Optimal. Leaf size=76 \[ -\frac {e^{-x}}{2}+\frac {e^x}{2}+e^{-x} \, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );e^{2 n x}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 76, 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 = {5721, 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 5721
Rubi steps
\begin {align*} \int \cosh (x) \coth (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\\ &=\frac {1}{2} \int e^{-x} \, dx+\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. \(156\) vs. \(2(76)=152\).
time = 0.12, size = 156, normalized size = 2.05 \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 \, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};e^{2 n x}\right )-e^{3 x} \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};e^{2 n x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.64, size = 0, normalized size = 0.00 \[\int \cosh \left (x \right ) \coth \left (n x \right )\, dx\]
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 \cosh {\left (x \right )} \coth {\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 {coth}\left (n\,x\right )\,\mathrm {cosh}\left (x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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