Integrand size = 21, antiderivative size = 24 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {a \coth (c+d x)}{d}+\frac {b \tanh (c+d x)}{d} \]
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Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3744, 14} \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {b \tanh (c+d x)}{d}-\frac {a \coth (c+d x)}{d} \]
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Rule 14
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b x^2}{x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (b+\frac {a}{x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = -\frac {a \coth (c+d x)}{d}+\frac {b \tanh (c+d x)}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {a \coth (c+d x)}{d}+\frac {b \tanh (c+d x)}{d} \]
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Time = 0.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {b \tanh \left (d x +c \right )-\frac {a}{\tanh \left (d x +c \right )}}{d}\) | \(25\) |
default | \(\frac {b \tanh \left (d x +c \right )-\frac {a}{\tanh \left (d x +c \right )}}{d}\) | \(25\) |
risch | \(-\frac {2 \left ({\mathrm e}^{2 d x +2 c} a +b \,{\mathrm e}^{2 d x +2 c}+a -b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left ({\mathrm e}^{2 d x +2 c}+1\right )}\) | \(59\) |
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.67 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (a \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )}}{d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right ) + {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )} \]
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\[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \operatorname {csch}^{2}{\left (c + d x \right )}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {2 \, b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} + \frac {2 \, a}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b\right )}}{d {\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \text {csch}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {\frac {2\,\left (a-b\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1} \]
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