Optimal. Leaf size=29 \[ \frac {x}{a}-\frac {\tanh (c+d x)}{d (a+a \text {sech}(c+d x))} \]
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Rubi [A]
time = 0.01, antiderivative size = 29, 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 = {3862, 8}
\begin {gather*} \frac {x}{a}-\frac {\tanh (c+d x)}{d (a \text {sech}(c+d x)+a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3862
Rubi steps
\begin {align*} \int \frac {1}{a+a \text {sech}(c+d x)} \, dx &=-\frac {\tanh (c+d x)}{d (a+a \text {sech}(c+d x))}+\frac {\int a \, dx}{a^2}\\ &=\frac {x}{a}-\frac {\tanh (c+d x)}{d (a+a \text {sech}(c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 58, normalized size = 2.00 \begin {gather*} \frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (d x \cosh \left (\frac {d x}{2}\right )+d x \cosh \left (c+\frac {d x}{2}\right )-2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.16, size = 46, normalized size = 1.59
method | result | size |
risch | \(\frac {x}{a}+\frac {2}{a d \left ({\mathrm e}^{d x +c}+1\right )}\) | \(25\) |
derivativedivides | \(\frac {-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}\) | \(46\) |
default | \(\frac {-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 33, normalized size = 1.14 \begin {gather*} \frac {d x + c}{a d} - \frac {2}{{\left (a e^{\left (-d x - c\right )} + a\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 48, normalized size = 1.66 \begin {gather*} \frac {d x \cosh \left (d x + c\right ) + d x \sinh \left (d x + c\right ) + d x + 2}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right ) + a d} \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*} \frac {\int \frac {1}{\operatorname {sech}{\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 29, normalized size = 1.00 \begin {gather*} \frac {\frac {d x + c}{a} + \frac {2}{a {\left (e^{\left (d x + c\right )} + 1\right )}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 24, normalized size = 0.83 \begin {gather*} \frac {x}{a}+\frac {2}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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