Integrand size = 23, antiderivative size = 36 \[ \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=(a+b)^2 x-\frac {a^2 \coth (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d} \]
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Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 472, 213} \[ \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {a^2 \coth (c+d x)}{d}+x (a+b)^2-\frac {b^2 \tanh (c+d x)}{d} \]
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Rule 213
Rule 472
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-b^2+\frac {a^2}{x^2}-\frac {(a+b)^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = -\frac {a^2 \coth (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d}-\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = (a+b)^2 x-\frac {a^2 \coth (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.78 \[ \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=2 a b x+\frac {b^2 \text {arctanh}(\tanh (c+d x))}{d}-\frac {a^2 \coth (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\tanh ^2(c+d x)\right )}{d}-\frac {b^2 \tanh (c+d x)}{d} \]
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {-\coth \left (d x +c \right ) a^{2}-b^{2} \tanh \left (d x +c \right )+d x \left (a +b \right )^{2}}{d}\) | \(36\) |
derivativedivides | \(\frac {a^{2} \left (d x +c -\coth \left (d x +c \right )\right )+2 a b \left (d x +c \right )+b^{2} \left (d x +c -\tanh \left (d x +c \right )\right )}{d}\) | \(49\) |
default | \(\frac {a^{2} \left (d x +c -\coth \left (d x +c \right )\right )+2 a b \left (d x +c \right )+b^{2} \left (d x +c -\tanh \left (d x +c \right )\right )}{d}\) | \(49\) |
risch | \(a^{2} x +2 a b x +b^{2} x -\frac {2 \left (a^{2} {\mathrm e}^{2 d x +2 c}-{\mathrm e}^{2 d x +2 c} b^{2}+a^{2}+b^{2}\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.69 \[ \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {{\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} d x + a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} - b^{2}}{2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )} \]
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\[ \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \coth ^{2}{\left (c + d x \right )}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.78 \[ \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=b^{2} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{2} {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + 2 \, a b x \]
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Time = 0.35 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.97 \[ \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (a^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + b^{2}\right )}}{e^{\left (4 \, d x + 4 \, c\right )} - 1}}{d} \]
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Time = 1.83 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.64 \[ \int \coth ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=x\,{\left (a+b\right )}^2-\frac {\frac {2\,\left (a^2+b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2-b^2\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1} \]
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