3.64 \(\int \frac {1}{(1-\sinh ^2(x))^2} \, dx\)

Optimal. Leaf size=37 \[ \frac {3 \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )} \]

[Out]

1/4*cosh(x)*sinh(x)/(1-sinh(x)^2)+3/8*arctanh(2^(1/2)*tanh(x))*2^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3184, 12, 3181, 206} \[ \frac {3 \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sinh[x]^2)^(-2),x]

[Out]

(3*ArcTanh[Sqrt[2]*Tanh[x]])/(4*Sqrt[2]) + (Cosh[x]*Sinh[x])/(4*(1 - Sinh[x]^2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 3184

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
e + f*x]^2)^(p + 1))/(2*a*f*(p + 1)*(a + b)), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^(p
+ 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ
[a + b, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx &=\frac {\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )}-\frac {1}{4} \int -\frac {3}{1-\sinh ^2(x)} \, dx\\ &=\frac {\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )}+\frac {3}{4} \int \frac {1}{1-\sinh ^2(x)} \, dx\\ &=\frac {\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {3 \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 35, normalized size = 0.95 \[ \frac {3 \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}-\frac {\sinh (2 x)}{4 (\cosh (2 x)-3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sinh[x]^2)^(-2),x]

[Out]

(3*ArcTanh[Sqrt[2]*Tanh[x]])/(4*Sqrt[2]) - Sinh[2*x]/(4*(-3 + Cosh[2*x]))

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fricas [B]  time = 1.42, size = 216, normalized size = 5.84 \[ -\frac {24 \, \cosh \relax (x)^{2} - 3 \, {\left (\sqrt {2} \cosh \relax (x)^{4} + 4 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{3} + \sqrt {2} \sinh \relax (x)^{4} + 6 \, {\left (\sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x)^{2} - 6 \, \sqrt {2} \cosh \relax (x)^{2} + 4 \, {\left (\sqrt {2} \cosh \relax (x)^{3} - 3 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) + \sqrt {2}\right )} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \relax (x)^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \relax (x) \sinh \relax (x) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {2} + 3}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}\right ) + 48 \, \cosh \relax (x) \sinh \relax (x) + 24 \, \sinh \relax (x)^{2} - 8}{16 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 6 \, {\left (\cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^2)^2,x, algorithm="fricas")

[Out]

-1/16*(24*cosh(x)^2 - 3*(sqrt(2)*cosh(x)^4 + 4*sqrt(2)*cosh(x)*sinh(x)^3 + sqrt(2)*sinh(x)^4 + 6*(sqrt(2)*cosh
(x)^2 - sqrt(2))*sinh(x)^2 - 6*sqrt(2)*cosh(x)^2 + 4*(sqrt(2)*cosh(x)^3 - 3*sqrt(2)*cosh(x))*sinh(x) + sqrt(2)
)*log(-(3*(2*sqrt(2) - 3)*cosh(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)) + 48*cosh(x)*sinh(x) + 24*sinh(x)^2 - 8)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^
3 + sinh(x)^4 + 6*(cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x))*sinh(x) + 1)

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giac [B]  time = 0.12, size = 62, normalized size = 1.68 \[ -\frac {3}{16} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - \frac {3 \, e^{\left (2 \, x\right )} - 1}{2 \, {\left (e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^2)^2,x, algorithm="giac")

[Out]

-3/16*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x) - 6)) - 1/2*(3*e^(2*x) - 1)/(e^(4*
x) - 6*e^(2*x) + 1)

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maple [B]  time = 0.04, size = 92, normalized size = 2.49 \[ -\frac {-\frac {\tanh \left (\frac {x}{2}\right )}{4}-\frac {1}{4}}{\tanh ^{2}\left (\frac {x}{2}\right )-2 \tanh \left (\frac {x}{2}\right )-1}+\frac {3 \sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{8}-\frac {-\frac {\tanh \left (\frac {x}{2}\right )}{4}+\frac {1}{4}}{\tanh ^{2}\left (\frac {x}{2}\right )+2 \tanh \left (\frac {x}{2}\right )-1}+\frac {3 \sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-sinh(x)^2)^2,x)

[Out]

-(-1/4*tanh(1/2*x)-1/4)/(tanh(1/2*x)^2-2*tanh(1/2*x)-1)+3/8*2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)-2)*2^(1/2))-(-1
/4*tanh(1/2*x)+1/4)/(tanh(1/2*x)^2+2*tanh(1/2*x)-1)+3/8*2^(1/2)*arctanh(1/4*(2*tanh(1/2*x)+2)*2^(1/2))

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maxima [B]  time = 0.41, size = 87, normalized size = 2.35 \[ \frac {3}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {3}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) - \frac {3 \, e^{\left (-2 \, x\right )} - 1}{2 \, {\left (6 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^2)^2,x, algorithm="maxima")

[Out]

3/16*sqrt(2)*log(-(sqrt(2) - e^(-x) + 1)/(sqrt(2) + e^(-x) - 1)) - 3/16*sqrt(2)*log(-(sqrt(2) - e^(-x) - 1)/(s
qrt(2) + e^(-x) + 1)) - 1/2*(3*e^(-2*x) - 1)/(6*e^(-2*x) - e^(-4*x) - 1)

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mupad [B]  time = 0.69, size = 77, normalized size = 2.08 \[ \frac {3\,\sqrt {2}\,\ln \left (3\,{\mathrm {e}}^{2\,x}+\frac {3\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{16}\right )}{16}-\frac {3\,\sqrt {2}\,\ln \left (3\,{\mathrm {e}}^{2\,x}-\frac {3\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{16}\right )}{16}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{2}-\frac {1}{2}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2 - 1)^2,x)

[Out]

(3*2^(1/2)*log(3*exp(2*x) + (3*2^(1/2)*(12*exp(2*x) - 4))/16))/16 - (3*2^(1/2)*log(3*exp(2*x) - (3*2^(1/2)*(12
*exp(2*x) - 4))/16))/16 - ((3*exp(2*x))/2 - 1/2)/(exp(4*x) - 6*exp(2*x) + 1)

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sympy [B]  time = 9.29, size = 2052, normalized size = 55.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)**2)**2,x)

[Out]

525888*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**4/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 118994
88*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 371859*sqrt(2)*log(tanh(x/2) - 1
+ sqrt(2))*tanh(x/2)**4/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 841420
8*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 2231154*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**
2/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2
+ 1402368*sqrt(2) + 1983248) - 3155328*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**2/(1402368*sqrt(2)*tanh(x/2)**4
+ 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) +
525888*log(tanh(x/2) - 1 + sqrt(2))/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)*
*2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 371859*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))/(
1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1
402368*sqrt(2) + 1983248) + 525888*log(tanh(x/2) + 1 + sqrt(2))*tanh(x/2)**4/(1402368*sqrt(2)*tanh(x/2)**4 + 1
983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 3718
59*sqrt(2)*log(tanh(x/2) + 1 + sqrt(2))*tanh(x/2)**4/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11
899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 2231154*sqrt(2)*log(tanh(x/2
) + 1 + sqrt(2))*tanh(x/2)**2/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8
414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 3155328*log(tanh(x/2) + 1 + sqrt(2))*tanh(x/2)**2/(
1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1
402368*sqrt(2) + 1983248) + 525888*log(tanh(x/2) + 1 + sqrt(2))/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x
/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 371859*sqrt(2)*lo
g(tanh(x/2) + 1 + sqrt(2))/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414
208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 371859*sqrt(2)*log(tanh(x/2) - sqrt(2) - 1)*tanh(x/2)*
*4/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2
+ 1402368*sqrt(2) + 1983248) - 525888*log(tanh(x/2) - sqrt(2) - 1)*tanh(x/2)**4/(1402368*sqrt(2)*tanh(x/2)**4
+ 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) +
3155328*log(tanh(x/2) - sqrt(2) - 1)*tanh(x/2)**2/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899
488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 2231154*sqrt(2)*log(tanh(x/2) -
sqrt(2) - 1)*tanh(x/2)**2/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414
208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 371859*sqrt(2)*log(tanh(x/2) - sqrt(2) - 1)/(1402368*s
qrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sq
rt(2) + 1983248) - 525888*log(tanh(x/2) - sqrt(2) - 1)/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 -
11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 371859*sqrt(2)*log(tanh(x/
2) - sqrt(2) + 1)*tanh(x/2)**4/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 -
8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 525888*log(tanh(x/2) - sqrt(2) + 1)*tanh(x/2)**4/(
1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1
402368*sqrt(2) + 1983248) + 3155328*log(tanh(x/2) - sqrt(2) + 1)*tanh(x/2)**2/(1402368*sqrt(2)*tanh(x/2)**4 +
1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 223
1154*sqrt(2)*log(tanh(x/2) - sqrt(2) + 1)*tanh(x/2)**2/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 -
11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 371859*sqrt(2)*log(tanh(x/
2) - sqrt(2) + 1)/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(
2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) - 525888*log(tanh(x/2) - sqrt(2) + 1)/(1402368*sqrt(2)*tanh(x/2)*
*4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248)
+ 701184*sqrt(2)*tanh(x/2)**3/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8
414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 991624*tanh(x/2)**3/(1402368*sqrt(2)*tanh(x/2)**4 +
1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 70
1184*sqrt(2)*tanh(x/2)/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*
sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 991624*tanh(x/2)/(1402368*sqrt(2)*tanh(x/2)**4 + 1983248*t
anh(x/2)**4 - 11899488*tanh(x/2)**2 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248)

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