Optimal. Leaf size=22 \[ \frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {1}{2} \cosh ^2(x) \coth (x) \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {290, 325, 206} \[ \frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {1}{2} \cosh ^2(x) \coth (x) \]
Antiderivative was successfully verified.
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Rule 206
Rule 290
Rule 325
Rubi steps
\begin {align*} \int (\text {csch}(x)+\sinh (x))^2 \, dx &=\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right )^2} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \cosh ^2(x) \coth (x)+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac {3 \coth (x)}{2}+\frac {1}{2} \cosh ^2(x) \coth (x)+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {1}{2} \cosh ^2(x) \coth (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 0.82 \[ \frac {3 x}{2}+\frac {1}{4} \sinh (2 x)-\coth (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 32, normalized size = 1.45 \[ \frac {\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + 4 \, {\left (3 \, x + 2\right )} \sinh \relax (x) - 9 \, \cosh \relax (x)}{8 \, \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 39, normalized size = 1.77 \[ \frac {3}{2} \, x - \frac {3 \, e^{\left (4 \, x\right )} + 14 \, e^{\left (2 \, x\right )} - 1}{8 \, {\left (e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}\right )}} + \frac {1}{8} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 15, normalized size = 0.68 \[ -\coth \relax (x )+\frac {3 x}{2}+\frac {\cosh \relax (x ) \sinh \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 26, normalized size = 1.18 \[ \frac {3}{2} \, x + \frac {2}{e^{\left (-2 \, x\right )} - 1} + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 26, normalized size = 1.18 \[ \frac {3\,x}{2}-\frac {{\mathrm {e}}^{-2\,x}}{8}+\frac {{\mathrm {e}}^{2\,x}}{8}-\frac {2}{{\mathrm {e}}^{2\,x}-1} \]
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 \[ \int \left (\sinh {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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