Integrand size = 14, antiderivative size = 107 \[ \int \frac {e^x}{a-\tanh (2 x)} \, dx=-\frac {e^x}{1-a}+\frac {\arctan \left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{(1-a) \sqrt {1+a} \sqrt [4]{1-a^2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{(1-a) \sqrt {1+a} \sqrt [4]{1-a^2}} \] Output:
-exp(x)/(1-a)+arctan((1-a)^(1/4)*exp(x)/(1+a)^(1/4))/(1-a)/(1+a)^(1/2)/(-a ^2+1)^(1/4)+arctanh((1-a)^(1/4)*exp(x)/(1+a)^(1/4))/(1-a)/(1+a)^(1/2)/(-a^ 2+1)^(1/4)
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {e^x}{a-\tanh (2 x)} \, dx=\frac {-\sqrt [4]{1-a} (1+a)^{3/4} e^x+\arctan \left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{(1-a)^{5/4} (1+a)^{3/4}} \] Input:
Integrate[E^x/(a - Tanh[2*x]),x]
Output:
(-((1 - a)^(1/4)*(1 + a)^(3/4)*E^x) + ArcTan[((1 - a)^(1/4)*E^x)/(1 + a)^( 1/4)] + ArcTanh[((1 - a)^(1/4)*E^x)/(1 + a)^(1/4)])/((1 - a)^(5/4)*(1 + a) ^(3/4))
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2720, 913, 756, 218, 221}
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.
\(\displaystyle \int \frac {e^x}{a-\tanh (2 x)} \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \frac {e^{4 x}+1}{(1-a) \left (-e^{4 x}\right )+a+1}de^x\) |
\(\Big \downarrow \) 913 |
\(\displaystyle \frac {2 \int \frac {1}{-e^{4 x} (1-a)+a+1}de^x}{1-a}-\frac {e^x}{1-a}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {1}{\sqrt {a+1}-\sqrt {1-a} e^{2 x}}de^x}{2 \sqrt {a+1}}+\frac {\int \frac {1}{e^{2 x} \sqrt {1-a}+\sqrt {a+1}}de^x}{2 \sqrt {a+1}}\right )}{1-a}-\frac {e^x}{1-a}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {1}{\sqrt {a+1}-\sqrt {1-a} e^{2 x}}de^x}{2 \sqrt {a+1}}+\frac {\arctan \left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}\right )}{1-a}-\frac {e^x}{1-a}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \left (\frac {\arctan \left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{2 \sqrt {a+1} \sqrt [4]{1-a^2}}\right )}{1-a}-\frac {e^x}{1-a}\) |
Input:
Int[E^x/(a - Tanh[2*x]),x]
Output:
-(E^x/(1 - a)) + (2*(ArcTan[((1 - a)^(1/4)*E^x)/(1 + a)^(1/4)]/(2*Sqrt[1 + a]*(1 - a^2)^(1/4)) + ArcTanh[((1 - a)^(1/4)*E^x)/(1 + a)^(1/4)]/(2*Sqrt[ 1 + a]*(1 - a^2)^(1/4))))/(1 - a)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.94 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{a -1}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (16 a^{8}-32 a^{7}-32 a^{6}+96 a^{5}-96 a^{3}+32 a^{2}+32 a -16\right ) \textit {\_Z}^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}+\left (-2 a^{2}+2\right ) \textit {\_R} \right )\right )\) | \(70\) |
Input:
int(exp(x)/(a-tanh(2*x)),x,method=_RETURNVERBOSE)
Output:
exp(x)/(a-1)+sum(_R*ln(exp(x)+(-2*a^2+2)*_R),_R=RootOf(1+(16*a^8-32*a^7-32 *a^6+96*a^5-96*a^3+32*a^2+32*a-16)*_Z^4))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.79 \[ \int \frac {e^x}{a-\tanh (2 x)} \, dx=-\frac {{\left (a - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \log \left ({\left (a^{2} - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\left (a - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (a^{2} - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\left (i \, a - i\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (i \, a^{2} - i\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\left (-i \, a + i\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (-i \, a^{2} + i\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} + \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, \cosh \left (x\right ) - 2 \, \sinh \left (x\right )}{2 \, {\left (a - 1\right )}} \] Input:
integrate(exp(x)/(a-tanh(2*x)),x, algorithm="fricas")
Output:
-1/2*((a - 1)*(-1/(a^8 - 2*a^7 - 2*a^6 + 6*a^5 - 6*a^3 + 2*a^2 + 2*a - 1)) ^(1/4)*log((a^2 - 1)*(-1/(a^8 - 2*a^7 - 2*a^6 + 6*a^5 - 6*a^3 + 2*a^2 + 2* a - 1))^(1/4) + cosh(x) + sinh(x)) - (a - 1)*(-1/(a^8 - 2*a^7 - 2*a^6 + 6* a^5 - 6*a^3 + 2*a^2 + 2*a - 1))^(1/4)*log(-(a^2 - 1)*(-1/(a^8 - 2*a^7 - 2* a^6 + 6*a^5 - 6*a^3 + 2*a^2 + 2*a - 1))^(1/4) + cosh(x) + sinh(x)) - (I*a - I)*(-1/(a^8 - 2*a^7 - 2*a^6 + 6*a^5 - 6*a^3 + 2*a^2 + 2*a - 1))^(1/4)*lo g(-(I*a^2 - I)*(-1/(a^8 - 2*a^7 - 2*a^6 + 6*a^5 - 6*a^3 + 2*a^2 + 2*a - 1) )^(1/4) + cosh(x) + sinh(x)) - (-I*a + I)*(-1/(a^8 - 2*a^7 - 2*a^6 + 6*a^5 - 6*a^3 + 2*a^2 + 2*a - 1))^(1/4)*log(-(-I*a^2 + I)*(-1/(a^8 - 2*a^7 - 2* a^6 + 6*a^5 - 6*a^3 + 2*a^2 + 2*a - 1))^(1/4) + cosh(x) + sinh(x)) - 2*cos h(x) - 2*sinh(x))/(a - 1)
\[ \int \frac {e^x}{a-\tanh (2 x)} \, dx=\int \frac {e^{x}}{a - \tanh {\left (2 x \right )}}\, dx \] Input:
integrate(exp(x)/(a-tanh(2*x)),x)
Output:
Integral(exp(x)/(a - tanh(2*x)), x)
Exception generated. \[ \int \frac {e^x}{a-\tanh (2 x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(exp(x)/(a-tanh(2*x)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (83) = 166\).
Time = 0.11 (sec) , antiderivative size = 328, normalized size of antiderivative = 3.07 \[ \int \frac {e^x}{a-\tanh (2 x)} \, dx=-\frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + 2 \, e^{x}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} - \sqrt {2} a^{2} - \sqrt {2} a + \sqrt {2}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} - 2 \, e^{x}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} - \sqrt {2} a^{2} - \sqrt {2} a + \sqrt {2}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} e^{x} + \sqrt {\frac {a + 1}{a - 1}} + e^{\left (2 \, x\right )}\right )}{2 \, {\left (\sqrt {2} a^{3} - \sqrt {2} a^{2} - \sqrt {2} a + \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (-\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} e^{x} + \sqrt {\frac {a + 1}{a - 1}} + e^{\left (2 \, x\right )}\right )}{2 \, {\left (\sqrt {2} a^{3} - \sqrt {2} a^{2} - \sqrt {2} a + \sqrt {2}\right )}} + \frac {e^{x}}{a - 1} \] Input:
integrate(exp(x)/(a-tanh(2*x)),x, algorithm="giac")
Output:
-(a^4 - 2*a^3 + 2*a - 1)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*((a + 1)/(a - 1 ))^(1/4) + 2*e^x)/((a + 1)/(a - 1))^(1/4))/(sqrt(2)*a^3 - sqrt(2)*a^2 - sq rt(2)*a + sqrt(2)) - (a^4 - 2*a^3 + 2*a - 1)^(1/4)*arctan(-1/2*sqrt(2)*(sq rt(2)*((a + 1)/(a - 1))^(1/4) - 2*e^x)/((a + 1)/(a - 1))^(1/4))/(sqrt(2)*a ^3 - sqrt(2)*a^2 - sqrt(2)*a + sqrt(2)) - 1/2*(a^4 - 2*a^3 + 2*a - 1)^(1/4 )*log(sqrt(2)*((a + 1)/(a - 1))^(1/4)*e^x + sqrt((a + 1)/(a - 1)) + e^(2*x ))/(sqrt(2)*a^3 - sqrt(2)*a^2 - sqrt(2)*a + sqrt(2)) + 1/2*(a^4 - 2*a^3 + 2*a - 1)^(1/4)*log(-sqrt(2)*((a + 1)/(a - 1))^(1/4)*e^x + sqrt((a + 1)/(a - 1)) + e^(2*x))/(sqrt(2)*a^3 - sqrt(2)*a^2 - sqrt(2)*a + sqrt(2)) + e^x/( a - 1)
Time = 2.44 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.52 \[ \int \frac {e^x}{a-\tanh (2 x)} \, dx=\frac {\ln \left (8\,a\,{\left (-a-1\right )}^{1/4}+8\,{\mathrm {e}}^x\,{\left (a-1\right )}^{5/4}-8\,{\left (-a-1\right )}^{1/4}\right )-\ln \left (8\,{\mathrm {e}}^x\,{\left (a-1\right )}^{5/4}-8\,a\,{\left (-a-1\right )}^{1/4}+8\,{\left (-a-1\right )}^{1/4}\right )+2\,{\mathrm {e}}^x\,{\left (a-1\right )}^{1/4}\,{\left (-a-1\right )}^{3/4}-\ln \left (8\,{\mathrm {e}}^x\,{\left (a-1\right )}^{5/4}-a\,{\left (-a-1\right )}^{1/4}\,8{}\mathrm {i}+{\left (-a-1\right )}^{1/4}\,8{}\mathrm {i}\right )\,1{}\mathrm {i}+\ln \left (a\,{\left (-a-1\right )}^{1/4}\,8{}\mathrm {i}+8\,{\mathrm {e}}^x\,{\left (a-1\right )}^{5/4}-{\left (-a-1\right )}^{1/4}\,8{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,{\left (a-1\right )}^{5/4}\,{\left (-a-1\right )}^{3/4}} \] Input:
int(exp(x)/(a - tanh(2*x)),x)
Output:
(log(8*a*(- a - 1)^(1/4) + 8*exp(x)*(a - 1)^(5/4) - 8*(- a - 1)^(1/4)) - l og(8*exp(x)*(a - 1)^(5/4) - 8*a*(- a - 1)^(1/4) + 8*(- a - 1)^(1/4)) - log (8*exp(x)*(a - 1)^(5/4) - a*(- a - 1)^(1/4)*8i + (- a - 1)^(1/4)*8i)*1i + log(a*(- a - 1)^(1/4)*8i + 8*exp(x)*(a - 1)^(5/4) - (- a - 1)^(1/4)*8i)*1i + 2*exp(x)*(a - 1)^(1/4)*(- a - 1)^(3/4))/(2*(a - 1)^(5/4)*(- a - 1)^(3/4 ))
Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.15 \[ \int \frac {e^x}{a-\tanh (2 x)} \, dx=\frac {2 \left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}-2 e^{x} \sqrt {a -1}}{\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}}\right )-2 \left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}+2 e^{x} \sqrt {a -1}}{\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}}\right )+\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-e^{x} \left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}+e^{2 x} \sqrt {a -1}+\sqrt {a +1}\right )-\left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (e^{x} \left (a +1\right )^{\frac {1}{4}} \left (a -1\right )^{\frac {1}{4}} \sqrt {2}+e^{2 x} \sqrt {a -1}+\sqrt {a +1}\right )+4 e^{x} a^{2}-4 e^{x}}{4 a^{3}-4 a^{2}-4 a +4} \] Input:
int(exp(x)/(a-tanh(2*x)),x)
Output:
(2*(a + 1)**(1/4)*(a - 1)**(3/4)*sqrt(2)*atan(((a + 1)**(1/4)*(a - 1)**(1/ 4)*sqrt(2) - 2*e**x*sqrt(a - 1))/((a + 1)**(1/4)*(a - 1)**(1/4)*sqrt(2))) - 2*(a + 1)**(1/4)*(a - 1)**(3/4)*sqrt(2)*atan(((a + 1)**(1/4)*(a - 1)**(1 /4)*sqrt(2) + 2*e**x*sqrt(a - 1))/((a + 1)**(1/4)*(a - 1)**(1/4)*sqrt(2))) + (a + 1)**(1/4)*(a - 1)**(3/4)*sqrt(2)*log( - e**x*(a + 1)**(1/4)*(a - 1 )**(1/4)*sqrt(2) + e**(2*x)*sqrt(a - 1) + sqrt(a + 1)) - (a + 1)**(1/4)*(a - 1)**(3/4)*sqrt(2)*log(e**x*(a + 1)**(1/4)*(a - 1)**(1/4)*sqrt(2) + e**( 2*x)*sqrt(a - 1) + sqrt(a + 1)) + 4*e**x*a**2 - 4*e**x)/(4*(a**3 - a**2 - a + 1))