Optimal. Leaf size=137 \[ \frac{4 c (1-a x)^{1-\frac{n}{2}} (a x+1)^{\frac{n-2}{2}} \text{Hypergeometric2F1}\left (2,1-\frac{n}{2},2-\frac{n}{2},\frac{1-a x}{a x+1}\right )}{a (2-n)}-\frac{c 2^{\frac{n}{2}+1} (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a (2-n)} \]
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Rubi [C] time = 0.0968612, antiderivative size = 70, normalized size of antiderivative = 0.51, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, 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 6157
Rule 6150
Rule 136
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*}
Mathematica [A] time = 0.229702, size = 126, normalized size = 0.92 \[ \frac{c e^{n \tanh ^{-1}(a x)} \left (a n x e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,\frac{n}{2}+1,\frac{n}{2}+2,e^{2 \tanh ^{-1}(a x)}\right )+a (n+2) x \text{Hypergeometric2F1}\left (1,\frac{n}{2},\frac{n}{2}+1,e^{2 \tanh ^{-1}(a x)}\right )+4 a x e^{2 \tanh ^{-1}(a x)} \text{Hypergeometric2F1}\left (2,\frac{n}{2}+1,\frac{n}{2}+2,-e^{2 \tanh ^{-1}(a x)}\right )+n+2\right )}{a^2 (n+2) x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c \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\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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