Optimal. Leaf size=96 \[ \frac {2 x \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \, _2F_1\left (-\frac {1}{2},\frac {n+1}{2};\frac {1}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{\sqrt {c-a c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6176, 6181, 132} \[ \frac {2 x \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {n+1}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \, _2F_1\left (-\frac {1}{2},\frac {n+1}{2};\frac {1}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{\sqrt {c-a c x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 132
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=\frac {\left (\sqrt {1-\frac {1}{a x}} \sqrt {x}\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \, dx}{\sqrt {c-a c x}}\\ &=-\frac {\sqrt {1-\frac {1}{a x}} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\\ &=\frac {2 \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1+n}{2}} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x \, _2F_1\left (-\frac {1}{2},\frac {1+n}{2};\frac {1}{2};\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{\sqrt {c-a c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 96, normalized size = 1.00 \[ \frac {2 (a x+1) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \left (\frac {a x-1}{a x+1}\right )^{\frac {n+1}{2}} \, _2F_1\left (-\frac {1}{2},\frac {n+1}{2};\frac {1}{2};\frac {2}{a x+1}\right )}{a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a c x - c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a c x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\sqrt {-a c x +c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{\sqrt {-a c x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{\sqrt {c-a\,c\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________