Optimal. Leaf size=32 \[ \frac {e^x}{1-e^{4 x}}-\frac {\text {ArcTan}\left (e^x\right )}{2}-\frac {1}{2} \tanh ^{-1}\left (e^x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2320, 12, 294,
218, 212, 209} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (e^x\right )+\frac {e^x}{1-e^{4 x}}-\frac {1}{2} \tanh ^{-1}\left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 212
Rule 218
Rule 294
Rule 2320
Rubi steps
\begin {align*} \int e^x \text {csch}^2(2 x) \, dx &=\text {Subst}\left (\int \frac {4 x^4}{\left (1-x^4\right )^2} \, dx,x,e^x\right )\\ &=4 \text {Subst}\left (\int \frac {x^4}{\left (1-x^4\right )^2} \, dx,x,e^x\right )\\ &=\frac {e^x}{1-e^{4 x}}-\text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,e^x\right )\\ &=\frac {e^x}{1-e^{4 x}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=\frac {e^x}{1-e^{4 x}}-\frac {1}{2} \tan ^{-1}\left (e^x\right )-\frac {1}{2} \tanh ^{-1}\left (e^x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 32, normalized size = 1.00 \begin {gather*} \frac {e^x}{1-e^{4 x}}-\frac {\text {ArcTan}\left (e^x\right )}{2}-\frac {1}{2} \tanh ^{-1}\left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.65, size = 46, normalized size = 1.44
method | result | size |
risch | \(-\frac {{\mathrm e}^{x}}{{\mathrm e}^{4 x}-1}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{4}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{4}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 32, normalized size = 1.00 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs.
\(2 (23) = 46\).
time = 0.51, size = 182, normalized size = 5.69 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \operatorname {csch}^{2}{\left (2 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 33, normalized size = 1.03 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.72, size = 36, normalized size = 1.12 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________