Optimal. Leaf size=15 \[ -\frac {\text {sech}^2(a+b x)}{2 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2686, 30}
\begin {gather*} -\frac {\text {sech}^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rubi steps
\begin {align*} \int \text {sech}^2(a+b x) \tanh (a+b x) \, dx &=-\frac {\text {Subst}(\int x \, dx,x,\text {sech}(a+b x))}{b}\\ &=-\frac {\text {sech}^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {\text {sech}^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 14, normalized size = 0.93
method | result | size |
derivativedivides | \(-\frac {\mathrm {sech}\left (b x +a \right )^{2}}{2 b}\) | \(14\) |
default | \(-\frac {\mathrm {sech}\left (b x +a \right )^{2}}{2 b}\) | \(14\) |
risch | \(-\frac {2 \,{\mathrm e}^{2 b x +2 a}}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 13, normalized size = 0.87 \begin {gather*} \frac {\tanh \left (b x + a\right )^{2}}{2 \, b} \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. 84 vs.
\(2 (13) = 26\).
time = 0.35, size = 84, normalized size = 5.60 \begin {gather*} -\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) + {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 22, normalized size = 1.47 \begin {gather*} \begin {cases} - \frac {\operatorname {sech}^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \tanh {\left (a \right )} \operatorname {sech}^{2}{\left (a \right )} & \text {otherwise} \end {cases} \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. 27 vs.
\(2 (13) = 26\).
time = 0.40, size = 27, normalized size = 1.80 \begin {gather*} -\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 13, normalized size = 0.87 \begin {gather*} -\frac {1}{2\,b\,{\mathrm {cosh}\left (a+b\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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