3.19 \(\int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx\)

Optimal. Leaf size=98 \[ \frac {\log \left (3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}\right )}{12 \sqrt [3]{6}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{6} \cosh (x)+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}} \]

[Out]

1/12*arctan(1/3*(1+6^(1/3)*cosh(x))*3^(1/2))*2^(2/3)*3^(1/6)-1/36*ln(2^(2/3)-3^(1/3)*cosh(x))*6^(2/3)+1/72*ln(
2*2^(1/3)+2^(2/3)*3^(1/3)*cosh(x)+3^(2/3)*cosh(x)^2)*6^(2/3)

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Rubi [A]  time = 0.12, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3223, 200, 31, 634, 617, 204, 628} \[ \frac {\log \left (3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}\right )}{12 \sqrt [3]{6}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\tan ^{-1}\left (\frac {\sqrt [3]{6} \cosh (x)+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(4 - 3*Cosh[x]^3),x]

[Out]

ArcTan[(1 + 6^(1/3)*Cosh[x])/Sqrt[3]]/(2*2^(1/3)*3^(5/6)) - Log[2^(2/3) - 3^(1/3)*Cosh[x]]/(6*6^(1/3)) + Log[2
*2^(1/3) + 2^(2/3)*3^(1/3)*Cosh[x] + 3^(2/3)*Cosh[x]^2]/(12*6^(1/3))

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 3223

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

Rubi steps

\begin {align*} \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{4-3 x^3} \, dx,x,\cosh (x)\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{3} x} \, dx,x,\cosh (x)\right )}{6 \sqrt [3]{2}}+\frac {\operatorname {Subst}\left (\int \frac {2\ 2^{2/3}+\sqrt [3]{3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{6 \sqrt [3]{2}}\\ &=-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{2\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{3}+2\ 3^{2/3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{12 \sqrt [3]{6}}\\ &=-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cosh (x)+3^{2/3} \cosh ^2(x)\right )}{12 \sqrt [3]{6}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{6} \cosh (x)\right )}{2 \sqrt [3]{6}}\\ &=\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{6} \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cosh (x)+3^{2/3} \cosh ^2(x)\right )}{12 \sqrt [3]{6}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 77, normalized size = 0.79 \[ \frac {1}{72} \left (6^{2/3} \left (\log \left (6^{2/3} \cosh ^2(x)+2 \sqrt [3]{6} \cosh (x)+4\right )-2 \log \left (2-\sqrt [3]{6} \cosh (x)\right )\right )+6\ 2^{2/3} \sqrt [6]{3} \tan ^{-1}\left (\frac {\sqrt [3]{6} \cosh (x)+1}{\sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(4 - 3*Cosh[x]^3),x]

[Out]

(6*2^(2/3)*3^(1/6)*ArcTan[(1 + 6^(1/3)*Cosh[x])/Sqrt[3]] + 6^(2/3)*(-2*Log[2 - 6^(1/3)*Cosh[x]] + Log[4 + 2*6^
(1/3)*Cosh[x] + 6^(2/3)*Cosh[x]^2]))/72

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fricas [B]  time = 0.52, size = 305, normalized size = 3.11 \[ \frac {1}{12} \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{12} \cdot 6^{\frac {1}{6}} {\left (6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \cosh \relax (x)^{3} + 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \sinh \relax (x)^{3} + {\left (3 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \cosh \relax (x) + 4 \cdot 6^{\frac {1}{3}} \sqrt {2}\right )} \sinh \relax (x)^{2} + 4 \cdot 6^{\frac {1}{3}} \sqrt {2} \cosh \relax (x)^{2} + {\left (6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} - 16 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}}\right )} \cosh \relax (x) + {\left (3 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \cosh \relax (x)^{2} + 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} + 8 \cdot 6^{\frac {1}{3}} \sqrt {2} \cosh \relax (x) - 16 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}}\right )} \sinh \relax (x) + 2 \cdot 6^{\frac {1}{3}} \sqrt {2}\right )}\right ) - \frac {1}{12} \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{12} \cdot 6^{\frac {1}{6}} {\left (6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \cosh \relax (x) + 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \sinh \relax (x) + 2 \cdot 6^{\frac {1}{3}} \sqrt {2}\right )}\right ) - \frac {1}{72} \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2 \, {\left (2 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \cosh \relax (x) - 3 \, \cosh \relax (x)^{2} - 3 \, \sinh \relax (x)^{2} - 4 \cdot 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} - 3\right )}}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) + \frac {1}{36} \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {2 \, {\left (6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \, \cosh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(4-3*cosh(x)^3),x, algorithm="fricas")

[Out]

1/12*6^(1/6)*sqrt(2)*(-1)^(1/3)*arctan(1/12*6^(1/6)*(6^(2/3)*sqrt(2)*(-1)^(2/3)*cosh(x)^3 + 6^(2/3)*sqrt(2)*(-
1)^(2/3)*sinh(x)^3 + (3*6^(2/3)*sqrt(2)*(-1)^(2/3)*cosh(x) + 4*6^(1/3)*sqrt(2))*sinh(x)^2 + 4*6^(1/3)*sqrt(2)*
cosh(x)^2 + (6^(2/3)*sqrt(2)*(-1)^(2/3) - 16*sqrt(2)*(-1)^(1/3))*cosh(x) + (3*6^(2/3)*sqrt(2)*(-1)^(2/3)*cosh(
x)^2 + 6^(2/3)*sqrt(2)*(-1)^(2/3) + 8*6^(1/3)*sqrt(2)*cosh(x) - 16*sqrt(2)*(-1)^(1/3))*sinh(x) + 2*6^(1/3)*sqr
t(2))) - 1/12*6^(1/6)*sqrt(2)*(-1)^(1/3)*arctan(1/12*6^(1/6)*(6^(2/3)*sqrt(2)*(-1)^(2/3)*cosh(x) + 6^(2/3)*sqr
t(2)*(-1)^(2/3)*sinh(x) + 2*6^(1/3)*sqrt(2))) - 1/72*6^(2/3)*(-1)^(1/3)*log(-2*(2*6^(2/3)*(-1)^(1/3)*cosh(x) -
3*cosh(x)^2 - 3*sinh(x)^2 - 4*6^(1/3)*(-1)^(2/3) - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1/36*6^(
2/3)*(-1)^(1/3)*log(2*(6^(2/3)*(-1)^(1/3) + 3*cosh(x))/(cosh(x) - sinh(x)))

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giac [A]  time = 0.15, size = 80, normalized size = 0.82 \[ \frac {1}{12} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{4} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {2}{3}} {\left (\left (\frac {4}{3}\right )^{\frac {1}{3}} + e^{\left (-x\right )} + e^{x}\right )}\right ) + \frac {1}{72} \cdot 36^{\frac {1}{3}} \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 2 \, \left (\frac {4}{3}\right )^{\frac {1}{3}} {\left (e^{\left (-x\right )} + e^{x}\right )} + 4 \, \left (\frac {4}{3}\right )^{\frac {2}{3}}\right ) - \frac {1}{12} \, \left (\frac {4}{3}\right )^{\frac {1}{3}} \log \left ({\left | -2 \, \left (\frac {4}{3}\right )^{\frac {1}{3}} + e^{\left (-x\right )} + e^{x} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(4-3*cosh(x)^3),x, algorithm="giac")

[Out]

1/12*sqrt(3)*(4/3)^(1/3)*arctan(1/4*sqrt(3)*(4/3)^(2/3)*((4/3)^(1/3) + e^(-x) + e^x)) + 1/72*36^(1/3)*log((e^(
-x) + e^x)^2 + 2*(4/3)^(1/3)*(e^(-x) + e^x) + 4*(4/3)^(2/3)) - 1/12*(4/3)^(1/3)*log(abs(-2*(4/3)^(1/3) + e^(-x
) + e^x))

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maple [A]  time = 0.05, size = 80, normalized size = 0.82 \[ -\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh \relax (x )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh ^{2}\relax (x )+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cosh \relax (x )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}+\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cosh \relax (x )}{2}+1\right )}{3}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(4-3*cosh(x)^3),x)

[Out]

-1/36*4^(1/3)*3^(2/3)*ln(cosh(x)-1/3*4^(1/3)*3^(2/3))+1/72*4^(1/3)*3^(2/3)*ln(cosh(x)^2+1/3*4^(1/3)*3^(2/3)*co
sh(x)+1/3*4^(2/3)*3^(1/3))+1/12*4^(1/3)*3^(1/6)*arctan(1/3*3^(1/2)*(1/2*4^(2/3)*3^(1/3)*cosh(x)+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(4-3*cosh(x)^3),x, algorithm="maxima")

[Out]

-integrate(sinh(x)/(3*cosh(x)^3 - 4), x)

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mupad [B]  time = 3.51, size = 205, normalized size = 2.09 \[ -\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )}{36}-\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{36}+\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}-\frac {6^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}-\frac {6^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-sinh(x)/(3*cosh(x)^3 - 4),x)

[Out]

(6^(2/3)*log((256*exp(2*x))/81 - (128*exp(x))/27 - (6^(2/3)*((3^(1/2)*1i)/2 + 1/2)*((256*exp(2*x))/9 - (2048*e
xp(x))/27 - (6^(2/3)*((3^(1/2)*1i)/2 + 1/2)*(256*exp(2*x) - (2048*exp(x))/3 + 256))/36 + 256/9))/36 + 256/81)*
((3^(1/2)*1i)/2 + 1/2))/36 - (6^(2/3)*log((256*exp(2*x))/81 - (128*exp(x))/27 + (6^(2/3)*((3^(1/2)*1i)/2 - 1/2
)*((256*exp(2*x))/9 - (2048*exp(x))/27 + (6^(2/3)*((3^(1/2)*1i)/2 - 1/2)*(256*exp(2*x) - (2048*exp(x))/3 + 256
))/36 + 256/9))/36 + 256/81)*((3^(1/2)*1i)/2 - 1/2))/36 - (6^(2/3)*log((256*exp(2*x))/81 - (128*exp(x))/27 + (
6^(2/3)*((256*exp(2*x))/9 - (2048*exp(x))/27 + (6^(2/3)*(256*exp(2*x) - (2048*exp(x))/3 + 256))/36 + 256/9))/3
6 + 256/81))/36

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sympy [A]  time = 1.54, size = 85, normalized size = 0.87 \[ - \frac {6^{\frac {2}{3}} \log {\left (\cosh {\relax (x )} - \frac {6^{\frac {2}{3}}}{3} \right )}}{36} + \frac {6^{\frac {2}{3}} \log {\left (36 \cosh ^{2}{\relax (x )} + 12 \cdot 6^{\frac {2}{3}} \cosh {\relax (x )} + 24 \sqrt [3]{6} \right )}}{72} + \frac {2^{\frac {2}{3}} \sqrt [6]{3} \operatorname {atan}{\left (\frac {\sqrt [3]{2} \cdot 3^{\frac {5}{6}} \cosh {\relax (x )}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(4-3*cosh(x)**3),x)

[Out]

-6**(2/3)*log(cosh(x) - 6**(2/3)/3)/36 + 6**(2/3)*log(36*cosh(x)**2 + 12*6**(2/3)*cosh(x) + 24*6**(1/3))/72 +
2**(2/3)*3**(1/6)*atan(2**(1/3)*3**(5/6)*cosh(x)/3 + sqrt(3)/3)/12

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