Optimal. Leaf size=81 \[ \frac {e^{-x}}{2}+\frac {e^x}{2}-e^{-x} \, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};-e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-e^{2 n x}\right ) \]
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Rubi [A]
time = 0.06, 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 = {5720, 2225,
2283} \begin {gather*} -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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2283
Rule 5720
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] Leaf count is larger than twice the leaf count of optimal. \(164\) vs. \(2(81)=162\).
time = 0.12, size = 164, normalized size = 2.02 \begin {gather*} \frac {1}{2} e^{-2 x} \left (-\frac {e^{x+2 n x} \, _2F_1\left (1,1-\frac {1}{2 n};2-\frac {1}{2 n};-e^{2 n x}\right )}{-1+2 n}+\frac {e^{(3+2 n) x} \, _2F_1\left (1,1+\frac {1}{2 n};2+\frac {1}{2 n};-e^{2 n x}\right )}{1+2 n}-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 ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \sinh \left (x \right ) \tanh \left (n x \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sinh {\left (x \right )} \tanh {\left (n x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {tanh}\left (n\,x\right )\,\mathrm {sinh}\left (x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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