Integrand size = 10, antiderivative size = 16 \[ \int \frac {1}{\sqrt {a \sinh ^4(x)}} \, dx=-\frac {\cosh (x) \sinh (x)}{\sqrt {a \sinh ^4(x)}} \]
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Time = 0.01 (sec) , antiderivative size = 16, 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 = {3286, 3852, 8} \[ \int \frac {1}{\sqrt {a \sinh ^4(x)}} \, dx=-\frac {\sinh (x) \cosh (x)}{\sqrt {a \sinh ^4(x)}} \]
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Rule 8
Rule 3286
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\sinh ^2(x) \int \text {csch}^2(x) \, dx}{\sqrt {a \sinh ^4(x)}} \\ & = -\frac {\left (i \sinh ^2(x)\right ) \text {Subst}(\int 1 \, dx,x,-i \coth (x))}{\sqrt {a \sinh ^4(x)}} \\ & = -\frac {\cosh (x) \sinh (x)}{\sqrt {a \sinh ^4(x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a \sinh ^4(x)}} \, dx=-\frac {\cosh (x) \sinh (x)}{\sqrt {a \sinh ^4(x)}} \]
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Time = 1.55 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{-2 x} \left ({\mathrm e}^{2 x}-1\right )}{\sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}\) | \(29\) |
default | \(-\frac {\sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}\, \sqrt {a \sinh \left (2 x \right )^{2}}}{a \sinh \left (2 x \right ) \sqrt {\left (-1+\cosh \left (2 x \right )\right )^{2} a}}\) | \(50\) |
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (14) = 28\).
Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 7.62 \[ \int \frac {1}{\sqrt {a \sinh ^4(x)}} \, dx=-\frac {2 \, \sqrt {a e^{\left (8 \, x\right )} - 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} - 4 \, a e^{\left (2 \, x\right )} + a}}{a \cosh \left (x\right )^{2} + {\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (a \cosh \left (x\right )^{2} - a\right )} e^{\left (4 \, x\right )} - 2 \, {\left (a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \left (x\right ) e^{\left (4 \, x\right )} - 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a} \]
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\[ \int \frac {1}{\sqrt {a \sinh ^4(x)}} \, dx=\int \frac {1}{\sqrt {a \sinh ^{4}{\left (x \right )}}}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {a \sinh ^4(x)}} \, dx=\frac {2}{\sqrt {a} e^{\left (-2 \, x\right )} - \sqrt {a}} \]
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none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {a \sinh ^4(x)}} \, dx=-\frac {2}{\sqrt {a} {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
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Time = 1.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.38 \[ \int \frac {1}{\sqrt {a \sinh ^4(x)}} \, dx=\frac {{\mathrm {e}}^{-x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}{a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^3} \]
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