\(\int \frac {1}{(1-\sinh ^2(x))^2} \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 37 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx=\frac {3 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\cosh (x) \sinh (x)}{4 \left (1-\sinh ^2(x)\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx=\frac {3 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}-\frac {\sinh (2 x)}{4 (-3+\cosh (2 x))} \] Input:

Integrate[(1 - Sinh[x]^2)^(-2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3663, 27, 3042, 3660, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (1+\sin (i x)^2\right )^2}dx\)

\(\Big \downarrow \) 3663

\(\displaystyle \frac {\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}-\frac {1}{4} \int -\frac {3}{1-\sinh ^2(x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{4} \int \frac {1}{1-\sinh ^2(x)}dx+\frac {\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}+\frac {3}{4} \int \frac {1}{\sin (i x)^2+1}dx\)

\(\Big \downarrow \) 3660

\(\displaystyle \frac {3}{4} \int \frac {1}{1-2 \tanh ^2(x)}d\tanh (x)+\frac {\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}\)

Input:

Int[(1 - Sinh[x]^2)^(-2),x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3660

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = 
FreeFactors[Tan[e + f*x], x]}, Simp[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 3663

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C 
os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a 
+ b))), x] + Simp[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(28)=56\).

Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62

method	result	size
risch	\(-\frac {3 \,{\mathrm e}^{2 x}-1}{2 \left ({\mathrm e}^{4 x}-6 \,{\mathrm e}^{2 x}+1\right )}+\frac {3 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3+2 \sqrt {2}\right )}{16}-\frac {3 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{16}\)	\(60\)
default	\(-\frac {-\frac {\tanh \left (\frac {x}{2}\right )}{4}+\frac {1}{4}}{\tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right )-1}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{8}-\frac {-\frac {\tanh \left (\frac {x}{2}\right )}{4}-\frac {1}{4}}{\tanh \left (\frac {x}{2}\right )^{2}-2 \tanh \left (\frac {x}{2}\right )-1}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{8}\)	\(92\)

Input:

int(1/(1-sinh(x)^2)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (27) = 54\).

Time = 0.08 (sec) , antiderivative size = 216, normalized size of antiderivative = 5.84 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx=-\frac {24 \, \cosh \left (x\right )^{2} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} \sinh \left (x\right )^{4} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{2} - 6 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} + 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}\right ) + 48 \, \cosh \left (x\right ) \sinh \left (x\right ) + 24 \, \sinh \left (x\right )^{2} - 8}{16 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 6 \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \] Input:

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

Output:

-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*c 
osh(x))*sinh(x) + 1)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2052 vs. \(2 (32) = 64\).

Time = 4.77 (sec) , antiderivative size = 2052, normalized size of antiderivative = 55.46 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

525888*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 - 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 - 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/(1 
402368*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 + 198 
3248*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 - 841 
4208*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 - 8414208*sqrt(2)*tanh(x/2)**2 + 1402368*sqr 
t(2) + 1983248) + 525888*log(tanh(x/2) + 1 + sqrt(2))*tanh(x/2)**4/(140236 
8*sqrt(2)*tanh(x/2)**4 + 1983248*tanh(x/2)**4 - 11899488*tanh(x/2)**2 - 84 
14208*sqrt(2)*tanh(x/2)**2 + 1402368*sqrt(2) + 1983248) + 371859*sqrt(2)*l 
og(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)...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (27) = 54\).

Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.35 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx=\frac {3}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {3}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) - \frac {3 \, e^{\left (-2 \, x\right )} - 1}{2 \, {\left (6 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1\right )}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).

Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.68 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx=-\frac {3}{16} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - \frac {3 \, e^{\left (2 \, x\right )} - 1}{2 \, {\left (e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1\right )}} \] Input:

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

Output:

-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)
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.08 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx=\frac {3\,\sqrt {2}\,\ln \left (3\,{\mathrm {e}}^{2\,x}+\frac {3\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{16}\right )}{16}-\frac {3\,\sqrt {2}\,\ln \left (3\,{\mathrm {e}}^{2\,x}-\frac {3\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{16}\right )}{16}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{2}-\frac {1}{2}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \] Input:

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

Output:

(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)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 224, normalized size of antiderivative = 6.05 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^2} \, dx=\frac {-3 e^{4 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )+3 e^{4 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )+3 e^{4 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )-3 e^{4 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )-4 e^{4 x}+18 e^{2 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )-18 e^{2 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )-18 e^{2 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )+18 e^{2 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )-3 \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )+3 \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )+3 \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )-3 \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )+4}{16 e^{4 x}-96 e^{2 x}+16} \] Input:

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

Output:

( - 3*e**(4*x)*sqrt(2)*log(e**x - sqrt(2) - 1) + 3*e**(4*x)*sqrt(2)*log(e* 
*x - sqrt(2) + 1) + 3*e**(4*x)*sqrt(2)*log(e**x + sqrt(2) - 1) - 3*e**(4*x 
)*sqrt(2)*log(e**x + sqrt(2) + 1) - 4*e**(4*x) + 18*e**(2*x)*sqrt(2)*log(e 
**x - sqrt(2) - 1) - 18*e**(2*x)*sqrt(2)*log(e**x - sqrt(2) + 1) - 18*e**( 
2*x)*sqrt(2)*log(e**x + sqrt(2) - 1) + 18*e**(2*x)*sqrt(2)*log(e**x + sqrt 
(2) + 1) - 3*sqrt(2)*log(e**x - sqrt(2) - 1) + 3*sqrt(2)*log(e**x - sqrt(2 
) + 1) + 3*sqrt(2)*log(e**x + sqrt(2) - 1) - 3*sqrt(2)*log(e**x + sqrt(2) 
+ 1) + 4)/(16*(e**(4*x) - 6*e**(2*x) + 1))