Optimal. Leaf size=104 \[ \frac {\left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{a c^3 (n+4)}-\frac {(n+3) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{a c^3 (n+2) (n+4)} \]
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Rubi [A] time = 0.15, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6175, 6180, 79, 37} \[ \frac {\left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{a c^3 (n+4)}-\frac {(n+3) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{a c^3 (n+2) (n+4)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 79
Rule 6175
Rule 6180
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\frac {\int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^3 x^3} \, dx}{a^3 c^3}\\ &=\frac {\operatorname {Subst}\left (\int x \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^3 c^3}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^3 (4+n)}-\frac {(3+n) \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^3 (4+n)}\\ &=\frac {\left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^3 (4+n)}-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^3 (2+n) (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 64, normalized size = 0.62 \[ \frac {(-a x+n+3) e^{n \coth ^{-1}(a x)} \left (\cosh \left (3 \coth ^{-1}(a x)\right )+\sinh \left (3 \coth ^{-1}(a x)\right )\right )}{a^2 c^3 (n+2) (n+4) x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 125, normalized size = 1.20 \[ -\frac {{\left (a^{2} x^{2} + {\left (a n - 2 \, a\right )} x + n - 3\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a c^{3} n^{2} - 6 \, a c^{3} n + 8 \, a c^{3} + {\left (a^{3} c^{3} n^{2} - 6 \, a^{3} c^{3} n + 8 \, a^{3} c^{3}\right )} x^{2} - 2 \, {\left (a^{2} c^{3} n^{2} - 6 \, a^{2} c^{3} n + 8 \, a^{2} c^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 0.44 \[ -\frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (a x -n -3\right ) \left (a x +1\right )}{\left (a x -1\right )^{2} c^{3} \left (n^{2}+6 n +8\right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 113, normalized size = 1.09 \[ \frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {n+3}{a^3\,c^3\,\left (n^2+6\,n+8\right )}-\frac {x^2}{a\,c^3\,\left (n^2+6\,n+8\right )}+\frac {x\,\left (n+2\right )}{a^2\,c^3\,\left (n^2+6\,n+8\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^2}-\frac {2\,x}{a}+x^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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