Optimal. Leaf size=111 \[ -\frac {2^{\frac {n}{2}+1} (n+1) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a c (2-n) n}-\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-n/2}}{a c n} \]
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Rubi [A] time = 0.13, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6131, 6129, 79, 69} \[ -\frac {2^{\frac {n}{2}+1} (n+1) (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a c (2-n) n}-\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-n/2}}{a c n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 79
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\frac {a \int \frac {e^{n \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac {a \int x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2} \, dx}{c}\\ &=-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {2+n}{2}}}{a c n}+\frac {(1+n) \int (1-a x)^{-n/2} (1+a x)^{n/2} \, dx}{c n}\\ &=-\frac {(1-a x)^{-n/2} (1+a x)^{\frac {2+n}{2}}}{a c n}-\frac {2^{1+\frac {n}{2}} (1+n) (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 c (2-n) n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 95, normalized size = 0.86 \[ \frac {(1-a x)^{-n/2} \left (-2^{\frac {n}{2}+1} (n+1) (a x-1) \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )-\left ((n-2) (a x+1)^{\frac {n}{2}+1}\right )\right )}{a c (n-2) n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c x - c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{c -\frac {c}{a x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-\frac {c}{a\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a \int \frac {x e^{n \operatorname {atanh}{\left (a x \right )}}}{a x - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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