Optimal. Leaf size=20 \[ \frac {\coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2157, 30} \[ \frac {\coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rubi steps
\begin {align*} \int \coth ^{-1}(\tanh (a+b x))^n \, dx &=\frac {\operatorname {Subst}\left (\int x^n \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b}\\ &=\frac {\coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \[ \frac {\coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 164, normalized size = 8.20 \[ \frac {2 \, {\left (b x + a\right )} {\left (b^{2} x^{2} + 2 \, a b x + \frac {1}{4} \, \pi ^{2} + a^{2}\right )}^{\frac {1}{2} \, n} \cos \left (2 \, n \arctan \left (-\frac {2 \, b x}{\pi } - \frac {2 \, a}{\pi } + \frac {\sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right )\right ) - \pi {\left (b^{2} x^{2} + 2 \, a b x + \frac {1}{4} \, \pi ^{2} + a^{2}\right )}^{\frac {1}{2} \, n} \sin \left (2 \, n \arctan \left (-\frac {2 \, b x}{\pi } - \frac {2 \, a}{\pi } + \frac {\sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right )\right )}{2 \, {\left (b n + b\right )}} \]
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 \[ \int \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 21, normalized size = 1.05 \[ \frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{1+n}}{b \left (1+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.53, size = 65, normalized size = 3.25 \[ \frac {{\left (-i \, \pi + 2 \, b x + 2 \, a\right )} {\left (\cosh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right ) - \sinh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right )\right )}}{{\left (2^{n + 1} n + 2^{n + 1}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 121, normalized size = 6.05 \[ {\left (\frac {1}{2}\right )}^n\,\left (\frac {x}{n+1}-\frac {\frac {\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2}-\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2}+b\,x}{b\,\left (n+1\right )}\right )\,{\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )}^n \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 51, normalized size = 2.55 \[ \begin {cases} \frac {x}{\operatorname {acoth}{\left (\tanh {\relax (a )} \right )}} & \text {for}\: b = 0 \wedge n = -1 \\x \operatorname {acoth}^{n}{\left (\tanh {\relax (a )} \right )} & \text {for}\: b = 0 \\\frac {\log {\left (\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )} \right )}}{b} & \text {for}\: n = -1 \\\frac {\operatorname {acoth}{\left (\tanh {\left (a + b x \right )} \right )} \operatorname {acoth}^{n}{\left (\tanh {\left (a + b x \right )} \right )}}{b n + b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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