Optimal. Leaf size=27 \[ a^2 x+a b \sinh (x)-\text{csch}(x) (a \cosh (x)+b) (a+b \cosh (x)) \]
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Rubi [A] time = 0.0683047, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4392, 2691, 2637} \[ a^2 x+a b \sinh (x)-\text{csch}(x) (a \cosh (x)+b) (a+b \cosh (x)) \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2691
Rule 2637
Rubi steps
\begin{align*} \int (a \coth (x)+b \text{csch}(x))^2 \, dx &=-\int (i b+i a \cosh (x))^2 \text{csch}^2(x) \, dx\\ &=-(b+a \cosh (x)) (a+b \cosh (x)) \text{csch}(x)-\int \left (-a^2-a b \cosh (x)\right ) \, dx\\ &=a^2 x-(b+a \cosh (x)) (a+b \cosh (x)) \text{csch}(x)+(a b) \int \cosh (x) \, dx\\ &=a^2 x-(b+a \cosh (x)) (a+b \cosh (x)) \text{csch}(x)+a b \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.120875, size = 23, normalized size = 0.85 \[ a (a x-2 b \text{csch}(x))-\left (a^2+b^2\right ) \coth (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 36, normalized size = 1.3 \begin{align*}{a}^{2} \left ( x-{\rm coth} \left (x\right ) \right ) +2\,ab \left ( -{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\sinh \left ( x \right ) }}+\sinh \left ( x \right ) \right ) -{b}^{2}{\rm coth} \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0443, size = 61, normalized size = 2.26 \begin{align*} a^{2}{\left (x + \frac{2}{e^{\left (-2 \, x\right )} - 1}\right )} + \frac{4 \, a b}{e^{\left (-x\right )} - e^{x}} + \frac{2 \, b^{2}}{e^{\left (-2 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9817, size = 95, normalized size = 3.52 \begin{align*} -\frac{2 \, a b +{\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) -{\left (a^{2} x + a^{2} + b^{2}\right )} \sinh \left (x\right )}{\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 (a \coth{\left (x \right )} + b \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 [A] time = 1.13322, size = 39, normalized size = 1.44 \begin{align*} a^{2} x - \frac{2 \,{\left (2 \, a b e^{x} + a^{2} + b^{2}\right )}}{e^{\left (2 \, x\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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