Optimal. Leaf size=111 \[ -\frac {2^{-\frac {1}{2}-\frac {n}{2}} \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} F_1\left (\frac {2+n}{2};\frac {3+n}{2},2;\frac {4+n}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n) \left (c-\frac {c}{a x}\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6317, 6314,
141} \begin {gather*} -\frac {2^{-\frac {n}{2}-\frac {1}{2}} \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} F_1\left (\frac {n+2}{2};\frac {n+3}{2},2;\frac {n+4}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2) \left (c-\frac {c}{a x}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 141
Rule 6314
Rule 6317
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=\frac {\left (1-\frac {1}{a x}\right )^{3/2} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{3/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{\left (c-\frac {c}{a x}\right )^{3/2}}\\ &=-\frac {2^{-\frac {1}{2}-\frac {n}{2}} \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} F_1\left (\frac {2+n}{2};\frac {3+n}{2},2;\frac {4+n}{2};\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n) \left (c-\frac {c}{a x}\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\left (c -\frac {c}{a x}\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________