Optimal. Leaf size=105 \[ -\frac {2 a^2 n (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )}{2-n}-\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6126, 96, 131} \[ -\frac {2 a^2 n (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )}{2-n}-\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 96
Rule 131
Rule 6126
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{x^3} \, dx\\ &=-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{2 x^2}+\frac {1}{2} (a n) \int \frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{x^2} \, dx\\ &=-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{2 x^2}-\frac {2 a^2 n (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{1+a x}\right )}{2-n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 91, normalized size = 0.87 \[ \frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n}{2}-1} \left (4 a^2 n x^2 \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )-(n-2) (a x+1)^2\right )}{2 (n-2) x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________