3.988 \(\int \frac {\text {sech}^2(x)}{3-4 \tanh ^3(x)} \, dx\)

Optimal. Leaf size=102 \[ \frac {\log \left (2 \sqrt [3]{2} \tanh ^2(x)+2^{2/3} \sqrt [3]{3} \tanh (x)+3^{2/3}\right )}{6\ 6^{2/3}}-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac {\tan ^{-1}\left (\frac {2\ 2^{2/3} \tanh (x)+\sqrt [3]{3}}{3^{5/6}}\right )}{3\ 2^{2/3} \sqrt [6]{3}} \]

[Out]

1/18*arctan(1/3*(3^(1/3)+2*2^(2/3)*tanh(x))*3^(1/6))*2^(1/3)*3^(5/6)-1/18*ln(3^(1/3)-2^(2/3)*tanh(x))*6^(1/3)+
1/36*ln(3^(2/3)+2^(2/3)*3^(1/3)*tanh(x)+2*2^(1/3)*tanh(x)^2)*6^(1/3)

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3675, 200, 31, 634, 617, 204, 628} \[ \frac {\log \left (2 \sqrt [3]{2} \tanh ^2(x)+2^{2/3} \sqrt [3]{3} \tanh (x)+3^{2/3}\right )}{6\ 6^{2/3}}-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac {\tan ^{-1}\left (\frac {2\ 2^{2/3} \tanh (x)+\sqrt [3]{3}}{3^{5/6}}\right )}{3\ 2^{2/3} \sqrt [6]{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^2/(3 - 4*Tanh[x]^3),x]

[Out]

ArcTan[(3^(1/3) + 2*2^(2/3)*Tanh[x])/3^(5/6)]/(3*2^(2/3)*3^(1/6)) - Log[3^(1/3) - 2^(2/3)*Tanh[x]]/(3*6^(2/3))
+ Log[3^(2/3) + 2^(2/3)*3^(1/3)*Tanh[x] + 2*2^(1/3)*Tanh[x]^2]/(6*6^(2/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

\begin {align*} \int \frac {\text {sech}^2(x)}{3-4 \tanh ^3(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{3-4 x^3} \, dx,x,\tanh (x)\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{3}-2^{2/3} x} \, dx,x,\tanh (x)\right )}{3\ 3^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {2 \sqrt [3]{3}+2^{2/3} x}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tanh (x)\right )}{3\ 3^{2/3}}\\ &=-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tanh (x)\right )}{2 \sqrt [3]{3}}+\frac {\operatorname {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{3}+4 \sqrt [3]{2} x}{3^{2/3}+2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{2} x^2} \, dx,x,\tanh (x)\right )}{6\ 6^{2/3}}\\ &=-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac {\log \left (3^{2/3}+2^{2/3} \sqrt [3]{3} \tanh (x)+2 \sqrt [3]{2} \tanh ^2(x)\right )}{6\ 6^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2\ 2^{2/3} \tanh (x)}{\sqrt [3]{3}}\right )}{6^{2/3}}\\ &=\frac {\tan ^{-1}\left (\frac {3+2\ 6^{2/3} \tanh (x)}{3 \sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [6]{3}}-\frac {\log \left (\sqrt [3]{3}-2^{2/3} \tanh (x)\right )}{3\ 6^{2/3}}+\frac {\log \left (3^{2/3}+2^{2/3} \sqrt [3]{3} \tanh (x)+2 \sqrt [3]{2} \tanh ^2(x)\right )}{6\ 6^{2/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 74, normalized size = 0.73 \[ \frac {\log \left (2 \sqrt [3]{6} \tanh ^2(x)+6^{2/3} \tanh (x)+3\right )-2 \log \left (3-6^{2/3} \tanh (x)\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2\ 6^{2/3} \tanh (x)+3}{3 \sqrt {3}}\right )}{6\ 6^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^2/(3 - 4*Tanh[x]^3),x]

[Out]

(2*Sqrt[3]*ArcTan[(3 + 2*6^(2/3)*Tanh[x])/(3*Sqrt[3])] - 2*Log[3 - 6^(2/3)*Tanh[x]] + Log[3 + 6^(2/3)*Tanh[x]
+ 2*6^(1/3)*Tanh[x]^2])/(6*6^(2/3))

________________________________________________________________________________________

fricas [B]  time = 0.46, size = 309, normalized size = 3.03 \[ -\frac {1}{18} \cdot 36^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{54} \cdot 36^{\frac {1}{6}} {\left ({\left (36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} + 3 \cdot 36^{\frac {1}{3}} \sqrt {3} - 9 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \cosh \relax (x)^{2} + 2 \, {\left (36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} + 3 \cdot 36^{\frac {1}{3}} \sqrt {3} - 9 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} + 3 \cdot 36^{\frac {1}{3}} \sqrt {3} - 9 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )} \sinh \relax (x)^{2} - 36^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} - 9 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}}\right )}\right ) - \frac {1}{216} \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {2 \, {\left ({\left (36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + 3\right )} \cosh \relax (x)^{2} - 2 \, {\left (36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + 3\right )} \sinh \relax (x)^{2} - 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} - 21\right )}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) + \frac {1}{108} \cdot 36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {2 \, {\left ({\left (36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} - 3 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} - 9\right )} \cosh \relax (x) - {\left (36^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} - 3 \cdot 36^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} - 12\right )} \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(3-4*tanh(x)^3),x, algorithm="fricas")

[Out]

-1/18*36^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/54*36^(1/6)*((36^(2/3)*sqrt(3)*(-1)^(2/3) + 3*36^(1/3)*sqrt(3) - 9*
sqrt(3)*(-1)^(1/3))*cosh(x)^2 + 2*(36^(2/3)*sqrt(3)*(-1)^(2/3) + 3*36^(1/3)*sqrt(3) - 9*sqrt(3)*(-1)^(1/3))*co
sh(x)*sinh(x) + (36^(2/3)*sqrt(3)*(-1)^(2/3) + 3*36^(1/3)*sqrt(3) - 9*sqrt(3)*(-1)^(1/3))*sinh(x)^2 - 36^(2/3)
*sqrt(3)*(-1)^(2/3) - 9*sqrt(3)*(-1)^(1/3))) - 1/216*36^(2/3)*(-1)^(1/3)*log(2*((36^(2/3)*(-1)^(1/3) + 3*36^(1
/3)*(-1)^(2/3) + 3)*cosh(x)^2 - 2*(36^(2/3)*(-1)^(1/3) + 3*36^(1/3)*(-1)^(2/3))*cosh(x)*sinh(x) + (36^(2/3)*(-
1)^(1/3) + 3*36^(1/3)*(-1)^(2/3) + 3)*sinh(x)^2 - 36^(2/3)*(-1)^(1/3) + 3*36^(1/3)*(-1)^(2/3) - 21)/(cosh(x)^2
- 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1/108*36^(2/3)*(-1)^(1/3)*log(2*((36^(2/3)*(-1)^(1/3) - 3*36^(1/3)*(-1)^(
2/3) - 9)*cosh(x) - (36^(2/3)*(-1)^(1/3) - 3*36^(1/3)*(-1)^(2/3) - 12)*sinh(x))/(cosh(x) - sinh(x)))

________________________________________________________________________________________

giac [A]  time = 0.14, size = 1, normalized size = 0.01 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(3-4*tanh(x)^3),x, algorithm="giac")

[Out]

0

________________________________________________________________________________________

maple [C]  time = 0.24, size = 34, normalized size = 0.33 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (36 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (-24 \tanh \left (\frac {x}{2}\right ) \textit {\_R}^{2}+\tanh ^{2}\left (\frac {x}{2}\right )+1\right )\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^2/(3-4*tanh(x)^3),x)

[Out]

1/3*sum(_R*ln(-24*tanh(1/2*x)*_R^2+tanh(1/2*x)^2+1),_R=RootOf(36*_Z^3+1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\operatorname {sech}\relax (x)^{2}}{4 \, \tanh \relax (x)^{3} - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(3-4*tanh(x)^3),x, algorithm="maxima")

[Out]

-integrate(sech(x)^2/(4*tanh(x)^3 - 3), x)

________________________________________________________________________________________

mupad [B]  time = 3.34, size = 169, normalized size = 1.66 \[ -\frac {6^{1/3}\,\ln \left (\frac {6^{1/3}\,\left (29856\,{\mathrm {e}}^{2\,x}-\frac {6^{1/3}\,\left (109440\,{\mathrm {e}}^{2\,x}+153216\right )}{18}+672\right )}{18}-\frac {5696\,{\mathrm {e}}^{2\,x}}{3}+\frac {4480}{3}\right )}{18}-\frac {6^{1/3}\,\ln \left (\frac {4480}{3}+\frac {6^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (29856\,{\mathrm {e}}^{2\,x}-\frac {6^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (109440\,{\mathrm {e}}^{2\,x}+153216\right )}{18}+672\right )}{18}-\frac {5696\,{\mathrm {e}}^{2\,x}}{3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18}+\frac {6^{1/3}\,\ln \left (\frac {4480}{3}-\frac {6^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (29856\,{\mathrm {e}}^{2\,x}+\frac {6^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (109440\,{\mathrm {e}}^{2\,x}+153216\right )}{18}+672\right )}{18}-\frac {5696\,{\mathrm {e}}^{2\,x}}{3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(cosh(x)^2*(4*tanh(x)^3 - 3)),x)

[Out]

(6^(1/3)*log(4480/3 - (6^(1/3)*((3^(1/2)*1i)/2 + 1/2)*(29856*exp(2*x) + (6^(1/3)*((3^(1/2)*1i)/2 + 1/2)*(10944
0*exp(2*x) + 153216))/18 + 672))/18 - (5696*exp(2*x))/3)*((3^(1/2)*1i)/2 + 1/2))/18 - (6^(1/3)*log((6^(1/3)*((
3^(1/2)*1i)/2 - 1/2)*(29856*exp(2*x) - (6^(1/3)*((3^(1/2)*1i)/2 - 1/2)*(109440*exp(2*x) + 153216))/18 + 672))/
18 - (5696*exp(2*x))/3 + 4480/3)*((3^(1/2)*1i)/2 - 1/2))/18 - (6^(1/3)*log((6^(1/3)*(29856*exp(2*x) - (6^(1/3)
*(109440*exp(2*x) + 153216))/18 + 672))/18 - (5696*exp(2*x))/3 + 4480/3))/18

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {sech}^{2}{\relax (x )}}{4 \tanh ^{3}{\relax (x )} - 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2/(3-4*tanh(x)**3),x)

[Out]

-Integral(sech(x)**2/(4*tanh(x)**3 - 3), x)

________________________________________________________________________________________