Integrand size = 12, antiderivative size = 34 \[ \int e^x \coth (2 x) \text {csch}(2 x) \, dx=\frac {e^{3 x}}{1-e^{4 x}}+\frac {\arctan \left (e^x\right )}{2}-\frac {\text {arctanh}\left (e^x\right )}{2} \] Output:
exp(3*x)/(1-exp(4*x))+1/2*arctan(exp(x))-1/2*arctanh(exp(x))
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int e^x \coth (2 x) \text {csch}(2 x) \, dx=\frac {1}{2} \left (-\frac {2 e^{3 x}}{-1+e^{4 x}}+\arctan \left (e^x\right )-\text {arctanh}\left (e^x\right )\right ) \] Input:
Integrate[E^x*Coth[2*x]*Csch[2*x],x]
Output:
((-2*E^(3*x))/(-1 + E^(4*x)) + ArcTan[E^x] - ArcTanh[E^x])/2
Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2720, 27, 957, 827, 216, 219}
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.
\(\displaystyle \int e^x \coth (2 x) \text {csch}(2 x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \frac {2 e^{2 x} \left (e^{4 x}+1\right )}{\left (1-e^{4 x}\right )^2}de^x\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {e^{2 x} \left (1+e^{4 x}\right )}{\left (1-e^{4 x}\right )^2}de^x\) |
\(\Big \downarrow \) 957 |
\(\displaystyle 2 \left (\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )}-\frac {1}{2} \int \frac {e^{2 x}}{1-e^{4 x}}de^x\right )\) |
\(\Big \downarrow \) 827 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{1+e^{2 x}}de^x-\frac {1}{2} \int \frac {1}{1-e^{2 x}}de^x\right )+\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\arctan \left (e^x\right )}{2}-\frac {1}{2} \int \frac {1}{1-e^{2 x}}de^x\right )+\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {1}{2} \left (\frac {\arctan \left (e^x\right )}{2}-\frac {\text {arctanh}\left (e^x\right )}{2}\right )+\frac {e^{3 x}}{2 \left (1-e^{4 x}\right )}\right )\) |
Input:
Int[E^x*Coth[2*x]*Csch[2*x],x]
Output:
2*(E^(3*x)/(2*(1 - E^(4*x))) + (ArcTan[E^x]/2 - ArcTanh[E^x]/2)/2)
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
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 complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41
method | result | size |
risch | \(-\frac {{\mathrm e}^{3 x}}{{\mathrm e}^{4 x}-1}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{4}-\frac {\ln \left (1+{\mathrm e}^{x}\right )}{4}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}\) | \(48\) |
Input:
int(exp(x)*coth(2*x)*csch(2*x),x,method=_RETURNVERBOSE)
Output:
-exp(x)^3/(exp(x)^4-1)+1/4*I*ln(exp(x)+I)-1/4*I*ln(exp(x)-I)-1/4*ln(1+exp( x))+1/4*ln(exp(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 5.94 \[ \int e^x \coth (2 x) \text {csch}(2 x) \, dx=-\frac {4 \, \cosh \left (x\right )^{3} + 12 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \, \sinh \left (x\right )^{3} - 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )}} \] Input:
integrate(exp(x)*coth(2*x)*csch(2*x),x, algorithm="fricas")
Output:
-1/4*(4*cosh(x)^3 + 12*cosh(x)^2*sinh(x) + 12*cosh(x)*sinh(x)^2 + 4*sinh(x )^3 - 2*(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh( x)*sinh(x)^3 + sinh(x)^4 - 1)*arctan(cosh(x) + sinh(x)) + (cosh(x)^4 + 4*c osh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 - 1)*log(cosh(x) + sinh(x) + 1) - (cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*co sh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 - 1)*log(cosh(x) + sin h(x) - 1))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*co sh(x)*sinh(x)^3 + sinh(x)^4 - 1)
\[ \int e^x \coth (2 x) \text {csch}(2 x) \, dx=\int e^{x} \coth {\left (2 x \right )} \operatorname {csch}{\left (2 x \right )}\, dx \] Input:
integrate(exp(x)*coth(2*x)*csch(2*x),x)
Output:
Integral(exp(x)*coth(2*x)*csch(2*x), x)
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int e^x \coth (2 x) \text {csch}(2 x) \, dx=-\frac {e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - 1} + \frac {1}{2} \, \arctan \left (e^{x}\right ) - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left (e^{x} - 1\right ) \] Input:
integrate(exp(x)*coth(2*x)*csch(2*x),x, algorithm="maxima")
Output:
-e^(3*x)/(e^(4*x) - 1) + 1/2*arctan(e^x) - 1/4*log(e^x + 1) + 1/4*log(e^x - 1)
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int e^x \coth (2 x) \text {csch}(2 x) \, dx=-\frac {e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - 1} + \frac {1}{2} \, \arctan \left (e^{x}\right ) - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(exp(x)*coth(2*x)*csch(2*x),x, algorithm="giac")
Output:
-e^(3*x)/(e^(4*x) - 1) + 1/2*arctan(e^x) - 1/4*log(e^x + 1) + 1/4*log(abs( e^x - 1))
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int e^x \coth (2 x) \text {csch}(2 x) \, dx=\frac {\ln \left ({\mathrm {e}}^x-1\right )}{4}-\frac {\mathrm {atan}\left ({\mathrm {e}}^{-x}\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{4}-\frac {{\mathrm {e}}^{3\,x}}{{\mathrm {e}}^{4\,x}-1} \] Input:
int((coth(2*x)*exp(x))/sinh(2*x),x)
Output:
log(exp(x) - 1)/4 - atan(exp(-x))/2 - log(- exp(x) - 1)/4 - exp(3*x)/(exp( 4*x) - 1)
Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24 \[ \int e^x \coth (2 x) \text {csch}(2 x) \, dx=\frac {2 e^{4 x} \mathit {atan} \left (e^{x}\right )-2 \mathit {atan} \left (e^{x}\right )+e^{4 x} \mathrm {log}\left (e^{x}-1\right )-e^{4 x} \mathrm {log}\left (e^{x}+1\right )-4 e^{3 x}-\mathrm {log}\left (e^{x}-1\right )+\mathrm {log}\left (e^{x}+1\right )}{4 e^{4 x}-4} \] Input:
int(exp(x)*coth(2*x)*csch(2*x),x)
Output:
(2*e**(4*x)*atan(e**x) - 2*atan(e**x) + e**(4*x)*log(e**x - 1) - e**(4*x)* log(e**x + 1) - 4*e**(3*x) - log(e**x - 1) + log(e**x + 1))/(4*(e**(4*x) - 1))