Optimal. Leaf size=20 \[ \frac {2 \tanh ^{-1}\left (\frac {3+2 \tanh (x)}{\sqrt {17}}\right )}{\sqrt {17}} \]
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Rubi [A]
time = 0.09, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {632, 212}
\begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {2 \tanh (x)+3}{\sqrt {17}}\right )}{\sqrt {17}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x)}{1+\text {sech}^2(x)-3 \tanh (x)} \, dx &=\text {Subst}\left (\int \frac {1}{2-3 x-x^2} \, dx,x,\tanh (x)\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{17-x^2} \, dx,x,-3-2 \tanh (x)\right )\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {3+2 \tanh (x)}{\sqrt {17}}\right )}{\sqrt {17}}\\ \end {align*}
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Mathematica [F]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {sech}^2(x)}{1+\text {sech}^2(x)-3 \tanh (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs.
\(2(17)=34\).
time = 1.48, size = 63, normalized size = 3.15
method | result | size |
risch | \(\frac {\sqrt {17}\, \ln \left ({\mathrm e}^{2 x}+\frac {\sqrt {17}}{2}-\frac {3}{2}\right )}{17}-\frac {\sqrt {17}\, \ln \left ({\mathrm e}^{2 x}-\frac {\sqrt {17}}{2}-\frac {3}{2}\right )}{17}\) | \(36\) |
default | \(\frac {\sqrt {17}\, \ln \left (\sqrt {17}\, \tanh \left (\frac {x}{2}\right )+2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-3 \tanh \left (\frac {x}{2}\right )+2\right )}{17}-\frac {\sqrt {17}\, \ln \left (-\sqrt {17}\, \tanh \left (\frac {x}{2}\right )+2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-3 \tanh \left (\frac {x}{2}\right )+2\right )}{17}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs.
\(2 (17) = 34\).
time = 0.35, size = 67, normalized size = 3.35 \begin {gather*} \frac {1}{17} \, \sqrt {17} \log \left (\frac {3 \, {\left (\sqrt {17} - 5\right )} \cosh \left (x\right )^{2} - 2 \, {\left (3 \, \sqrt {17} - 11\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (\sqrt {17} - 5\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {17} + 6}{\cosh \left (x\right )^{2} - 6 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{2}{\left (x \right )}}{- 3 \tanh {\left (x \right )} + \operatorname {sech}^{2}{\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (17) = 34\).
time = 0.41, size = 35, normalized size = 1.75 \begin {gather*} -\frac {1}{17} \, \sqrt {17} \log \left (\frac {{\left | -\sqrt {17} + 2 \, e^{\left (2 \, x\right )} - 3 \right |}}{{\left | \sqrt {17} + 2 \, e^{\left (2 \, x\right )} - 3 \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.79, size = 50, normalized size = 2.50 \begin {gather*} -\frac {\sqrt {17}\,\left (\ln \left (2\,{\mathrm {e}}^{2\,x}-\frac {\sqrt {17}\,\left (6\,{\mathrm {e}}^{2\,x}+8\right )}{17}\right )-\ln \left (2\,{\mathrm {e}}^{2\,x}+\frac {\sqrt {17}\,\left (6\,{\mathrm {e}}^{2\,x}+8\right )}{17}\right )\right )}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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