Optimal. Leaf size=19 \[ \frac {\tanh ^{n+1}(a+b x)}{b (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 19, 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} \[ \frac {\tanh ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2607
Rubi steps
\begin {align*} \int \text {sech}^2(a+b x) \tanh ^n(a+b x) \, dx &=-\frac {i \operatorname {Subst}\left (\int (-i x)^n \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\tanh ^{1+n}(a+b x)}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.00 \[ \frac {\tanh ^{n+1}(a+b x)}{b (n+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 69, normalized size = 3.63 \[ \frac {\cosh \left (n \log \left (\frac {\sinh \left (b x + a\right )}{\cosh \left (b x + a\right )}\right )\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right ) \sinh \left (n \log \left (\frac {\sinh \left (b x + a\right )}{\cosh \left (b x + a\right )}\right )\right )}{{\left (b n + b\right )} \cosh \left (b x + a\right )} \]
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 \[ \int \tanh \left (b x + a\right )^{n} \operatorname {sech}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 20, normalized size = 1.05 \[ \frac {\tanh ^{n +1}\left (b x +a \right )}{b \left (n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 19, normalized size = 1.00 \[ \frac {\tanh \left (b x + a\right )^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 42, normalized size = 2.21 \[ \frac {\mathrm {tanh}\left (a+b\,x\right )\,{\left (\frac {{\mathrm {e}}^{2\,a+2\,b\,x}-1}{{\mathrm {e}}^{2\,a+2\,b\,x}+1}\right )}^n}{b\,\left (n+1\right )} \]
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 \[ \int \tanh ^{n}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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