Optimal. Leaf size=16 \[ -\frac {\text {sech}^n(a+b x)}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 16, 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 = {2622, 30} \[ -\frac {\text {sech}^n(a+b x)}{b n} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2622
Rubi steps
\begin {align*} \int \text {sech}^{1+n}(a+b x) \sinh (a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x^{-1+n} \, dx,x,\text {sech}(a+b x)\right )}{b}\\ &=-\frac {\text {sech}^n(a+b x)}{b n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 16, normalized size = 1.00 \[ -\frac {\text {sech}^n(a+b x)}{b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 115, normalized size = 7.19 \[ -\frac {\cosh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}\right )\right ) + \sinh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}\right )\right )}{b n} \]
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 \[ \int \operatorname {sech}\left (b x + a\right )^{n} \tanh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 17, normalized size = 1.06 \[ -\frac {\mathrm {sech}\left (b x +a \right )^{n}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.83, size = 36, normalized size = 2.25 \[ -\frac {2^{n} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )\right )}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 31, normalized size = 1.94 \[ -\frac {{\left (\frac {2\,{\mathrm {e}}^{a+b\,x}}{{\mathrm {e}}^{2\,a+2\,b\,x}+1}\right )}^n}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 39, normalized size = 2.44 \[ \begin {cases} x \tanh {\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\x \tanh {\relax (a )} \operatorname {sech}^{n}{\relax (a )} & \text {for}\: b = 0 \\x - \frac {\log {\left (\tanh {\left (a + b x \right )} + 1 \right )}}{b} & \text {for}\: n = 0 \\- \frac {\operatorname {sech}^{n}{\left (a + b x \right )}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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