Optimal. Leaf size=187 \[ -\frac {2 c (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1,\frac {n-2}{2};\frac {n}{2};\frac {a x+1}{1-a x}\right )}{a (2-n)}+\frac {c 2^{n/2} (1-n) (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (\frac {2-n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (2-n) (4-n)}+\frac {c (a x+1)^{\frac {n-2}{2}} (1-a x)^{2-\frac {n}{2}}}{a (2-n)} \]
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Rubi [A] time = 0.13, antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6131, 6129, 105, 69, 131} \[ -\frac {c 2^{\frac {n}{2}+1} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (2-n)}-\frac {2 c (a x+1)^{n/2} (1-a x)^{-n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {1-a x}{a x+1}\right )}{a n}+\frac {c 2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a n} \]
Warning: Unable to verify antiderivative.
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Rule 69
Rule 105
Rule 131
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\frac {c \int \frac {e^{n \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac {c \int \frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{n/2}}{x} \, dx}{a}\\ &=c \int (1-a x)^{-n/2} (1+a x)^{n/2} \, dx-\frac {c \int \frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{x} \, dx}{a}\\ &=-\frac {2^{1+\frac {n}{2}} c (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 (2-n)}+c \int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2} \, dx-\frac {c \int \frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2}}{x} \, dx}{a}\\ &=-\frac {2 c (1-a x)^{-n/2} (1+a x)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {1-a x}{1+a x}\right )}{a n}-\frac {2^{1+\frac {n}{2}} c (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 (2-n)}+\frac {2^{1+\frac {n}{2}} c (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 n}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 180, normalized size = 0.96 \[ \frac {c e^{n \tanh ^{-1}(a x)} \left (e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a x)}\right )+e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \tanh ^{-1}(a x)}\right )-\frac {(n+2) \left (\, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;-e^{2 \tanh ^{-1}(a x)}\right )-\, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \tanh ^{-1}(a x)}\right )\right )}{n}+4 e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (2,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a x)}\right )\right )}{a (n+2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c x - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a x}, x\right ) \]
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 \[ \int {\left (c - \frac {c}{a x}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (c -\frac {c}{a x}\right )\, dx \]
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 \[ \int {\left (c - \frac {c}{a x}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (c-\frac {c}{a\,x}\right ) \,d x \]
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 \[ \frac {c \left (\int a e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x}\right )\, dx\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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