Optimal. Leaf size=152 \[ -\frac {(a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{2 \left (1-a^2\right ) x^2}-\frac {2 b^2 (2 a+n) (a+b x+1)^{\frac {n-2}{2}} (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(a+1) (-a-b x+1)}{(1-a) (a+b x+1)}\right )}{(1-a)^3 (a+1) (2-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6163, 96, 131} \[ -\frac {(a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{2 \left (1-a^2\right ) x^2}-\frac {2 b^2 (2 a+n) (a+b x+1)^{\frac {n-2}{2}} (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(a+1) (-a-b x+1)}{(1-a) (a+b x+1)}\right )}{(1-a)^3 (a+1) (2-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 96
Rule 131
Rule 6163
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {(1-a-b x)^{-n/2} (1+a+b x)^{n/2}}{x^3} \, dx\\ &=-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 \left (1-a^2\right ) x^2}+\frac {(b (2 a+n)) \int \frac {(1-a-b x)^{-n/2} (1+a+b x)^{n/2}}{x^2} \, dx}{2 \left (1-a^2\right )}\\ &=-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 \left (1-a^2\right ) x^2}-\frac {2 b^2 (2 a+n) (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(1+a) (1-a-b x)}{(1-a) (1+a+b x)}\right )}{(1-a)^3 (1+a) (2-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 123, normalized size = 0.81 \[ \frac {(-a-b x+1)^{1-\frac {n}{2}} (a+b x+1)^{\frac {n}{2}-1} \left ((a-1)^2 (n-2) (a+b x+1)^2-4 b^2 x^2 (2 a+n) \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(a+1) (a+b x-1)}{(a-1) (a+b x+1)}\right )\right )}{2 (a-1)^3 (a+1) (n-2) x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (b x +a \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \operatorname {atanh}{\left (a + b x \right )}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________