Optimal. Leaf size=137 \[ \frac {4 c (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )}{a (2-n)}-\frac {c 2^{\frac {n}{2}+1} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (2-n)} \]
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Rubi [C] time = 0.10, antiderivative size = 70, normalized size of antiderivative = 0.51, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6157, 6150, 136} \[ -\frac {c 2^{2-\frac {n}{2}} (a x+1)^{\frac {n+4}{2}} F_1\left (\frac {n+4}{2};\frac {n-2}{2},2;\frac {n+6}{2};\frac {1}{2} (a x+1),a x+1\right )}{a (n+4)} \]
Warning: Unable to verify antiderivative.
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Rule 136
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\frac {c \int \frac {e^{n \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{1+\frac {n}{2}}}{x^2} \, dx}{a^2}\\ &=-\frac {2^{2-\frac {n}{2}} c (1+a x)^{\frac {4+n}{2}} F_1\left (\frac {4+n}{2};\frac {1}{2} (-2+n),2;\frac {6+n}{2};\frac {1}{2} (1+a x),1+a x\right )}{a (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 126, normalized size = 0.92 \[ \frac {c e^{n \tanh ^{-1}(a x)} \left (a n x e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \tanh ^{-1}(a x)}\right )+a (n+2) x \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \tanh ^{-1}(a x)}\right )+4 a x e^{2 \tanh ^{-1}(a x)} \, _2F_1\left (2,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a x)}\right )+n+2\right )}{a^2 (n+2) x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.37, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c x^{2} - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} x^{2}}, x\right ) \]
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 \[ \int {\left (c - \frac {c}{a^{2} x^{2}}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (c -\frac {c}{a^{2} x^{2}}\right )\, dx \]
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 \[ \int {\left (c - \frac {c}{a^{2} x^{2}}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (c-\frac {c}{a^2\,x^2}\right ) \,d x \]
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 \[ \frac {c \left (\int a^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{2}}\right )\, dx\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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