Optimal. Leaf size=125 \[ -\frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a c}+\frac {x (a x+1)^{n/2} (1-a x)^{-n/2}}{c}-\frac {(1-n) (a x+1)^{n/2} (1-a x)^{-n/2}}{a c n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 125, 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 = {6157, 6150, 90, 79, 69} \[ -\frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a c}+\frac {x (a x+1)^{n/2} (1-a x)^{-n/2}}{c}-\frac {(1-n) (a x+1)^{n/2} (1-a x)^{-n/2}}{a c n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 79
Rule 90
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {a^2 \int \frac {e^{n \tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \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}}{c}+\frac {\int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} (-1-a n x) \, dx}{c}\\ &=-\frac {(1-n) (1-a x)^{-n/2} (1+a x)^{n/2}}{a c n}+\frac {x (1-a x)^{-n/2} (1+a x)^{n/2}}{c}-\frac {n \int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2} \, dx}{c}\\ &=-\frac {(1-n) (1-a x)^{-n/2} (1+a x)^{n/2}}{a c n}+\frac {x (1-a x)^{-n/2} (1+a x)^{n/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 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 82, normalized size = 0.66 \[ \frac {(1-a x)^{-n/2} \left ((a x+1)^{n/2} (a n x+n-1)-2^{\frac {n}{2}+1} n \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )\right )}{a c n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} x^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a^{2} x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{c -\frac {c}{a^{2} x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a^{2} x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-\frac {c}{a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \int \frac {x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________