Integrand size = 8, antiderivative size = 13 \[ \int \tanh ^2(a+b x) \, dx=x-\frac {\tanh (a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 8} \[ \int \tanh ^2(a+b x) \, dx=x-\frac {\tanh (a+b x)}{b} \]
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Rule 8
Rule 3554
Rubi steps \begin{align*} \text {integral}& = -\frac {\tanh (a+b x)}{b}+\int 1 \, dx \\ & = x-\frac {\tanh (a+b x)}{b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.77 \[ \int \tanh ^2(a+b x) \, dx=\frac {\text {arctanh}(\tanh (a+b x))}{b}-\frac {\tanh (a+b x)}{b} \]
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Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31
method | result | size |
parallelrisch | \(\frac {b x -\tanh \left (b x +a \right )}{b}\) | \(17\) |
derivativedivides | \(\frac {b x +a -\tanh \left (b x +a \right )}{b}\) | \(18\) |
default | \(\frac {b x +a -\tanh \left (b x +a \right )}{b}\) | \(18\) |
risch | \(x +\frac {2}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )}\) | \(21\) |
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (13) = 26\).
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.54 \[ \int \tanh ^2(a+b x) \, dx=\frac {{\left (b x + 1\right )} \cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )} \]
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\[ \int \tanh ^2(a+b x) \, dx=\int \sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.92 \[ \int \tanh ^2(a+b x) \, dx=x + \frac {a}{b} - \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int \tanh ^2(a+b x) \, dx=\frac {b x + a + \frac {2}{e^{\left (2 \, b x + 2 \, a\right )} + 1}}{b} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \tanh ^2(a+b x) \, dx=x+\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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