Optimal. Leaf size=130 \[ \frac {c^2 2^{n/2} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},2-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)}+\frac {4 c^2 (a x+1)^{n/2} (1-a x)^{-n/2} \, _2F_1\left (2,\frac {n}{2};\frac {n+2}{2};\frac {a x+1}{1-a x}\right )}{a n} \]
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Rubi [C] time = 0.12, antiderivative size = 71, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6131, 6129, 136} \[ \frac {c^2 2^{3-\frac {n}{2}} (a x+1)^{\frac {n+2}{2}} F_1\left (\frac {n+2}{2};\frac {n-4}{2},2;\frac {n+4}{2};\frac {1}{2} (a x+1),a x+1\right )}{a (n+2)} \]
Warning: Unable to verify antiderivative.
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Rule 136
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{n \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \frac {(1-a x)^{2-\frac {n}{2}} (1+a x)^{n/2}}{x^2} \, dx}{a^2}\\ &=\frac {2^{3-\frac {n}{2}} c^2 (1+a x)^{\frac {2+n}{2}} F_1\left (\frac {2+n}{2};\frac {1}{2} (-4+n),2;\frac {4+n}{2};\frac {1}{2} (1+a x),1+a x\right )}{a (2+n)}\\ \end {align*}
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Mathematica [B] time = 0.50, size = 262, normalized size = 2.02 \[ -\frac {c^2 e^{n \tanh ^{-1}(a x)} \left (a n^2 x \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \tanh ^{-1}(a x)}\right )-2 a n x e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a x)}\right )+a (n-2) n x e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \tanh ^{-1}(a x)}\right )+2 a n x \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;-e^{2 \tanh ^{-1}(a x)}\right )-4 a n x e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (2,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a x)}\right )+4 a x \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;-e^{2 \tanh ^{-1}(a x)}\right )-4 a x \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \tanh ^{-1}(a x)}\right )+n^2+2 n\right )}{a^2 n (n+2) x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} x^{2}}, x\right ) \]
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 \[ \int {\left (c - \frac {c}{a x}\right )}^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (c -\frac {c}{a x}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c - \frac {c}{a x}\right )}^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-\frac {c}{a\,x}\right )}^2 \,d x \]
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 \[ \frac {c^{2} \left (\int a^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{2}}\, dx + \int \left (- \frac {2 a e^{n \operatorname {atanh}{\left (a x \right )}}}{x}\right )\, dx\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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