Integrand size = 13, antiderivative size = 54 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^3} \, dx=-x+\frac {7 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )} \]
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Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {425, 541, 536, 212} \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^3} \, dx=\frac {7 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}-x-\frac {\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )}+\frac {\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2} \]
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Rule 212
Rule 425
Rule 536
Rule 541
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-2 x^2\right )^3 \left (1-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}+\frac {1}{4} \text {Subst}\left (\int \frac {2-6 x^2}{\left (1-2 x^2\right )^2 \left (1-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )}+\frac {1}{8} \text {Subst}\left (\int \frac {6+2 x^2}{\left (1-2 x^2\right ) \left (1-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = \frac {\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )}+\frac {7}{4} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\tanh (x)\right )-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right ) \\ & = -x+\frac {7 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {\tanh (x)}{2 \left (1-2 \tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (1-2 \tanh ^2(x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.75 (sec) , antiderivative size = 765, normalized size of antiderivative = 14.17 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^3} \, dx=\frac {\text {sech}(x) \left (-7 \text {arctanh}\left (\frac {\sqrt {(-1-i) (i+\sinh (x))}}{\sqrt {(1+i)-(1-i) \sinh (x)}}\right ) (-3+\cosh (2 x))^2 \sqrt {(2+2 i)-(2-2 i) \sinh (x)} \sqrt {(-1-i) (i+\sinh (x))}+7 (-1)^{3/4} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {(-1-i) (i+\sinh (x))}}{\sqrt {(1+i)-(1-i) \sinh (x)}}\right ) (-3+\cosh (2 x))^2 \sqrt {(2+2 i)-(2-2 i) \sinh (x)} \sqrt {(-1-i) (i+\sinh (x))}+4 \left (2 \sqrt {2} \sinh (x)+8 \sqrt {2} \sinh ^3(x)+6 \sqrt {2} \sinh ^5(x)+(7-i) \sqrt {1+i} \arcsin \left (\frac {1}{2} \sqrt {1+i} \sqrt {(-1-i) (i+\sinh (x))}\right ) \sqrt {1+i \sinh (x)} \sqrt {(-1-i) (i+\sinh (x))}-(14-2 i) \sqrt {1+i} \arcsin \left (\frac {1}{2} \sqrt {1+i} \sqrt {(-1-i) (i+\sinh (x))}\right ) \sqrt {1+i \sinh (x)} \sinh ^2(x) \sqrt {(-1-i) (i+\sinh (x))}+(7-i) \sqrt {1+i} \arcsin \left (\frac {1}{2} \sqrt {1+i} \sqrt {(-1-i) (i+\sinh (x))}\right ) \sqrt {1+i \sinh (x)} \sinh ^4(x) \sqrt {(-1-i) (i+\sinh (x))}+\left (\frac {7}{4}+\frac {i}{4}\right ) \sqrt {-1+i} \text {arcsinh}\left (\frac {1}{2} \sqrt {-1+i} \sqrt {(1-i) (i+\sinh (x))}\right ) (-3+\cosh (2 x))^2 \sqrt {1+i \sinh (x)} \sqrt {(1-i) (i+\sinh (x))}-(7+7 i) \text {arctanh}\left (\frac {(1+i) \sqrt {(1-i) (i+\sinh (x))}}{\sqrt {2} \sqrt {(1+i) (-i+\sinh (x))}}\right ) \sqrt {(1+i) (-i+\sinh (x))} \sqrt {(1-i) (i+\sinh (x))}+\frac {7 \text {arctanh}\left (\frac {\sqrt {(1-i) (i+\sinh (x))}}{\sqrt {(1+i) (-i+\sinh (x))}}\right ) (-3+\cosh (2 x))^2 \sqrt {(1+i) (-i+\sinh (x))} \sqrt {(1-i) (i+\sinh (x))}}{2 \sqrt {2}}+(14+14 i) \text {arctanh}\left (\frac {(1+i) \sqrt {(1-i) (i+\sinh (x))}}{\sqrt {2} \sqrt {(1+i) (-i+\sinh (x))}}\right ) \sinh ^2(x) \sqrt {(1+i) (-i+\sinh (x))} \sqrt {(1-i) (i+\sinh (x))}-(7+7 i) \text {arctanh}\left (\frac {(1+i) \sqrt {(1-i) (i+\sinh (x))}}{\sqrt {2} \sqrt {(1+i) (-i+\sinh (x))}}\right ) \sinh ^4(x) \sqrt {(1+i) (-i+\sinh (x))} \sqrt {(1-i) (i+\sinh (x))}\right )\right )}{8 \sqrt {2} (-3+\cosh (2 x))^2} \]
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Time = 210.08 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.04
method | result | size |
parallelrisch | \(0\) | \(2\) |
risch | \(-x +\frac {17 \,{\mathrm e}^{6 x}-57 \,{\mathrm e}^{4 x}+19 \,{\mathrm e}^{2 x}-3}{2 \left ({\mathrm e}^{4 x}-6 \,{\mathrm e}^{2 x}+1\right )^{2}}+\frac {7 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+2 \sqrt {2}-3\right )}{16}-\frac {7 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{16}\) | \(75\) |
default | \(-\frac {2 \left (-\frac {\tanh \left (\frac {x}{2}\right )^{3}}{8}+\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {5 \tanh \left (\frac {x}{2}\right )}{8}+\frac {1}{8}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{8}-\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )-\frac {2 \left (-\frac {\tanh \left (\frac {x}{2}\right )^{3}}{8}-\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {5 \tanh \left (\frac {x}{2}\right )}{8}-\frac {1}{8}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2}-2 \tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{8}+\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(140\) |
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Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (44) = 88\).
Time = 0.28 (sec) , antiderivative size = 717, normalized size of antiderivative = 13.28 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^3} \, dx=\int \frac {1}{\left (- \tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{3} \left (\tanh {\left (x \right )} + \operatorname {sech}{\left (x \right )}\right )^{3}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (44) = 88\).
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^3} \, dx=\frac {7}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {7}{16} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) - x + \frac {19 \, e^{\left (-2 \, x\right )} - 57 \, e^{\left (-4 \, x\right )} + 17 \, e^{\left (-6 \, x\right )} - 3}{2 \, {\left (12 \, e^{\left (-2 \, x\right )} - 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^3} \, dx=-\frac {7}{16} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - x + \frac {17 \, e^{\left (6 \, x\right )} - 57 \, e^{\left (4 \, x\right )} + 19 \, e^{\left (2 \, x\right )} - 3}{2 \, {\left (e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\left (\text {sech}^2(x)-\tanh ^2(x)\right )^3} \, dx=\frac {136\,{\mathrm {e}}^{2\,x}-24}{38\,{\mathrm {e}}^{4\,x}-12\,{\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-x-\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}-\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{16}\right )}{16}+\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}+\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{16}\right )}{16}+\frac {\frac {17\,{\mathrm {e}}^{2\,x}}{2}+\frac {45}{2}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \]
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