Optimal. Leaf size=19 \[ \frac {(a+b \tanh (x))^{n+1}}{b (n+1)} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 32} \[ \frac {(a+b \tanh (x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3506
Rubi steps
\begin {align*} \int \text {sech}^2(x) (a+b \tanh (x))^n \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^n \, dx,x,b \tanh (x)\right )}{b}\\ &=\frac {(a+b \tanh (x))^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 18, normalized size = 0.95 \[ \frac {(a+b \tanh (x))^{n+1}}{b n+b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 69, normalized size = 3.63 \[ \frac {{\left (a \cosh \relax (x) + b \sinh \relax (x)\right )} \cosh \left (n \log \left (\frac {a \cosh \relax (x) + b \sinh \relax (x)}{\cosh \relax (x)}\right )\right ) + {\left (a \cosh \relax (x) + b \sinh \relax (x)\right )} \sinh \left (n \log \left (\frac {a \cosh \relax (x) + b \sinh \relax (x)}{\cosh \relax (x)}\right )\right )}{{\left (b n + b\right )} \cosh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 39, normalized size = 2.05 \[ \frac {\left (\frac {a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b}{e^{\left (2 \, x\right )} + 1}\right )^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 20, normalized size = 1.05 \[ \frac {\left (a +b \tanh \relax (x )\right )^{n +1}}{b \left (n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 19, normalized size = 1.00 \[ \frac {{\left (b \tanh \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 54, normalized size = 2.84 \[ \frac {{\left (a+\frac {b\,\left ({\mathrm {e}}^{2\,x}-1\right )}{{\mathrm {e}}^{2\,x}+1}\right )}^n\,\left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{b\,\left ({\mathrm {e}}^{2\,x}+1\right )\,\left (n+1\right )} \]
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 \[ \int \left (a + b \tanh {\relax (x )}\right )^{n} \operatorname {sech}^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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