Optimal. Leaf size=22 \[ \frac {\text {ArcTan}\left (\sqrt {2} \tanh (2+3 x)\right )}{3 \sqrt {2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3756, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\sqrt {2} \tanh (3 x+2)\right )}{3 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3756
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(2+3 x)}{1+2 \tanh ^2(2+3 x)} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\tanh (2+3 x)\right )\\ &=\frac {\tan ^{-1}\left (\sqrt {2} \tanh (2+3 x)\right )}{3 \sqrt {2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
time = 0.06, size = 47, normalized size = 2.14 \begin {gather*} \frac {\text {ArcTan}\left (\frac {3 \left (-1+e^8\right )+\left (3+2 e^4+3 e^8\right ) \tanh (3 x)}{4 \sqrt {2} e^4}\right )}{3 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs.
\(2(16)=32\).
time = 3.69, size = 92, normalized size = 4.18
method | result | size |
risch | \(\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{4+6 x}-\frac {1}{3}+\frac {2 i \sqrt {2}}{3}\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{4+6 x}-\frac {1}{3}-\frac {2 i \sqrt {2}}{3}\right )}{12}\) | \(44\) |
derivativedivides | \(-\frac {\left (3+\sqrt {6}\right ) \sqrt {6}\, \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}+2 \sqrt {2}}\right )}{9 \left (2 \sqrt {3}+2 \sqrt {2}\right )}-\frac {\sqrt {6}\, \left (-3+\sqrt {6}\right ) \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}-2 \sqrt {2}}\right )}{9 \left (2 \sqrt {3}-2 \sqrt {2}\right )}\) | \(92\) |
default | \(-\frac {\left (3+\sqrt {6}\right ) \sqrt {6}\, \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}+2 \sqrt {2}}\right )}{9 \left (2 \sqrt {3}+2 \sqrt {2}\right )}-\frac {\sqrt {6}\, \left (-3+\sqrt {6}\right ) \arctan \left (\frac {2 \tanh \left (1+\frac {3 x}{2}\right )}{2 \sqrt {3}-2 \sqrt {2}}\right )}{9 \left (2 \sqrt {3}-2 \sqrt {2}\right )}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 21, normalized size = 0.95 \begin {gather*} -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, e^{\left (-6 \, x - 4\right )} - 1\right )}\right ) \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. 47 vs.
\(2 (16) = 32\).
time = 0.41, size = 47, normalized size = 2.14 \begin {gather*} -\frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} \cosh \left (3 \, x + 2\right ) + 2 \, \sqrt {2} \sinh \left (3 \, x + 2\right )}{2 \, {\left (\cosh \left (3 \, x + 2\right ) - \sinh \left (3 \, x + 2\right )\right )}}\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 (3 x + 2 \right )}}{2 \tanh ^{2}{\left (3 x + 2 \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, e^{\left (6 \, x + 4\right )} - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.49, size = 21, normalized size = 0.95 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (3\,{\mathrm {e}}^{6\,x+4}-1\right )}{4}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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