3.6.88 \(\int \frac {1}{1-\sinh ^4(x)} \, dx\) [588]

Optimal. Leaf size=25 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3288, 396, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Sinh[x]^4)^(-1),x]

[Out]

ArcTanh[Sqrt[2]*Tanh[x]]/(2*Sqrt[2]) + Tanh[x]/2

Rule 212

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

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 3288

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}{1-\sinh ^4(x)} \, dx &=\text {Subst}\left (\int \frac {1-x^2}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{2 \sqrt {2}}+\frac {\tanh (x)}{2}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{4} \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )+2 \tanh (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sinh[x]^4)^(-1),x]

[Out]

(Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]] + 2*Tanh[x])/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(17)=34\).
time = 0.06, size = 55, normalized size = 2.20

method	result	size
risch	\(-\frac {1}{1+{\mathrm e}^{2 x}}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+2 \sqrt {2}-3\right )}{8}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{8}\)	\(46\)
default	\(\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{4}+\frac {\tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{4}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-sinh(x)^4),x,method=_RETURNVERBOSE)

[Out]

1/4*2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)-2)*2^(1/2))+tanh(1/2*x)/(tanh(1/2*x)^2+1)+1/4*2^(1/2)*arctanh(1/4*(2*ta
nh(1/2*x)+2)*2^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (17) = 34\).
time = 1.80, size = 69, normalized size = 2.76 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) + \frac {1}{e^{\left (-2 \, x\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^4),x, algorithm="maxima")

[Out]

1/8*sqrt(2)*log(-(sqrt(2) - e^(-x) + 1)/(sqrt(2) + e^(-x) - 1)) - 1/8*sqrt(2)*log(-(sqrt(2) - e^(-x) - 1)/(sqr
t(2) + e^(-x) + 1)) + 1/(e^(-2*x) + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (17) = 34\).
time = 0.80, size = 113, normalized size = 4.52 \begin {gather*} \frac {{\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) - 8}{8 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^4),x, algorithm="fricas")

[Out]

1/8*((sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*log(-(3*(2*sqrt(2) - 3)*cos
h(x)^2 - 4*(3*sqrt(2) - 4)*cosh(x)*sinh(x) + 3*(2*sqrt(2) - 3)*sinh(x)^2 - 2*sqrt(2) + 3)/(cosh(x)^2 + sinh(x)
^2 - 3)) - 8)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 908 vs. \(2 (20) = 40\).
time = 2.77, size = 908, normalized size = 36.32 \begin {gather*} \frac {3064704 \log {\left (\tanh {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {2167073 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {3064704 \log {\left (\tanh {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {2167073 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {3064704 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {2167073 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {3064704 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {2167073 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} - \frac {2167073 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} - \frac {3064704 \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} - \frac {2167073 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} - \frac {3064704 \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} - \frac {2167073 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} - \frac {3064704 \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} - \frac {2167073 \sqrt {2} \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} - \frac {3064704 \log {\left (\tanh {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {12258816 \sqrt {2} \tanh {\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} + \frac {17336584 \tanh {\left (\frac {x}{2} \right )}}{12258816 \sqrt {2} \tanh ^{2}{\left (\frac {x}{2} \right )} + 17336584 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12258816 \sqrt {2} + 17336584} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)**4),x)

[Out]

3064704*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**2/(12258816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 122
58816*sqrt(2) + 17336584) + 2167073*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**2/(12258816*sqrt(2)*tanh(x
/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) + 3064704*log(tanh(x/2) - 1 + sqrt(2))/(12258816
*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) + 2167073*sqrt(2)*log(tanh(x/2) -
1 + sqrt(2))/(12258816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) + 3064704*
log(tanh(x/2) + 1 + sqrt(2))*tanh(x/2)**2/(12258816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sq
rt(2) + 17336584) + 2167073*sqrt(2)*log(tanh(x/2) + 1 + sqrt(2))*tanh(x/2)**2/(12258816*sqrt(2)*tanh(x/2)**2 +
17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) + 3064704*log(tanh(x/2) + 1 + sqrt(2))/(12258816*sqrt(2)
*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) + 2167073*sqrt(2)*log(tanh(x/2) + 1 + sqr
t(2))/(12258816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) - 2167073*sqrt(2)*
log(tanh(x/2) - sqrt(2) - 1)*tanh(x/2)**2/(12258816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sq
rt(2) + 17336584) - 3064704*log(tanh(x/2) - sqrt(2) - 1)*tanh(x/2)**2/(12258816*sqrt(2)*tanh(x/2)**2 + 1733658
4*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) - 2167073*sqrt(2)*log(tanh(x/2) - sqrt(2) - 1)/(12258816*sqrt(2)
*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) - 3064704*log(tanh(x/2) - sqrt(2) - 1)/(1
2258816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) - 2167073*sqrt(2)*log(tanh
(x/2) - sqrt(2) + 1)*tanh(x/2)**2/(12258816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) +
17336584) - 3064704*log(tanh(x/2) - sqrt(2) + 1)*tanh(x/2)**2/(12258816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x
/2)**2 + 12258816*sqrt(2) + 17336584) - 2167073*sqrt(2)*log(tanh(x/2) - sqrt(2) + 1)/(12258816*sqrt(2)*tanh(x/
2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) - 3064704*log(tanh(x/2) - sqrt(2) + 1)/(12258816*
sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) + 12258816*sqrt(2)*tanh(x/2)/(1225
8816*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584) + 17336584*tanh(x/2)/(1225881
6*sqrt(2)*tanh(x/2)**2 + 17336584*tanh(x/2)**2 + 12258816*sqrt(2) + 17336584)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (17) = 34\).
time = 1.28, size = 48, normalized size = 1.92 \begin {gather*} -\frac {1}{8} \, \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 ) - \frac {1}{e^{\left (2 \, x\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^4),x, algorithm="giac")

[Out]

-1/8*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x) - 6)) - 1/(e^(2*x) + 1)

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Mupad [B]
time = 0.40, size = 63, normalized size = 2.52 \begin {gather*} \frac {\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{8}\right )}{8}-\frac {\sqrt {2}\,\ln \left (2\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{8}\right )}{8}-\frac {1}{{\mathrm {e}}^{2\,x}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(sinh(x)^4 - 1),x)

[Out]

(2^(1/2)*log(2*exp(2*x) + (2^(1/2)*(12*exp(2*x) - 4))/8))/8 - (2^(1/2)*log(2*exp(2*x) - (2^(1/2)*(12*exp(2*x)
- 4))/8))/8 - 1/(exp(2*x) + 1)

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