Optimal. Leaf size=209 \[ \frac{2^{\frac{n}{2}-1} n \left (n^2+8\right ) (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (1-\frac{n}{2},1-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{3 a^5 c (2-n)}+\frac{(a x+1)^{n/2} \left (-a \left (n^2+6\right ) n x+n^3+n^2+8 n+6\right ) (1-a x)^{-n/2}}{6 a^5 c n}-\frac{x^3 (a x+1)^{n/2} (1-a x)^{-n/2}}{3 a^2 c}-\frac{n x^2 (a x+1)^{n/2} (1-a x)^{-n/2}}{6 a^3 c} \]
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Rubi [A] time = 0.257294, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6150, 100, 153, 143, 69} \[ \frac{2^{\frac{n}{2}-1} n \left (n^2+8\right ) (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{3 a^5 c (2-n)}+\frac{(a x+1)^{n/2} \left (-a \left (n^2+6\right ) n x+n^3+n^2+8 n+6\right ) (1-a x)^{-n/2}}{6 a^5 c n}-\frac{x^3 (a x+1)^{n/2} (1-a x)^{-n/2}}{3 a^2 c}-\frac{n x^2 (a x+1)^{n/2} (1-a x)^{-n/2}}{6 a^3 c} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 100
Rule 153
Rule 143
Rule 69
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)} x^4}{c-a^2 c x^2} \, dx &=\frac{\int x^4 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}} \, dx}{c}\\ &=-\frac{x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}-\frac{\int x^2 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}} (-3-a n x) \, dx}{3 a^2 c}\\ &=-\frac{n x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^3 c}-\frac{x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}+\frac{\int x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{-1+\frac{n}{2}} \left (2 a n+a^2 \left (6+n^2\right ) x\right ) \, dx}{6 a^4 c}\\ &=-\frac{n x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^3 c}-\frac{x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}+\frac{(1-a x)^{-n/2} (1+a x)^{n/2} \left (6+8 n+n^2+n^3-a n \left (6+n^2\right ) x\right )}{6 a^5 c n}-\frac{\left (n \left (8+n^2\right )\right ) \int (1-a x)^{-n/2} (1+a x)^{-1+\frac{n}{2}} \, dx}{6 a^4 c}\\ &=-\frac{n x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^3 c}-\frac{x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}+\frac{(1-a x)^{-n/2} (1+a x)^{n/2} \left (6+8 n+n^2+n^3-a n \left (6+n^2\right ) x\right )}{6 a^5 c n}+\frac{2^{-1+\frac{n}{2}} n \left (8+n^2\right ) (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (1-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{3 a^5 c (2-n)}\\ \end{align*}
Mathematica [A] time = 0.202656, size = 172, normalized size = 0.82 \[ \frac{(1-a x)^{-n/2} \left (2^{\frac{n}{2}+1} n \left (n^2-2 n+2\right ) (a x-1) \text{Hypergeometric2F1}\left (1-\frac{n}{2},-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )-(n-2) \left ((a x+1)^{n/2} \left (2 n \left (a^3 x^3+2 a x+3\right )+(a n x+n)^2-6\right )-2^{\frac{n}{2}+2} n (n+3) \text{Hypergeometric2F1}\left (-\frac{n}{2},-\frac{n}{2},1-\frac{n}{2},\frac{1}{2} (1-a x)\right )\right )\right )}{6 a^5 c (n-2) n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.203, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{4}}{-{a}^{2}c{x}^{2}+c}}\, 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 \frac{x^{4} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}\,{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{x^{4} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}, 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{\int \frac{x^{4} e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \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 -\frac{x^{4} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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