Optimal. Leaf size=139 \[ -\frac {2^{\frac {n}{2}+1} (n+2) (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a c^2 n}+\frac {(n+3) (a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a c^2 (n+2)}-\frac {x (a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-1}}{c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6131, 6129, 90, 79, 69} \[ -\frac {2^{\frac {n}{2}+1} (n+2) (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a c^2 n}+\frac {(n+3) (a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a c^2 (n+2)}-\frac {x (a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-1}}{c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 79
Rule 90
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=\frac {a^2 \int \frac {e^{n \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac {a^2 \int x^2 (1-a x)^{-2-\frac {n}{2}} (1+a x)^{n/2} \, dx}{c^2}\\ &=-\frac {x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{c^2}-\frac {\int (1-a x)^{-2-\frac {n}{2}} (1+a x)^{n/2} (-1-a (2+n) x) \, dx}{c^2}\\ &=\frac {(3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^2 (2+n)}-\frac {x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{c^2}-\frac {(2+n) \int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2} \, dx}{c^2}\\ &=\frac {(3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^2 (2+n)}-\frac {x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{c^2}-\frac {2^{1+\frac {n}{2}} (2+n) (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 c^2 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.47, size = 194, normalized size = 1.40 \[ \frac {e^{n \tanh ^{-1}(a x)} \left (-2 n (a x-1) e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a x)}\right )-4 n e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (2,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a x)}\right )+4 a n x e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (2,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a x)}\right )+2 (n+2) (a x-1) \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;-e^{2 \tanh ^{-1}(a x)}\right )-3 a n x-4 a x+n+4\right )}{a c^2 n (n+2) (a x-1)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{\left (c -\frac {c}{a x}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (c - \frac {c}{a x}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a\,x}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \int \frac {x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{2} - 2 a x + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________