Optimal. Leaf size=224 \[ -\frac{\left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-3}}{a^2 c^4 x}-\frac{\left (n^2+8 n+14\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2}}{a c^4 (n+4) (n+6)}-\frac{\left (n^2+8 n+14\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{a c^4 (n+6) \left (n^2+6 n+8\right )}+\frac{(n+5) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-3}}{a c^4 (n+6)} \]
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Rubi [A] time = 0.261659, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6175, 6180, 90, 79, 45, 37} \[ -\frac{\left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-3}}{a^2 c^4 x}-\frac{\left (n^2+8 n+14\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2}}{a c^4 (n+4) (n+6)}-\frac{\left (n^2+8 n+14\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1}}{a c^4 (n+6) \left (n^2+6 n+8\right )}+\frac{(n+5) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-3}}{a c^4 (n+6)} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6180
Rule 90
Rule 79
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac{\int \frac{e^{n \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^4 x^4} \, dx}{a^4 c^4}\\ &=-\frac{\operatorname{Subst}\left (\int x^2 \left (1-\frac{x}{a}\right )^{-4-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{a^4 c^4}\\ &=-\frac{\left (1-\frac{1}{a x}\right )^{-3-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a^2 c^4 x}-\frac{\operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-4-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \left (-1-\frac{(4+n) x}{a}\right ) \, dx,x,\frac{1}{x}\right )}{a^2 c^4}\\ &=\frac{(5+n) \left (1-\frac{1}{a x}\right )^{-3-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^4 (6+n)}-\frac{\left (1-\frac{1}{a x}\right )^{-3-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a^2 c^4 x}-\frac{\left (14+8 n+n^2\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-3-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{a^2 c^4 (6+n)}\\ &=\frac{(5+n) \left (1-\frac{1}{a x}\right )^{-3-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^4 (6+n)}-\frac{\left (14+8 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^4 (4+n) (6+n)}-\frac{\left (1-\frac{1}{a x}\right )^{-3-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a^2 c^4 x}-\frac{\left (14+8 n+n^2\right ) \operatorname{Subst}\left (\int \left (1-\frac{x}{a}\right )^{-2-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{a^2 c^4 (4+n) (6+n)}\\ &=\frac{(5+n) \left (1-\frac{1}{a x}\right )^{-3-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^4 (6+n)}-\frac{\left (14+8 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^4 (4+n) (6+n)}-\frac{\left (14+8 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a c^4 (2+n) (4+n) (6+n)}-\frac{\left (1-\frac{1}{a x}\right )^{-3-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}}}{a^2 c^4 x}\\ \end{align*}
Mathematica [A] time = 0.274697, size = 83, normalized size = 0.37 \[ -\frac{e^{n \coth ^{-1}(a x)} \left (\cosh \left (4 \coth ^{-1}(a x)\right )+\sinh \left (4 \coth ^{-1}(a x)\right )\right ) \left ((n+4)^2 \cosh \left (2 \coth ^{-1}(a x)\right )-2 (n+4) \sinh \left (2 \coth ^{-1}(a x)\right )-n^2-8 n-12\right )}{2 a c^4 (n+2) (n+4) (n+6)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 68, normalized size = 0.3 \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{\rm arccoth} \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 = 1.79254, size = 485, normalized size = 2.17 \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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