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 = 0.0249942, antiderivative size = 22, normalized size of antiderivative = 1., 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 290
Rule 325
Rule 206
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*}
Mathematica [A] time = 0.0043097, 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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Maple [A] time = 0.011, size = 15, normalized size = 0.7 \begin{align*} -{\rm coth} \left (x\right )+{\frac{3\,x}{2}}+{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1756, size = 35, normalized size = 1.59 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76542, size = 109, normalized size = 4.95 \begin{align*} \frac{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \,{\left (3 \, x + 2\right )} \sinh \left (x\right ) - 9 \, \cosh \left (x\right )}{8 \, \sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\sinh{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14688, size = 53, normalized size = 2.41 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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