Integrand size = 10, antiderivative size = 20 \[ \int e^{2 x} \text {csch}^4(x) \, dx=\frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2320, 12, 270} \[ \int e^{2 x} \text {csch}^4(x) \, dx=\frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
[In]
[Out]
Rule 12
Rule 270
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {16 x^5}{\left (1-x^2\right )^4} \, dx,x,e^x\right ) \\ & = 16 \text {Subst}\left (\int \frac {x^5}{\left (1-x^2\right )^4} \, dx,x,e^x\right ) \\ & = \frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int e^{2 x} \text {csch}^4(x) \, dx=\frac {8 e^{6 x}}{3 \left (1-e^{2 x}\right )^3} \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(-\frac {{\mathrm e}^{2 x} \operatorname {csch}\left (x \right )^{2} \left (\coth \left (x \right )+1\right )}{3}\) | \(15\) |
default | \(-\frac {1}{3 \tanh \left (x \right )^{3}}-\frac {1}{\tanh \left (x \right )^{2}}-\frac {1}{\tanh \left (x \right )}\) | \(20\) |
risch | \(-\frac {8 \left (3 \,{\mathrm e}^{4 x}-3 \,{\mathrm e}^{2 x}+1\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{3}}\) | \(25\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (14) = 28\).
Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.75 \[ \int e^{2 x} \text {csch}^4(x) \, dx=-\frac {8 \, {\left (4 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sinh \left (x\right )^{2} - 3\right )}}{3 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}} \]
[In]
[Out]
\[ \int e^{2 x} \text {csch}^4(x) \, dx=\int \frac {e^{2 x}}{\sinh ^{4}{\left (x \right )}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int e^{2 x} \text {csch}^4(x) \, dx=\frac {8}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int e^{2 x} \text {csch}^4(x) \, dx=-\frac {8 \, {\left (3 \, e^{\left (4 \, x\right )} - 3 \, e^{\left (2 \, x\right )} + 1\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int e^{2 x} \text {csch}^4(x) \, dx=-\frac {8\,\left (3\,{\mathrm {e}}^{4\,x}-3\,{\mathrm {e}}^{2\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \]
[In]
[Out]