Optimal. Leaf size=36 \[ \frac {\coth (x)}{2 \sqrt {a \text {csch}^4(x)}}-\frac {x \text {csch}^2(x)}{2 \sqrt {a \text {csch}^4(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4123, 2635, 8} \[ \frac {\coth (x)}{2 \sqrt {a \text {csch}^4(x)}}-\frac {x \text {csch}^2(x)}{2 \sqrt {a \text {csch}^4(x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 4123
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \text {csch}^4(x)}} \, dx &=\frac {\text {csch}^2(x) \int \sinh ^2(x) \, dx}{\sqrt {a \text {csch}^4(x)}}\\ &=\frac {\coth (x)}{2 \sqrt {a \text {csch}^4(x)}}-\frac {\text {csch}^2(x) \int 1 \, dx}{2 \sqrt {a \text {csch}^4(x)}}\\ &=\frac {\coth (x)}{2 \sqrt {a \text {csch}^4(x)}}-\frac {x \text {csch}^2(x)}{2 \sqrt {a \text {csch}^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 24, normalized size = 0.67 \[ \frac {\coth (x)-x \text {csch}^2(x)}{2 \sqrt {a \text {csch}^4(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 253, normalized size = 7.03 \[ \frac {{\left ({\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{4} + \cosh \relax (x)^{4} + 4 \, {\left (\cosh \relax (x) e^{\left (4 \, x\right )} - 2 \, \cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x)^{3} - 4 \, x \cosh \relax (x)^{2} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + {\left (3 \, \cosh \relax (x)^{2} - 2 \, x\right )} e^{\left (4 \, x\right )} - 2 \, {\left (3 \, \cosh \relax (x)^{2} - 2 \, x\right )} e^{\left (2 \, x\right )} - 2 \, x\right )} \sinh \relax (x)^{2} + {\left (\cosh \relax (x)^{4} - 4 \, x \cosh \relax (x)^{2} - 1\right )} e^{\left (4 \, x\right )} - 2 \, {\left (\cosh \relax (x)^{4} - 4 \, x \cosh \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (\cosh \relax (x)^{3} - 2 \, x \cosh \relax (x) + {\left (\cosh \relax (x)^{3} - 2 \, x \cosh \relax (x)\right )} e^{\left (4 \, x\right )} - 2 \, {\left (\cosh \relax (x)^{3} - 2 \, x \cosh \relax (x)\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x) - 1\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )}}{8 \, {\left (a \cosh \relax (x)^{2} e^{\left (2 \, x\right )} + 2 \, a \cosh \relax (x) e^{\left (2 \, x\right )} \sinh \relax (x) + a e^{\left (2 \, x\right )} \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 26, normalized size = 0.72 \[ \frac {{\left (2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - 4 \, x + e^{\left (2 \, x\right )}}{8 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 89, normalized size = 2.47 \[ -\frac {{\mathrm e}^{2 x} x}{2 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {{\mathrm e}^{4 x}}{8 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {1}{8 \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 22, normalized size = 0.61 \[ -\frac {{\left (e^{\left (-4 \, x\right )} - 1\right )} e^{\left (2 \, x\right )}}{8 \, \sqrt {a}} - \frac {x}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {\frac {a}{{\mathrm {sinh}\relax (x)}^4}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \operatorname {csch}^{4}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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