Integrand size = 10, antiderivative size = 32 \[ \int e^x \text {csch}^2(2 x) \, dx=\frac {e^x}{1-e^{4 x}}-\frac {\arctan \left (e^x\right )}{2}-\frac {\text {arctanh}\left (e^x\right )}{2} \] Output:
exp(x)/(1-exp(4*x))-1/2*arctan(exp(x))-1/2*arctanh(exp(x))
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int e^x \text {csch}^2(2 x) \, dx=\frac {e^x}{1-e^{4 x}}-\frac {\arctan \left (e^x\right )}{2}-\frac {\text {arctanh}\left (e^x\right )}{2} \] Input:
Integrate[E^x*Csch[2*x]^2,x]
Output:
E^x/(1 - E^(4*x)) - ArcTan[E^x]/2 - ArcTanh[E^x]/2
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2720, 27, 817, 756, 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.
\(\displaystyle \int e^x \text {csch}^2(2 x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \frac {4 e^{4 x}}{\left (1-e^{4 x}\right )^2}de^x\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int \frac {e^{4 x}}{\left (1-e^{4 x}\right )^2}de^x\) |
\(\Big \downarrow \) 817 |
\(\displaystyle 4 \left (\frac {e^x}{4 \left (1-e^{4 x}\right )}-\frac {1}{4} \int \frac {1}{1-e^{4 x}}de^x\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle 4 \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {1}{1-e^{2 x}}de^x-\frac {1}{2} \int \frac {1}{1+e^{2 x}}de^x\right )+\frac {e^x}{4 \left (1-e^{4 x}\right )}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 4 \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {1}{1-e^{2 x}}de^x-\frac {1}{2} \arctan \left (e^x\right )\right )+\frac {e^x}{4 \left (1-e^{4 x}\right )}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 4 \left (\frac {1}{4} \left (-\frac {1}{2} \arctan \left (e^x\right )-\frac {\text {arctanh}\left (e^x\right )}{2}\right )+\frac {e^x}{4 \left (1-e^{4 x}\right )}\right )\) |
Input:
Int[E^x*Csch[2*x]^2,x]
Output:
4*(E^x/(4*(1 - E^(4*x))) + (-1/2*ArcTan[E^x] - ArcTanh[E^x]/2)/4)
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[((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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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]]]
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {1}{4 \cosh \left (x \right )}-\frac {\operatorname {arctanh}\left ({\mathrm e}^{x}\right )}{2}-\frac {1}{4 \sinh \left (x \right )}-\frac {\arctan \left ({\mathrm e}^{x}\right )}{2}\) | \(24\) |
risch | \(-\frac {{\mathrm e}^{x}}{{\mathrm e}^{4 x}-1}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{4}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{4}\) | \(46\) |
Input:
int(exp(x)*csch(2*x)^2,x,method=_RETURNVERBOSE)
Output:
1/4/cosh(x)-1/2*arctanh(exp(x))-1/4/sinh(x)-1/2*arctan(exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 5.69 \[ \int e^x \text {csch}^2(2 x) \, dx=-\frac {2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 4 \, \cosh \left (x\right ) + 4 \, \sinh \left (x\right )}{4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )}} \] Input:
integrate(exp(x)*csch(2*x)^2,x, algorithm="fricas")
Output:
-1/4*(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) + 4*cosh(x) + 4*sinh(x))/(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)
\[ \int e^x \text {csch}^2(2 x) \, dx=\int e^{x} \operatorname {csch}^{2}{\left (2 x \right )}\, dx \] Input:
integrate(exp(x)*csch(2*x)**2,x)
Output:
Integral(exp(x)*csch(2*x)**2, x)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int e^x \text {csch}^2(2 x) \, dx=-\frac {e^{x}}{e^{\left (4 \, x\right )} - 1} - \frac {1}{2} \, \arctan \left (e^{x}\right ) - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left (e^{x} - 1\right ) \] Input:
integrate(exp(x)*csch(2*x)^2,x, algorithm="maxima")
Output:
-e^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 = 33, normalized size of antiderivative = 1.03 \[ \int e^x \text {csch}^2(2 x) \, dx=-\frac {e^{x}}{e^{\left (4 \, x\right )} - 1} - \frac {1}{2} \, \arctan \left (e^{x}\right ) - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(exp(x)*csch(2*x)^2,x, algorithm="giac")
Output:
-e^x/(e^(4*x) - 1) - 1/2*arctan(e^x) - 1/4*log(e^x + 1) + 1/4*log(abs(e^x - 1))
Time = 1.76 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int e^x \text {csch}^2(2 x) \, dx=\frac {\ln \left (1-{\mathrm {e}}^x\right )}{4}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{4}-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{2}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}-1} \] Input:
int(exp(x)/sinh(2*x)^2,x)
Output:
log(1 - exp(x))/4 - log(- exp(x) - 1)/4 - atan(exp(x))/2 - exp(x)/(exp(4*x ) - 1)
Time = 0.15 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31 \[ \int e^x \text {csch}^2(2 x) \, dx=\frac {-2 e^{4 x} \mathit {atan} \left (e^{x}\right )+2 \mathit {atan} \left (e^{x}\right )+e^{4 x} \mathrm {log}\left (e^{x}-1\right )-e^{4 x} \mathrm {log}\left (e^{x}+1\right )-4 e^{x}-\mathrm {log}\left (e^{x}-1\right )+\mathrm {log}\left (e^{x}+1\right )}{4 e^{4 x}-4} \] Input:
int(exp(x)*csch(2*x)^2,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**x - log(e**x - 1) + log(e**x + 1))/(4*(e**(4*x) - 1))