Optimal. Leaf size=113 \[ \frac {x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{c}-\frac {2 (n+1) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6179, 96, 131} \[ \frac {x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{c}-\frac {2 (n+1) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 96
Rule 131
Rule 6179
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c}-\frac {(1+n) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c}-\frac {2 (1+n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.41, size = 97, normalized size = 0.86 \[ \frac {e^{n \coth ^{-1}(a x)} \left (n (n+1) e^{2 \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \coth ^{-1}(a x)}\right )+(n+2) \left ((n+1) \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \coth ^{-1}(a x)}\right )+a n x-1\right )\right )}{a c n (n+2)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a c x - c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{c -\frac {c}{a x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{c-\frac {c}{a\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a \int \frac {x e^{n \operatorname {acoth}{\left (a x \right )}}}{a x - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________