Optimal. Leaf size=376 \[ -\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \text{Hypergeometric2F1}\left (\frac{2-n}{2},1-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}-\frac{(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-\frac{n}{2}-1}}{a^5 c^2 \left (4-n^2\right )}-\frac{(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-n/2}}{a^5 c^2 n \left (4-n^2\right )}-\frac{x^3 (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^2 c^2}+\frac{(1-n) (n+3) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^5 c^2 (2-n)}+\frac{(n+3) x (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^4 c^2}+\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{a^5 c^2 (2-n)}-\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{a^5 c^2} \]
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Rubi [A] time = 0.38817, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {6150, 100, 159, 89, 79, 69, 90, 45, 37} \[ -\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (\frac{2-n}{2},1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}-\frac{(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-\frac{n}{2}-1}}{a^5 c^2 \left (4-n^2\right )}-\frac{(n+3) \left (2-n^2\right ) (a x+1)^{n/2} (1-a x)^{-n/2}}{a^5 c^2 n \left (4-n^2\right )}-\frac{x^3 (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^2 c^2}+\frac{(1-n) (n+3) (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^5 c^2 (2-n)}+\frac{(n+3) x (a x+1)^{\frac{n-2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a^4 c^2}+\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{1-\frac{n}{2}}}{a^5 c^2 (2-n)}-\frac{(a x+1)^{\frac{n-2}{2}} (1-a x)^{-n/2}}{a^5 c^2} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 100
Rule 159
Rule 89
Rule 79
Rule 69
Rule 90
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{\int x^4 (1-a x)^{-2-\frac{n}{2}} (1+a x)^{-2+\frac{n}{2}} \, dx}{c^2}\\ &=-\frac{x^3 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^2 c^2}-\frac{\int x^2 (1-a x)^{-2-\frac{n}{2}} (1+a x)^{-2+\frac{n}{2}} (-3-a n x) \, dx}{a^2 c^2}\\ &=-\frac{x^3 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^2 c^2}-\frac{n \int x^2 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{-2+\frac{n}{2}} \, dx}{a^2 c^2}+\frac{(3+n) \int x^2 (1-a x)^{-2-\frac{n}{2}} (1+a x)^{-2+\frac{n}{2}} \, dx}{a^2 c^2}\\ &=\frac{(3+n) x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^4 c^2}-\frac{x^3 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^2 c^2}-\frac{(1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2}+\frac{\int (1-a x)^{-n/2} (1+a x)^{-2+\frac{n}{2}} \left (-a (1-n)+a^2 n x\right ) \, dx}{a^5 c^2}+\frac{(3+n) \int (1-a x)^{-2-\frac{n}{2}} (1+a x)^{-2+\frac{n}{2}} (-1-a n x) \, dx}{a^4 c^2}\\ &=\frac{(1-n) (3+n) (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac{(3+n) x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^4 c^2}-\frac{x^3 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^2 c^2}+\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac{(1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2}+\frac{n \int (1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)} \, dx}{a^4 c^2}-\frac{\left ((3+n) \left (2-n^2\right )\right ) \int (1-a x)^{-2-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)} \, dx}{a^4 c^2 (2-n)}\\ &=\frac{(1-n) (3+n) (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac{(3+n) x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^4 c^2}-\frac{x^3 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^2 c^2}+\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac{(1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2}-\frac{(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac{n}{2}} (1+a x)^{n/2}}{a^5 c^2 \left (4-n^2\right )}-\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (\frac{2-n}{2},1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}-\frac{\left ((3+n) \left (2-n^2\right )\right ) \int (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)} \, dx}{a^4 c^2 \left (4-n^2\right )}\\ &=\frac{(1-n) (3+n) (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2 (2-n)}+\frac{(3+n) x (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^4 c^2}-\frac{x^3 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^2 c^2}+\frac{(1-a x)^{1-\frac{n}{2}} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2 (2-n)}-\frac{(1-a x)^{-n/2} (1+a x)^{\frac{1}{2} (-2+n)}}{a^5 c^2}-\frac{(3+n) \left (2-n^2\right ) (1-a x)^{-1-\frac{n}{2}} (1+a x)^{n/2}}{a^5 c^2 \left (4-n^2\right )}-\frac{(3+n) \left (2-n^2\right ) (1-a x)^{-n/2} (1+a x)^{n/2}}{a^5 c^2 n \left (4-n^2\right )}-\frac{2^{n/2} n (1-a x)^{1-\frac{n}{2}} \, _2F_1\left (\frac{2-n}{2},1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-a x)\right )}{a^5 c^2 (2-n)}\\ \end{align*}
Mathematica [A] time = 0.156389, size = 178, normalized size = 0.47 \[ -\frac{(1-a x)^{-\frac{n}{2}-1} \left ((a x+1)^{n/2} \left (n^2 \left (1-2 a^2 x^2\right )+n \left (-4 a^3 x^3+4 a^2 x^2+6 a x-4\right )+6 a^2 x^2+n^3 (a x-1)^2 (a x+1)-6\right )-2^{n/2} n^2 (n+2) (a x-1)^2 (a x+1) \text{Hypergeometric2F1}\left (1-\frac{n}{2},1-\frac{n}{2},2-\frac{n}{2},\frac{1}{2} (1-a x)\right )\right )}{a^5 c^2 (n-2) n (n+2) (a x+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n{\it Artanh} \left ( ax \right ) }}{x}^{4}}{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{2}}}\, 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}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{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^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{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{\int \frac{x^{4} e^{n \operatorname{atanh}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{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 \frac{x^{4} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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