Optimal. Leaf size=20 \[ -\frac {\coth ^{n+1}(a+b x)}{b (n+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2607, 32} \[ -\frac {\coth ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2607
Rubi steps
\begin {align*} \int \coth ^n(a+b x) \text {csch}^2(a+b x) \, dx &=\frac {i \operatorname {Subst}\left (\int (-i x)^n \, dx,x,i \coth (a+b x)\right )}{b}\\ &=-\frac {\coth ^{1+n}(a+b x)}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \[ -\frac {\coth ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 70, normalized size = 3.50 \[ -\frac {\cosh \left (b x + a\right ) \cosh \left (n \log \left (\frac {\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right )\right ) + \cosh \left (b x + a\right ) \sinh \left (n \log \left (\frac {\cosh \left (b x + a\right )}{\sinh \left (b x + a\right )}\right )\right )}{{\left (b n + b\right )} \sinh \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (b x + a\right )^{n} \operatorname {csch}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 21, normalized size = 1.05 \[ -\frac {\coth ^{n +1}\left (b x +a \right )}{b \left (n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 20, normalized size = 1.00 \[ -\frac {\coth \left (b x + a\right )^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 43, normalized size = 2.15 \[ -\frac {\mathrm {coth}\left (a+b\,x\right )\,{\left (\frac {{\mathrm {e}}^{2\,a+2\,b\,x}+1}{{\mathrm {e}}^{2\,a+2\,b\,x}-1}\right )}^n}{b\,\left (n+1\right )} \]
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 \[ \int \coth ^{n}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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