Optimal. Leaf size=20 \[ -\frac {(a+b \coth (x))^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 32} \[ -\frac {(a+b \coth (x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3506
Rubi steps
\begin {align*} \int (a+b \coth (x))^n \text {csch}^2(x) \, dx &=-\frac {\operatorname {Subst}\left (\int (a+x)^n \, dx,x,b \coth (x)\right )}{b}\\ &=-\frac {(a+b \coth (x))^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 19, normalized size = 0.95 \[ -\frac {(a+b \coth (x))^{n+1}}{b n+b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 70, normalized size = 3.50 \[ -\frac {{\left (b \cosh \relax (x) + a \sinh \relax (x)\right )} \cosh \left (n \log \left (\frac {b \cosh \relax (x) + a \sinh \relax (x)}{\sinh \relax (x)}\right )\right ) + {\left (b \cosh \relax (x) + a \sinh \relax (x)\right )} \sinh \left (n \log \left (\frac {b \cosh \relax (x) + a \sinh \relax (x)}{\sinh \relax (x)}\right )\right )}{{\left (b n + b\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 40, normalized size = 2.00 \[ -\frac {\left (\frac {a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b}{e^{\left (2 \, x\right )} - 1}\right )^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 21, normalized size = 1.05 \[ -\frac {\left (a +b \coth \relax (x )\right )^{n +1}}{b \left (n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 20, normalized size = 1.00 \[ -\frac {{\left (b \coth \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.99, size = 55, normalized size = 2.75 \[ -\frac {{\left (a+\frac {b\,\left ({\mathrm {e}}^{2\,x}+1\right )}{{\mathrm {e}}^{2\,x}-1}\right )}^n\,\left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{b\,\left ({\mathrm {e}}^{2\,x}-1\right )\,\left (n+1\right )} \]
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 \[ \int \left (a + b \coth {\relax (x )}\right )^{n} \operatorname {csch}^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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