Integrand size = 13, antiderivative size = 84 \[ \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx=\frac {\arctan \left (\frac {1+\sqrt [3]{6} \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}-\frac {\log \left (6^{2/3}-3 \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\log \left (2 \sqrt [3]{6}+6^{2/3} \cosh (x)+3 \cosh ^2(x)\right )}{12 \sqrt [3]{6}} \] Output:
1/12*arctan(1/3*(1+6^(1/3)*cosh(x))*3^(1/2))*2^(2/3)*3^(1/6)-1/36*ln(6^(2/ 3)-3*cosh(x))*6^(2/3)+1/72*ln(2*6^(1/3)+6^(2/3)*cosh(x)+3*cosh(x)^2)*6^(2/ 3)
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx=\frac {1}{72} \left (6\ 2^{2/3} \sqrt [6]{3} \arctan \left (\frac {1+\sqrt [3]{6} \cosh (x)}{\sqrt {3}}\right )+6^{2/3} \left (-2 \log \left (2-\sqrt [3]{6} \cosh (x)\right )+\log \left (4+2 \sqrt [3]{6} \cosh (x)+6^{2/3} \cosh ^2(x)\right )\right )\right ) \] Input:
Integrate[Sinh[x]/(4 - 3*Cosh[x]^3),x]
Output:
(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]))/7 2
Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 26, 3702, 750, 16, 1142, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \cos \left (\frac {\pi }{2}+i x\right )}{4-3 \sin \left (\frac {\pi }{2}+i x\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos \left (i x+\frac {\pi }{2}\right )}{4-3 \sin \left (i x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \int \frac {1}{4-3 \cosh ^3(x)}d\cosh (x)\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{3} \cosh (x)+2\ 2^{2/3}}{3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}}d\cosh (x)}{6 \sqrt [3]{2}}+\frac {\int \frac {1}{2^{2/3}-\sqrt [3]{3} \cosh (x)}d\cosh (x)}{6 \sqrt [3]{2}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\int \frac {\sqrt [3]{3} \cosh (x)+2\ 2^{2/3}}{3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}}d\cosh (x)}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}}d\cosh (x)}{\sqrt [3]{2}}+\frac {\int \frac {2^{2/3} \sqrt [3]{3} \left (\sqrt [3]{6} \cosh (x)+1\right )}{3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}}d\cosh (x)}{2 \sqrt [3]{3}}}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}}d\cosh (x)}{\sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{6} \cosh (x)+1}{3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}}d\cosh (x)}{\sqrt [3]{2}}}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{6} \cosh (x)+1}{3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}}d\cosh (x)}{\sqrt [3]{2}}-3^{2/3} \int \frac {1}{-\left (\sqrt [3]{6} \cosh (x)+1\right )^2-3}d\left (\sqrt [3]{6} \cosh (x)+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{6} \cosh (x)+1}{3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}}d\cosh (x)}{\sqrt [3]{2}}+\sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{6} \cosh (x)+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{6} \cosh (x)+1}{\sqrt {3}}\right )+\frac {\log \left (3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}\right )}{2 \sqrt [3]{3}}}{6 \sqrt [3]{2}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}\) |
Input:
Int[Sinh[x]/(4 - 3*Cosh[x]^3),x]
Output:
-1/6*Log[2^(2/3) - 3^(1/3)*Cosh[x]]/6^(1/3) + (3^(1/6)*ArcTan[(1 + 6^(1/3) *Cosh[x])/Sqrt[3]] + Log[2*2^(1/3) + 2^(2/3)*3^(1/3)*Cosh[x] + 3^(2/3)*Cos h[x]^2]/(2*3^(1/3)))/(6*2^(1/3))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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]}, Si mp[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])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.31
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1296 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (24 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+1\right )\) | \(26\) |
| derivativedivides | \(-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh \left (x \right )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh \left (x \right )^{2}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cosh \left (x \right )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}+\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cosh \left (x \right )}{2}+1\right )}{3}\right )}{12}\) | \(80\) |
| default | \(-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh \left (x \right )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh \left (x \right )^{2}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cosh \left (x \right )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}+\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cosh \left (x \right )}{2}+1\right )}{3}\right )}{12}\) | \(80\) |
Input:
int(sinh(x)/(4-3*cosh(x)^3),x,method=_RETURNVERBOSE)
Output:
sum(_R*ln(24*_R*exp(x)+exp(2*x)+1),_R=RootOf(1296*_Z^3+1))
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (61) = 122\).
Time = 0.08 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.50 \[ \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx=-\frac {1}{6} \cdot 6^{\frac {1}{6}} \sqrt {\frac {1}{2}} \arctan \left (\frac {1}{6} \cdot 6^{\frac {1}{6}} \sqrt {\frac {1}{2}} {\left (6^{\frac {2}{3}} \cosh \left (x\right )^{3} + 6^{\frac {2}{3}} \sinh \left (x\right )^{3} + {\left (3 \cdot 6^{\frac {2}{3}} \cosh \left (x\right ) + 4 \cdot 6^{\frac {1}{3}}\right )} \sinh \left (x\right )^{2} + 4 \cdot 6^{\frac {1}{3}} \cosh \left (x\right )^{2} + {\left (6^{\frac {2}{3}} + 16\right )} \cosh \left (x\right ) + {\left (3 \cdot 6^{\frac {2}{3}} \cosh \left (x\right )^{2} + 8 \cdot 6^{\frac {1}{3}} \cosh \left (x\right ) + 6^{\frac {2}{3}} + 16\right )} \sinh \left (x\right ) + 2 \cdot 6^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \cdot 6^{\frac {1}{6}} \sqrt {\frac {1}{2}} \arctan \left (\frac {1}{6} \cdot 6^{\frac {1}{6}} \sqrt {\frac {1}{2}} {\left (6^{\frac {2}{3}} \cosh \left (x\right ) + 6^{\frac {2}{3}} \sinh \left (x\right ) + 2 \cdot 6^{\frac {1}{3}}\right )}\right ) + \frac {1}{72} \cdot 6^{\frac {2}{3}} \log \left (\frac {2 \, {\left (3 \, \cosh \left (x\right )^{2} + 3 \, \sinh \left (x\right )^{2} + 2 \cdot 6^{\frac {2}{3}} \cosh \left (x\right ) + 4 \cdot 6^{\frac {1}{3}} + 3\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - \frac {1}{36} \cdot 6^{\frac {2}{3}} \log \left (-\frac {2 \, {\left (6^{\frac {2}{3}} - 3 \, \cosh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \] Input:
integrate(sinh(x)/(4-3*cosh(x)^3),x, algorithm="fricas")
Output:
-1/6*6^(1/6)*sqrt(1/2)*arctan(1/6*6^(1/6)*sqrt(1/2)*(6^(2/3)*cosh(x)^3 + 6 ^(2/3)*sinh(x)^3 + (3*6^(2/3)*cosh(x) + 4*6^(1/3))*sinh(x)^2 + 4*6^(1/3)*c osh(x)^2 + (6^(2/3) + 16)*cosh(x) + (3*6^(2/3)*cosh(x)^2 + 8*6^(1/3)*cosh( x) + 6^(2/3) + 16)*sinh(x) + 2*6^(1/3))) + 1/6*6^(1/6)*sqrt(1/2)*arctan(1/ 6*6^(1/6)*sqrt(1/2)*(6^(2/3)*cosh(x) + 6^(2/3)*sinh(x) + 2*6^(1/3))) + 1/7 2*6^(2/3)*log(2*(3*cosh(x)^2 + 3*sinh(x)^2 + 2*6^(2/3)*cosh(x) + 4*6^(1/3) + 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 1/36*6^(2/3)*log(-2*( 6^(2/3) - 3*cosh(x))/(cosh(x) - sinh(x)))
Time = 0.46 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01 \[ \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx=- \frac {6^{\frac {2}{3}} \log {\left (\cosh {\left (x \right )} - \frac {6^{\frac {2}{3}}}{3} \right )}}{36} + \frac {6^{\frac {2}{3}} \log {\left (36 \cosh ^{2}{\left (x \right )} + 12 \cdot 6^{\frac {2}{3}} \cosh {\left (x \right )} + 24 \cdot \sqrt [3]{6} \right )}}{72} + \frac {2^{\frac {2}{3}} \cdot \sqrt [6]{3} \operatorname {atan}{\left (\frac {\sqrt [3]{2} \cdot 3^{\frac {5}{6}} \cosh {\left (x \right )}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \] Input:
integrate(sinh(x)/(4-3*cosh(x)**3),x)
Output:
-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
\[ \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx=\int { -\frac {\sinh \left (x\right )}{3 \, \cosh \left (x\right )^{3} - 4} \,d x } \] Input:
integrate(sinh(x)/(4-3*cosh(x)^3),x, algorithm="maxima")
Output:
-integrate(sinh(x)/(3*cosh(x)^3 - 4), x)
Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95 \[ \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx=\frac {1}{12} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{4} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {2}{3}} {\left (\left (\frac {4}{3}\right )^{\frac {1}{3}} + e^{\left (-x\right )} + e^{x}\right )}\right ) + \frac {1}{72} \cdot 36^{\frac {1}{3}} \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 2 \, \left (\frac {4}{3}\right )^{\frac {1}{3}} {\left (e^{\left (-x\right )} + e^{x}\right )} + 4 \, \left (\frac {4}{3}\right )^{\frac {2}{3}}\right ) - \frac {1}{12} \, \left (\frac {4}{3}\right )^{\frac {1}{3}} \log \left ({\left | -2 \, \left (\frac {4}{3}\right )^{\frac {1}{3}} + e^{\left (-x\right )} + e^{x} \right |}\right ) \] Input:
integrate(sinh(x)/(4-3*cosh(x)^3),x, algorithm="giac")
Output:
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))
Time = 4.94 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.44 \[ \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx=-\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )}{36}-\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{36}+\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}-\frac {6^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}-\frac {6^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{36} \] Input:
int(-sinh(x)/(3*cosh(x)^3 - 4),x)
Output:
(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))/2 7 + (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))/3 6 + 256/9))/36 + 256/81)*((3^(1/2)*1i)/2 - 1/2))/36 - (6^(2/3)*log((256*ex p(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))/36 + 25 6/81))/36
Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx=\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \mathit {atan} \left (\frac {\left (2 \,3^{\frac {1}{3}} \cosh \left (x \right )+2^{\frac {2}{3}}\right ) 2^{\frac {1}{3}} \sqrt {3}}{6}\right )+\mathrm {log}\left (3^{\frac {2}{3}} \cosh \left (x \right )^{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} \cosh \left (x \right )+2 \,2^{\frac {1}{3}}\right )-2 \,\mathrm {log}\left (3^{\frac {1}{3}} \cosh \left (x \right )-2^{\frac {2}{3}}\right )\right ) 3^{\frac {2}{3}}}{72} \] Input:
int(sinh(x)/(4-3*cosh(x)^3),x)
Output:
(2**(2/3)*(2*3**(1/3)*3**(1/6)*atan((2*3**(1/3)*cosh(x) + 2**(2/3))/(2**(2 /3)*3**(1/3)*3**(1/6))) + log(3**(2/3)*cosh(x)**2 + 2**(2/3)*3**(1/3)*cosh (x) + 2*2**(1/3)) - 2*log(3**(1/3)*cosh(x) - 2**(2/3))))/(24*3**(1/3))