Optimal. Leaf size=76 \[ e^{-x} \, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};e^{2 n x}\right )-e^x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );e^{2 n x}\right )-\frac {e^{-x}}{2}+\frac {e^x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, 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 = {5602, 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.
[In]
[Out]
Rule 2194
Rule 2251
Rule 5602
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*}
________________________________________________________________________________________
Mathematica [B] time = 0.18, size = 156, normalized size = 2.05 \[ \frac {1}{2} e^{-2 x} \left (-\frac {e^{2 n x+x} \, _2F_1\left (1,1-\frac {1}{2 n};2-\frac {1}{2 n};e^{2 n x}\right )}{2 n-1}-\frac {e^{(2 n+3) x} \, _2F_1\left (1,1+\frac {1}{2 n};2+\frac {1}{2 n};e^{2 n x}\right )}{2 n+1}+e^x \, _2F_1\left (1,-\frac {1}{2 n};1-\frac {1}{2 n};e^{2 n x}\right )-e^{3 x} \, _2F_1\left (1,\frac {1}{2 n};1+\frac {1}{2 n};e^{2 n x}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \relax (x) \coth \left (n x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh \relax (x) \coth \left (n x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \cosh \relax (x ) \coth \left (n x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} - \frac {1}{2} \, \int \frac {e^{\left (2 \, x\right )} + 1}{e^{\left (n x + x\right )} + e^{x}}\,{d x} + \frac {1}{2} \, \int \frac {e^{\left (2 \, x\right )} + 1}{e^{\left (n x + x\right )} - e^{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {coth}\left (n\,x\right )\,\mathrm {cosh}\relax (x) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \coth {\left (n x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________