Optimal. Leaf size=42 \[ \frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 12, 294,
212} \begin {gather*} \frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 212
Rule 294
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \text {csch}^2(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {4 x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {4 \text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 38, normalized size = 0.90 \begin {gather*} \frac {2 \left (-\frac {e^{a+b x}}{-1+e^{2 (a+b x)}}-\tanh ^{-1}\left (e^{a+b x}\right )\right )}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.35, size = 25, normalized size = 0.60
method | result | size |
derivativedivides | \(\frac {-2 \arctanh \left ({\mathrm e}^{b x +a}\right )-\frac {1}{\sinh \left (b x +a \right )}}{b}\) | \(25\) |
default | \(\frac {-2 \arctanh \left ({\mathrm e}^{b x +a}\right )-\frac {1}{\sinh \left (b x +a \right )}}{b}\) | \(25\) |
risch | \(-\frac {2 \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 52, normalized size = 1.24 \begin {gather*} -\frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b} - \frac {2 \, e^{\left (b x + a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 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. 157 vs.
\(2 (37) = 74\).
time = 0.51, size = 157, normalized size = 3.74 \begin {gather*} -\frac {{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \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*} e^{a} \int e^{b x} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 48, normalized size = 1.14 \begin {gather*} -\frac {\frac {2 \, e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + \log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.62, size = 52, normalized size = 1.24 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________