Optimal. Leaf size=81 \[ -e^{-x} \, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};-e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-e^{2 n x}\right )+\frac {e^{-x}}{2}+\frac {e^x}{2} \]
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Rubi [A] time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5601, 2194, 2251} \[ -e^{-x} \, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};-e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-e^{2 n x}\right )+\frac {e^{-x}}{2}+\frac {e^x}{2} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 5601
Rubi steps
\begin {align*} \int \sinh (x) \tanh (n x) \, dx &=\int \left (-\frac {e^{-x}}{2}+\frac {e^x}{2}+\frac {e^{-x}}{1+e^{2 n x}}-\frac {e^x}{1+e^{2 n x}}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-x} \, dx\right )+\frac {\int e^x \, dx}{2}+\int \frac {e^{-x}}{1+e^{2 n x}} \, dx-\int \frac {e^x}{1+e^{2 n x}} \, dx\\ &=\frac {e^{-x}}{2}+\frac {e^x}{2}-e^{-x} \, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};-e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-e^{2 n x}\right )\\ \end {align*}
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Mathematica [B] time = 0.18, size = 164, normalized size = 2.02 \[ \frac {1}{2} e^{-2 x} \left (-\frac {e^{2 n x+x} \, _2F_1\left (1,1-\frac {1}{2 n};2-\frac {1}{2 n};-e^{2 n x}\right )}{2 n-1}+\frac {e^{(2 n+3) x} \, _2F_1\left (1,1+\frac {1}{2 n};2+\frac {1}{2 n};-e^{2 n x}\right )}{2 n+1}-e^x \left (\, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};-e^{2 n x}\right )+e^{2 x} \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};-e^{2 n x}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sinh \relax (x) \tanh \left (n x\right ), x\right ) \]
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 \[ \int \sinh \relax (x) \tanh \left (n x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \sinh \relax (x ) \tanh \left (n x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} - \frac {1}{2} \, \int \frac {2 \, {\left (e^{\left (2 \, x\right )} - 1\right )}}{e^{\left (2 \, n x + x\right )} + e^{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {tanh}\left (n\,x\right )\,\mathrm {sinh}\relax (x) \,d x \]
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 \[ \int \sinh {\relax (x )} \tanh {\left (n x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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