3.1.65 \(\int \frac {1}{(1-\sinh ^2(x))^3} \, dx\) [65]

Optimal. Leaf size=55 \[ \frac {19 \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{32 \sqrt {2}}+\frac {\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac {9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )} \]

[Out]

1/8*cosh(x)*sinh(x)/(1-sinh(x)^2)^2+9/32*cosh(x)*sinh(x)/(1-sinh(x)^2)+19/64*arctanh(2^(1/2)*tanh(x))*2^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3263, 3252, 12, 3260, 212} \begin {gather*} \frac {19 \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{32 \sqrt {2}}+\frac {9 \sinh (x) \cosh (x)}{32 \left (1-\sinh ^2(x)\right )}+\frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Sinh[x]^2)^(-3),x]

[Out]

(19*ArcTanh[Sqrt[2]*Tanh[x]])/(32*Sqrt[2]) + (Cosh[x]*Sinh[x])/(8*(1 - Sinh[x]^2)^2) + (9*Cosh[x]*Sinh[x])/(32
*(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 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 3252

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

Rule 3260

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 3263

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si
n[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] && N
eQ[a + b, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx &=\frac {\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}-\frac {1}{8} \int \frac {-7-2 \sinh ^2(x)}{\left (1-\sinh ^2(x)\right )^2} \, dx\\ &=\frac {\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac {9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )}-\frac {1}{32} \int -\frac {19}{1-\sinh ^2(x)} \, dx\\ &=\frac {\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac {9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )}+\frac {19}{32} \int \frac {1}{1-\sinh ^2(x)} \, dx\\ &=\frac {\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac {9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )}+\frac {19}{32} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {19 \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{32 \sqrt {2}}+\frac {\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac {9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 51, normalized size = 0.93 \begin {gather*} \frac {19 \tanh ^{-1}\left (\sqrt {2} \tanh (x)\right )}{32 \sqrt {2}}+\frac {\sinh (2 x)}{4 (-3+\cosh (2 x))^2}-\frac {9 \sinh (2 x)}{32 (-3+\cosh (2 x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sinh[x]^2)^(-3),x]

[Out]

(19*ArcTanh[Sqrt[2]*Tanh[x]])/(32*Sqrt[2]) + Sinh[2*x]/(4*(-3 + Cosh[2*x])^2) - (9*Sinh[2*x])/(32*(-3 + Cosh[2
*x]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(45)=90\).
time = 0.41, size = 124, normalized size = 2.25

method	result	size
risch	\(-\frac {19 \,{\mathrm e}^{6 x}-171 \,{\mathrm e}^{4 x}+89 \,{\mathrm e}^{2 x}-9}{16 \left ({\mathrm e}^{4 x}-6 \,{\mathrm e}^{2 x}+1\right )^{2}}+\frac {19 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3+2 \sqrt {2}\right )}{128}-\frac {19 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{128}\)	\(72\)
default	\(-\frac {-\frac {13 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{8}-\frac {11 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {31 \tanh \left (\frac {x}{2}\right )}{8}-\frac {11}{8}}{4 \left (\tanh ^{2}\left (\frac {x}{2}\right )+2 \tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {19 \sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{64}-\frac {-\frac {13 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{8}+\frac {11 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {31 \tanh \left (\frac {x}{2}\right )}{8}+\frac {11}{8}}{4 \left (\tanh ^{2}\left (\frac {x}{2}\right )-2 \tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {19 \sqrt {2}\, \arctanh \left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{64}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-sinh(x)^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/4*(-13/8*tanh(1/2*x)^3-11/8*tanh(1/2*x)^2+31/8*tanh(1/2*x)-11/8)/(tanh(1/2*x)^2+2*tanh(1/2*x)-1)^2+19/64*2^
(1/2)*arctanh(1/4*(2*tanh(1/2*x)+2)*2^(1/2))-1/4*(-13/8*tanh(1/2*x)^3+11/8*tanh(1/2*x)^2+31/8*tanh(1/2*x)+11/8
)/(tanh(1/2*x)^2-2*tanh(1/2*x)-1)^2+19/64*2^(1/2)*arctanh(1/4*(2*tanh(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. 111 vs. \(2 (41) = 82\).
time = 0.48, size = 111, normalized size = 2.02 \begin {gather*} \frac {19}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {19}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) - \frac {89 \, e^{\left (-2 \, x\right )} - 171 \, e^{\left (-4 \, x\right )} + 19 \, e^{\left (-6 \, x\right )} - 9}{16 \, {\left (12 \, e^{\left (-2 \, x\right )} - 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^2)^3,x, algorithm="maxima")

[Out]

19/128*sqrt(2)*log(-(sqrt(2) - e^(-x) + 1)/(sqrt(2) + e^(-x) - 1)) - 19/128*sqrt(2)*log(-(sqrt(2) - e^(-x) - 1
)/(sqrt(2) + e^(-x) + 1)) - 1/16*(89*e^(-2*x) - 171*e^(-4*x) + 19*e^(-6*x) - 9)/(12*e^(-2*x) - 38*e^(-4*x) + 1
2*e^(-6*x) - e^(-8*x) - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (41) = 82\).
time = 0.43, size = 575, normalized size = 10.45 \begin {gather*} -\frac {152 \, \cosh \left (x\right )^{6} + 912 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 152 \, \sinh \left (x\right )^{6} + 456 \, {\left (5 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{4} - 1368 \, \cosh \left (x\right )^{4} + 608 \, {\left (5 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 8 \, {\left (285 \, \cosh \left (x\right )^{4} - 1026 \, \cosh \left (x\right )^{2} + 89\right )} \sinh \left (x\right )^{2} + 712 \, \cosh \left (x\right )^{2} - 19 \, {\left (\sqrt {2} \cosh \left (x\right )^{8} + 8 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sqrt {2} \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{2} - 3 \, \sqrt {2}\right )} \sinh \left (x\right )^{6} - 12 \, \sqrt {2} \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{3} - 9 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \sqrt {2} \cosh \left (x\right )^{4} - 90 \, \sqrt {2} \cosh \left (x\right )^{2} + 19 \, \sqrt {2}\right )} \sinh \left (x\right )^{4} + 38 \, \sqrt {2} \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{5} - 30 \, \sqrt {2} \cosh \left (x\right )^{3} + 19 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{6} - 45 \, \sqrt {2} \cosh \left (x\right )^{4} + 57 \, \sqrt {2} \cosh \left (x\right )^{2} - 3 \, \sqrt {2}\right )} \sinh \left (x\right )^{2} - 12 \, \sqrt {2} \cosh \left (x\right )^{2} + 8 \, {\left (\sqrt {2} \cosh \left (x\right )^{7} - 9 \, \sqrt {2} \cosh \left (x\right )^{5} + 19 \, \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 ) + 16 \, {\left (57 \, \cosh \left (x\right )^{5} - 342 \, \cosh \left (x\right )^{3} + 89 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 72}{128 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{6} - 12 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 90 \, \cosh \left (x\right )^{2} + 19\right )} \sinh \left (x\right )^{4} + 38 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 30 \, \cosh \left (x\right )^{3} + 19 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 45 \, \cosh \left (x\right )^{4} + 57 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{2} - 12 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 9 \, \cosh \left (x\right )^{5} + 19 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^2)^3,x, algorithm="fricas")

[Out]

-1/128*(152*cosh(x)^6 + 912*cosh(x)*sinh(x)^5 + 152*sinh(x)^6 + 456*(5*cosh(x)^2 - 3)*sinh(x)^4 - 1368*cosh(x)
^4 + 608*(5*cosh(x)^3 - 9*cosh(x))*sinh(x)^3 + 8*(285*cosh(x)^4 - 1026*cosh(x)^2 + 89)*sinh(x)^2 + 712*cosh(x)
^2 - 19*(sqrt(2)*cosh(x)^8 + 8*sqrt(2)*cosh(x)*sinh(x)^7 + sqrt(2)*sinh(x)^8 + 4*(7*sqrt(2)*cosh(x)^2 - 3*sqrt
(2))*sinh(x)^6 - 12*sqrt(2)*cosh(x)^6 + 8*(7*sqrt(2)*cosh(x)^3 - 9*sqrt(2)*cosh(x))*sinh(x)^5 + 2*(35*sqrt(2)*
cosh(x)^4 - 90*sqrt(2)*cosh(x)^2 + 19*sqrt(2))*sinh(x)^4 + 38*sqrt(2)*cosh(x)^4 + 8*(7*sqrt(2)*cosh(x)^5 - 30*
sqrt(2)*cosh(x)^3 + 19*sqrt(2)*cosh(x))*sinh(x)^3 + 4*(7*sqrt(2)*cosh(x)^6 - 45*sqrt(2)*cosh(x)^4 + 57*sqrt(2)
*cosh(x)^2 - 3*sqrt(2))*sinh(x)^2 - 12*sqrt(2)*cosh(x)^2 + 8*(sqrt(2)*cosh(x)^7 - 9*sqrt(2)*cosh(x)^5 + 19*sqr
t(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)*c
osh(x)*sinh(x) + 3*(2*sqrt(2) - 3)*sinh(x)^2 - 2*sqrt(2) + 3)/(cosh(x)^2 + sinh(x)^2 - 3)) + 16*(57*cosh(x)^5
- 342*cosh(x)^3 + 89*cosh(x))*sinh(x) - 72)/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 3)
*sinh(x)^6 - 12*cosh(x)^6 + 8*(7*cosh(x)^3 - 9*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 90*cosh(x)^2 + 19)*sinh(
x)^4 + 38*cosh(x)^4 + 8*(7*cosh(x)^5 - 30*cosh(x)^3 + 19*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 45*cosh(x)^4 +
57*cosh(x)^2 - 3)*sinh(x)^2 - 12*cosh(x)^2 + 8*(cosh(x)^7 - 9*cosh(x)^5 + 19*cosh(x)^3 - 3*cosh(x))*sinh(x) +
1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 5666 vs. \(2 (51) = 102\).
time = 11.81, size = 5666, normalized size = 103.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)**2)**3,x)

[Out]

10001001174720*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**8/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072
*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)
*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/
2)**2 + 33687582904320*sqrt(2) + 47641436627072) + 7071775749331*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2
)**8/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 40425
0994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 5716
97239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320*sqrt(2) + 47641436627072) - 8
4861308991972*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**6/(33687582904320*sqrt(2)*tanh(x/2)**8 + 4764143
6627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*
sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*
tanh(x/2)**2 + 33687582904320*sqrt(2) + 47641436627072) - 120012014096640*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/
2)**6/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 4042
50994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 571
697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320*sqrt(2) + 47641436627072) +
380038044639360*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**4/(33687582904320*sqrt(2)*tanh(x/2)**8 + 4764143662707
2*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2
)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x
/2)**2 + 33687582904320*sqrt(2) + 47641436627072) + 268727478474578*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))*tanh(
x/2)**4/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 40
4250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 5
71697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320*sqrt(2) + 47641436627072)
- 84861308991972*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**2/(33687582904320*sqrt(2)*tanh(x/2)**8 + 4764
1436627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 12801281503641
60*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(
2)*tanh(x/2)**2 + 33687582904320*sqrt(2) + 47641436627072) - 120012014096640*log(tanh(x/2) - 1 + sqrt(2))*tanh
(x/2)**2/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 4
04250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 -
571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320*sqrt(2) + 47641436627072)
+ 10001001174720*log(tanh(x/2) - 1 + sqrt(2))/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)
**8 - 571697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)
**4 + 1810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33
687582904320*sqrt(2) + 47641436627072) + 7071775749331*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))/(33687582904320*sq
rt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh
(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**
2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320*sqrt(2) + 47641436627072) + 10001001174720*log(tanh(
x/2) + 1 + sqrt(2))*tanh(x/2)**8/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 57169723
9524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 18103745
91828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320*s
qrt(2) + 47641436627072) + 7071775749331*sqrt(2)*log(tanh(x/2) + 1 + sqrt(2))*tanh(x/2)**8/(33687582904320*sqr
t(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(
x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2
- 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320*sqrt(2) + 47641436627072) - 84861308991972*sqrt(2)*lo
g(tanh(x/2) + 1 + sqrt(2))*tanh(x/2)**6/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 5
71697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 1
810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 336875829
04320*sqrt(2) + 47641436627072) - 1200120140966...

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Giac [A]
time = 0.41, size = 74, normalized size = 1.35 \begin {gather*} -\frac {19}{128} \, \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 {19 \, e^{\left (6 \, x\right )} - 171 \, e^{\left (4 \, x\right )} + 89 \, e^{\left (2 \, x\right )} - 9}{16 \, {\left (e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-sinh(x)^2)^3,x, algorithm="giac")

[Out]

-19/128*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2*x) - 6)) - 1/16*(19*e^(6*x) - 171*e
^(4*x) + 89*e^(2*x) - 9)/(e^(4*x) - 6*e^(2*x) + 1)^2

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Mupad [B]
time = 0.61, size = 112, normalized size = 2.04 \begin {gather*} \frac {17\,{\mathrm {e}}^{2\,x}-3}{38\,{\mathrm {e}}^{4\,x}-12\,{\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {19\,\sqrt {2}\,\ln \left (\frac {19\,{\mathrm {e}}^{2\,x}}{8}-\frac {19\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{128}\right )}{128}+\frac {19\,\sqrt {2}\,\ln \left (\frac {19\,{\mathrm {e}}^{2\,x}}{8}+\frac {19\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{128}\right )}{128}-\frac {\frac {19\,{\mathrm {e}}^{2\,x}}{16}-\frac {57}{16}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(sinh(x)^2 - 1)^3,x)

[Out]

(17*exp(2*x) - 3)/(38*exp(4*x) - 12*exp(2*x) - 12*exp(6*x) + exp(8*x) + 1) - (19*2^(1/2)*log((19*exp(2*x))/8 -
(19*2^(1/2)*(12*exp(2*x) - 4))/128))/128 + (19*2^(1/2)*log((19*exp(2*x))/8 + (19*2^(1/2)*(12*exp(2*x) - 4))/1
28))/128 - ((19*exp(2*x))/16 - 57/16)/(exp(4*x) - 6*exp(2*x) + 1)

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