Optimal. Leaf size=86 \[ \frac {e^{n \tanh ^{-1}(a x)}}{a^3 c n}+\frac {2^{1+\frac {n}{2}} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^3 c (2-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 127, normalized size of antiderivative = 1.48, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6285, 92, 80,
71} \begin {gather*} \frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^3 c}+\frac {(1-n) (a x+1)^{n/2} (1-a x)^{-n/2}}{a^3 c n}-\frac {x (a x+1)^{n/2} (1-a x)^{-n/2}}{a^2 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 71
Rule 80
Rule 92
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx &=\frac {\int x^2 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \, dx}{c}\\ &=-\frac {x (1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c}-\frac {\int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} (-1-a n x) \, dx}{a^2 c}\\ &=\frac {(1-n) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^3 c n}-\frac {x (1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c}+\frac {n \int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2} \, dx}{a^2 c}\\ &=\frac {(1-n) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^3 c n}-\frac {x (1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c}+\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^3 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 63, normalized size = 0.73 \begin {gather*} \frac {e^{n \tanh ^{-1}(a x)} \left (2+n-4 e^{2 \tanh ^{-1}(a x)} n \, _2F_1\left (2,1+\frac {n}{2};2+\frac {n}{2};-e^{2 \tanh ^{-1}(a x)}\right )\right )}{a^3 c n (2+n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x^{2}}{-a^{2} c \,x^{2}+c}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \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*} - \frac {\int \frac {x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \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 {x^2\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-a^2\,c\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________