Optimal. Leaf size=43 \[ \frac{\tanh ^{n+1}(a+b x) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\tanh ^2(a+b x)\right )}{b (n+1)} \]
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Rubi [A] time = 0.0219585, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3476, 364} \[ \frac{\tanh ^{n+1}(a+b x) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\tanh ^2(a+b x)\right )}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \tanh ^n(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^n}{-1+x^2} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac{\, _2F_1\left (1,\frac{1+n}{2};\frac{3+n}{2};\tanh ^2(a+b x)\right ) \tanh ^{1+n}(a+b x)}{b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0447656, size = 45, normalized size = 1.05 \[ \frac{\tanh ^{n+1}(a+b x) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+1}{2}+1;\tanh ^2(a+b x)\right )}{b (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int \left ( \tanh \left ( bx+a \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh \left (b x + a\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\tanh \left (b x + a\right )^{n}, 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 \tanh ^{n}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \tanh \left (b x + a\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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