Optimal. Leaf size=135 \[ \frac{2 (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a c^4 (n+6) \left (n^2+6 n+8\right )}+\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-3}}{a c^4 (n+6)}+\frac{2 (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-2}}{a c^4 (n+4) (n+6)} \]
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Rubi [A] time = 0.0807233, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6129, 45, 37} \[ \frac{2 (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-1}}{a c^4 (n+6) \left (n^2+6 n+8\right )}+\frac{(a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-3}}{a c^4 (n+6)}+\frac{2 (a x+1)^{\frac{n+2}{2}} (1-a x)^{-\frac{n}{2}-2}}{a c^4 (n+4) (n+6)} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{n \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac{\int (1-a x)^{-4-\frac{n}{2}} (1+a x)^{n/2} \, dx}{c^4}\\ &=\frac{(1-a x)^{-3-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^4 (6+n)}+\frac{2 \int (1-a x)^{-3-\frac{n}{2}} (1+a x)^{n/2} \, dx}{c^4 (6+n)}\\ &=\frac{(1-a x)^{-3-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^4 (6+n)}+\frac{2 (1-a x)^{-2-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^4 (4+n) (6+n)}+\frac{2 \int (1-a x)^{-2-\frac{n}{2}} (1+a x)^{n/2} \, dx}{c^4 (4+n) (6+n)}\\ &=\frac{(1-a x)^{-3-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^4 (6+n)}+\frac{2 (1-a x)^{-2-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^4 (4+n) (6+n)}+\frac{2 (1-a x)^{-1-\frac{n}{2}} (1+a x)^{\frac{2+n}{2}}}{a c^4 (2+n) (4+n) (6+n)}\\ \end{align*}
Mathematica [A] time = 0.0438406, size = 74, normalized size = 0.55 \[ \frac{(1-a x)^{-\frac{n}{2}-3} (a x+1)^{\frac{n}{2}+1} \left (2 a^2 x^2-2 a n x-8 a x+n^2+8 n+14\right )}{a c^4 (n+2) (n+4) (n+6)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 68, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{a}^{2}{x}^{2}-2\,anx-8\,ax+{n}^{2}+8\,n+14 \right ) \left ( ax+1 \right ){{\rm e}^{n{\it Artanh} \left ( ax \right ) }}}{ \left ( ax-1 \right ) ^{3}{c}^{4}a \left ({n}^{2}+8\,n+12 \right ) \left ( 4+n \right ) }} \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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a c x - c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26367, size = 483, normalized size = 3.58 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} - 2 \,{\left (a^{2} n + 3 \, a^{2}\right )} x^{2} + n^{2} +{\left (a n^{2} + 6 \, a n + 6 \, a\right )} x + 8 \, n + 14\right )} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{a c^{4} n^{3} + 12 \, a c^{4} n^{2} + 44 \, a c^{4} n + 48 \, a c^{4} -{\left (a^{4} c^{4} n^{3} + 12 \, a^{4} c^{4} n^{2} + 44 \, a^{4} c^{4} n + 48 \, a^{4} c^{4}\right )} x^{3} + 3 \,{\left (a^{3} c^{4} n^{3} + 12 \, a^{3} c^{4} n^{2} + 44 \, a^{3} c^{4} n + 48 \, a^{3} c^{4}\right )} x^{2} - 3 \,{\left (a^{2} c^{4} n^{3} + 12 \, a^{2} c^{4} n^{2} + 44 \, a^{2} c^{4} n + 48 \, a^{2} c^{4}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}}{{\left (a c x - c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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