1/(1 - Si
n[e + f*x]^2/((-1)^((4*k)/n)*Rt[-(a/b), n/2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/
2]

Rubi steps

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

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Mathematica [C]  time = 0.45, size = 70, normalized size = 0.84 \[ \frac {1}{6} \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )+i \sqrt {3} \left (\tan ^{-1}\left (\frac {1-2 i \tanh (x)}{\sqrt {3}}\right )-\tan ^{-1}\left (\frac {1+2 i \tanh (x)}{\sqrt {3}}\right )\right )-\tan ^{-1}(\text {csch}(x) \text {sech}(x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sinh[x]^6)^(-1),x]

[Out]

(-ArcTan[Csch[x]*Sech[x]] + I*Sqrt[3]*(ArcTan[(1 - (2*I)*Tanh[x])/Sqrt[3]] - ArcTan[(1 + (2*I)*Tanh[x])/Sqrt[3
]]) + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]])/6

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fricas [B]  time = 0.95, size = 155, normalized size = 1.87 \[ -\frac {1}{12} \, \sqrt {3} \log \left (16 \, \sqrt {3} + 4 \, e^{\left (4 \, x\right )} + 28\right ) + \frac {1}{12} \, \sqrt {3} \log \left (-16 \, \sqrt {3} + 4 \, e^{\left (4 \, x\right )} + 28\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\frac {2 \, {\left (2 \, \sqrt {2} - 3\right )} e^{\left (2 \, x\right )} - 12 \, \sqrt {2} + e^{\left (4 \, x\right )} + 17}{e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1}\right ) - \frac {1}{3} \, \arctan \left (-{\left (\sqrt {3} + 2\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (\sqrt {3} + 2\right )} \sqrt {-16 \, \sqrt {3} + 4 \, e^{\left (4 \, x\right )} + 28}\right ) + \frac {1}{3} \, \arctan \left (-{\left (\sqrt {3} - 2\right )} e^{\left (2 \, x\right )} + \sqrt {4 \, \sqrt {3} + e^{\left (4 \, x\right )} + 7} {\left (\sqrt {3} - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^6),x, algorithm="fricas")

[Out]

-1/12*sqrt(3)*log(16*sqrt(3) + 4*e^(4*x) + 28) + 1/12*sqrt(3)*log(-16*sqrt(3) + 4*e^(4*x) + 28) + 1/12*sqrt(2)
*log((2*(2*sqrt(2) - 3)*e^(2*x) - 12*sqrt(2) + e^(4*x) + 17)/(e^(4*x) - 6*e^(2*x) + 1)) - 1/3*arctan(-(sqrt(3)
+ 2)*e^(2*x) + 1/2*(sqrt(3) + 2)*sqrt(-16*sqrt(3) + 4*e^(4*x) + 28)) + 1/3*arctan(-(sqrt(3) - 2)*e^(2*x) + sq
rt(4*sqrt(3) + e^(4*x) + 7)*(sqrt(3) - 2))

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giac [B]  time = 0.16, size = 143, normalized size = 1.72 \[ -\frac {1}{36} \, {\left ({\left (2 \, \sqrt {3} - 3\right )} e^{\left (4 \, x\right )} + 2 \, \sqrt {3} - 3\right )} \arctan \left (\frac {e^{\left (2 \, x\right )}}{\sqrt {3} + 2}\right ) + \frac {1}{36} \, {\left ({\left (2 \, \sqrt {3} + 3\right )} e^{\left (4 \, x\right )} + 2 \, \sqrt {3} + 3\right )} \arctan \left (-\frac {e^{\left (2 \, x\right )}}{\sqrt {3} - 2}\right ) - \frac {1}{12} \, \sqrt {3} \log \left ({\left (\sqrt {3} + 2\right )}^{2} + e^{\left (4 \, x\right )}\right ) + \frac {1}{12} \, \sqrt {3} \log \left ({\left (\sqrt {3} - 2\right )}^{2} + e^{\left (4 \, x\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^6),x, algorithm="giac")

[Out]

-1/36*((2*sqrt(3) - 3)*e^(4*x) + 2*sqrt(3) - 3)*arctan(e^(2*x)/(sqrt(3) + 2)) + 1/36*((2*sqrt(3) + 3)*e^(4*x)
+ 2*sqrt(3) + 3)*arctan(-e^(2*x)/(sqrt(3) - 2)) - 1/12*sqrt(3)*log((sqrt(3) + 2)^2 + e^(4*x)) + 1/12*sqrt(3)*l
og((sqrt(3) - 2)^2 + e^(4*x)) - 1/12*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x) - 6
))

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maple [C]  time = 0.06, size = 160, normalized size = 1.93 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}-\textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{3}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{6}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+\textit {\_R} +1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-sinh(x)^6),x)

[Out]

1/3*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))+1/6*2^(1/
2)*arctanh(1/4*(2*tanh(1/2*x)-2)*2^(1/2))+1/6*2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)+2)*2^(1/2))+1/3*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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{x} + 1}{\sqrt {2} + e^{x} - 1}\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{x} - 1}{\sqrt {2} + e^{x} + 1}\right ) + \int \frac {e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )} - e^{x}}{3 \, {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )}}\,{d x} - \int \frac {e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )} - e^{x}}{3 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 2 \, e^{x} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^6),x, algorithm="maxima")

[Out]

-1/12*sqrt(2)*log(-(sqrt(2) - e^x + 1)/(sqrt(2) + e^x - 1)) + 1/12*sqrt(2)*log(-(sqrt(2) - e^x - 1)/(sqrt(2) +
e^x + 1)) + integrate(1/3*(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) - int
egrate(1/3*(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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mupad [B]  time = 2.71, size = 285, normalized size = 3.43 \[ \frac {\mathrm {atan}\left (\frac {14009449395540459520\,{\mathrm {e}}^{2\,x}-955607545932677120\,\sqrt {3}+8088359377641144320\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-1655160823988879360}{6177144285775790080\,{\mathrm {e}}^{2\,x}+2167269359741829120\,\sqrt {3}+3566375915854233600\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+3753820658157486080}\right )}{6}-\frac {\sqrt {3}\,\ln \left ({\left (6177144285775790080\,{\mathrm {e}}^{2\,x}-2167269359741829120\,\sqrt {3}-3566375915854233600\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+3753820658157486080\right )}^2+{\left (14009449395540459520\,{\mathrm {e}}^{2\,x}+955607545932677120\,\sqrt {3}-8088359377641144320\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-1655160823988879360\right )}^2\right )}{12}+\frac {\sqrt {3}\,\ln \left ({\left (6177144285775790080\,{\mathrm {e}}^{2\,x}+2167269359741829120\,\sqrt {3}+3566375915854233600\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+3753820658157486080\right )}^2+{\left (14009449395540459520\,{\mathrm {e}}^{2\,x}-955607545932677120\,\sqrt {3}+8088359377641144320\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-1655160823988879360\right )}^2\right )}{12}+\frac {\sqrt {2}\,\ln \left (17674880313941032960\,{\mathrm {e}}^{2\,x}-2144322552070144000\,\sqrt {2}+12498027726650736640\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-3032530035220152320\right )}{12}-\frac {\sqrt {2}\,\ln \left (17674880313941032960\,{\mathrm {e}}^{2\,x}+2144322552070144000\,\sqrt {2}-12498027726650736640\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-3032530035220152320\right )}{12}-\frac {\ln \left ({\mathrm {e}}^{2\,x}\,\left (14009449395540459520-6177144285775790080{}\mathrm {i}\right )+\sqrt {3}\,\left (955607545932677120+2167269359741829120{}\mathrm {i}\right )+\sqrt {3}\,{\mathrm {e}}^{2\,x}\,\left (-8088359377641144320+3566375915854233600{}\mathrm {i}\right )-1655160823988879360-3753820658157486080{}\mathrm {i}\right )\,1{}\mathrm {i}}{12}+\frac {\ln \left ({\mathrm {e}}^{2\,x}\,\left (14009449395540459520+6177144285775790080{}\mathrm {i}\right )+\sqrt {3}\,\left (955607545932677120-2167269359741829120{}\mathrm {i}\right )+\sqrt {3}\,{\mathrm {e}}^{2\,x}\,\left (-8088359377641144320-3566375915854233600{}\mathrm {i}\right )-1655160823988879360+3753820658157486080{}\mathrm {i}\right )\,1{}\mathrm {i}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(sinh(x)^6 - 1),x)

[Out]

(log(exp(2*x)*(14009449395540459520 + 6177144285775790080i) + 3^(1/2)*(955607545932677120 - 216726935974182912
0i) - 3^(1/2)*exp(2*x)*(8088359377641144320 + 3566375915854233600i) - (1655160823988879360 - 37538206581574860
80i))*1i)/12 - (log(exp(2*x)*(14009449395540459520 - 6177144285775790080i) + 3^(1/2)*(955607545932677120 + 216
7269359741829120i) - 3^(1/2)*exp(2*x)*(8088359377641144320 - 3566375915854233600i) - (1655160823988879360 + 37
53820658157486080i))*1i)/12 + atan((14009449395540459520*exp(2*x) - 955607545932677120*3^(1/2) + 8088359377641
144320*3^(1/2)*exp(2*x) - 1655160823988879360)/(6177144285775790080*exp(2*x) + 2167269359741829120*3^(1/2) + 3
566375915854233600*3^(1/2)*exp(2*x) + 3753820658157486080))/6 - (3^(1/2)*log((6177144285775790080*exp(2*x) - 2
167269359741829120*3^(1/2) - 3566375915854233600*3^(1/2)*exp(2*x) + 3753820658157486080)^2 + (1400944939554045
9520*exp(2*x) + 955607545932677120*3^(1/2) - 8088359377641144320*3^(1/2)*exp(2*x) - 1655160823988879360)^2))/1
2 + (3^(1/2)*log((6177144285775790080*exp(2*x) + 2167269359741829120*3^(1/2) + 3566375915854233600*3^(1/2)*exp
(2*x) + 3753820658157486080)^2 + (14009449395540459520*exp(2*x) - 955607545932677120*3^(1/2) + 808835937764114
4320*3^(1/2)*exp(2*x) - 1655160823988879360)^2))/12 + (2^(1/2)*log(17674880313941032960*exp(2*x) - 21443225520
70144000*2^(1/2) + 12498027726650736640*2^(1/2)*exp(2*x) - 3032530035220152320))/12 - (2^(1/2)*log(17674880313
941032960*exp(2*x) + 2144322552070144000*2^(1/2) - 12498027726650736640*2^(1/2)*exp(2*x) - 3032530035220152320
))/12

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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/(1-sinh(x)**6),x)

[Out]

Timed out

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