Optimal. Leaf size=22 \[ \frac {\tanh (a+b x)}{b \sqrt {\text {sech}^2(a+b x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4207, 197}
\begin {gather*} \frac {\tanh (a+b x)}{b \sqrt {\text {sech}^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 4207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\text {sech}^2(a+b x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\tanh (a+b x)}{b \sqrt {\text {sech}^2(a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \frac {\tanh (a+b x)}{b \sqrt {\text {sech}^2(a+b x)}} \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. \(96\) vs.
\(2(20)=40\).
time = 2.41, size = 97, normalized size = 4.41
method | result | size |
risch | \(\frac {{\mathrm e}^{2 b x +2 a}}{2 b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}}-\frac {1}{2 b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 26, normalized size = 1.18 \begin {gather*} \frac {e^{\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 10, normalized size = 0.45 \begin {gather*} \frac {\sinh \left (b x + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.13, size = 29, normalized size = 1.32 \begin {gather*} \begin {cases} \frac {\tanh {\left (a + b x \right )}}{b \sqrt {\operatorname {sech}^{2}{\left (a + b x \right )}}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {\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 [A]
time = 0.39, size = 23, normalized size = 1.05 \begin {gather*} \frac {e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 53, normalized size = 2.41 \begin {gather*} \frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-1\right )\,\sqrt {\frac {4\,{\mathrm {e}}^{2\,a+2\,b\,x}}{{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^2}}}{4\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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