Optimal. Leaf size=42 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {1+\tanh (x)}}-2 \sqrt {1+\tanh (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3624, 3560,
3561, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}-2 \sqrt {\tanh (x)+1}-\frac {1}{\sqrt {\tanh (x)+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 3560
Rule 3561
Rule 3624
Rubi steps
\begin {align*} \int \frac {\tanh ^2(x)}{\sqrt {1+\tanh (x)}} \, dx &=-2 \sqrt {1+\tanh (x)}+\int \frac {1}{\sqrt {1+\tanh (x)}} \, dx\\ &=-\frac {1}{\sqrt {1+\tanh (x)}}-2 \sqrt {1+\tanh (x)}+\frac {1}{2} \int \sqrt {1+\tanh (x)} \, dx\\ &=-\frac {1}{\sqrt {1+\tanh (x)}}-2 \sqrt {1+\tanh (x)}+\text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (x)}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {1+\tanh (x)}}-2 \sqrt {1+\tanh (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 37, normalized size = 0.88 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {-3-2 \tanh (x)}{\sqrt {1+\tanh (x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.74, size = 35, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {1}{\sqrt {1+\tanh \left (x \right )}}-2 \sqrt {1+\tanh \left (x \right )}\) | \(35\) |
default | \(\frac {\arctanh \left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {1}{\sqrt {1+\tanh \left (x \right )}}-2 \sqrt {1+\tanh \left (x \right )}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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.
\(2 (34) = 68\).
time = 0.33, size = 182, normalized size = 4.33 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} + 10 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + 5 \, \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right ) + \sqrt {2} \cosh \left (x\right )\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right )}{4 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.14, size = 75, normalized size = 1.79 \begin {gather*} - 2 \sqrt {\tanh {\left (x \right )} + 1} - \begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2} \sqrt {\tanh {\left (x \right )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\left (x \right )} > 1 \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2} \sqrt {\tanh {\left (x \right )} + 1}}{2} \right )}}{2} & \text {for}\: \tanh {\left (x \right )} < 1 \end {cases} - \frac {1}{\sqrt {\tanh {\left (x \right )} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 54, normalized size = 1.29 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \log \left (-4 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 4 \, e^{\left (2 \, x\right )} + 2\right ) - \frac {5 \, \sqrt {2} e^{\left (2 \, x\right )} + \sqrt {2}}{2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.13, size = 36, normalized size = 0.86 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )}{2}-\frac {3}{\sqrt {\mathrm {tanh}\left (x\right )+1}}-\frac {2\,\mathrm {tanh}\left (x\right )}{\sqrt {\mathrm {tanh}\left (x\right )+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________