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 = 0.07, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 2637
Rule 2691
Rule 4392
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*}
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Mathematica [A] time = 0.13, 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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fricas [A] time = 0.40, size = 37, normalized size = 1.37 \[ -\frac {2 \, a b + {\left (a^{2} + b^{2}\right )} \cosh \relax (x) - {\left (a^{2} x + a^{2} + b^{2}\right )} \sinh \relax (x)}{\sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 29, normalized size = 1.07 \[ a^{2} x - \frac {2 \, {\left (2 \, a b e^{x} + a^{2} + b^{2}\right )}}{e^{\left (2 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 27, normalized size = 1.00 \[ a^{2} \left (x -\coth \relax (x )\right )-\frac {2 a b}{\sinh \relax (x )}-b^{2} \coth \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 45, normalized size = 1.67 \[ a^{2} {\left (x + \frac {2}{e^{\left (-2 \, x\right )} - 1}\right )} + \frac {4 \, a b}{e^{\left (-x\right )} - e^{x}} + \frac {2 \, b^{2}}{e^{\left (-2 \, x\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 33, normalized size = 1.22 \[ a^2\,x-\frac {2\,a^2+4\,{\mathrm {e}}^x\,a\,b+2\,b^2}{{\mathrm {e}}^{2\,x}-1} \]
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 \[ \int \left (a \coth {\relax (x )} + b \operatorname {csch}{\relax (x )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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