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 = 0.26, antiderivative size = 224, normalized size of antiderivative = 1.00, 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 37
Rule 45
Rule 79
Rule 90
Rule 6175
Rule 6180
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*}
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Mathematica [A] time = 0.31, 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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fricas [A] time = 0.91, size = 229, normalized size = 1.02 \[ -\frac {{\left (2 \, a^{3} x^{3} + 2 \, {\left (a^{2} n - 3 \, a^{2}\right )} x^{2} + n^{2} + {\left (a n^{2} - 6 \, a n + 6 \, a\right )} x - 8 \, n + 14\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a c^{4} n^{3} - 12 \, a c^{4} n^{2} + 44 \, a c^{4} n - 48 \, a c^{4} - {\left (a^{4} c^{4} n^{3} - 12 \, a^{4} c^{4} n^{2} + 44 \, a^{4} c^{4} n - 48 \, a^{4} c^{4}\right )} x^{3} + 3 \, {\left (a^{3} c^{4} n^{3} - 12 \, a^{3} c^{4} n^{2} + 44 \, a^{3} c^{4} n - 48 \, a^{3} c^{4}\right )} x^{2} - 3 \, {\left (a^{2} c^{4} n^{3} - 12 \, a^{2} c^{4} n^{2} + 44 \, a^{2} c^{4} n - 48 \, a^{2} c^{4}\right )} x} \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 68, normalized size = 0.30 \[ -\frac {\left (a x +1\right ) \left (2 a^{2} x^{2}-2 a n x -8 a x +n^{2}+8 n +14\right ) {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\left (a x -1\right )^{3} c^{4} a \left (n^{2}+8 n +12\right ) \left (4+n \right )} \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 180, normalized size = 0.80 \[ -\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {2\,x^3}{a\,c^4\,\left (n^3+12\,n^2+44\,n+48\right )}+\frac {n^2+8\,n+14}{a^4\,c^4\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {x^2\,\left (2\,n+6\right )}{a^2\,c^4\,\left (n^3+12\,n^2+44\,n+48\right )}+\frac {x\,\left (n^2+6\,n+6\right )}{a^3\,c^4\,\left (n^3+12\,n^2+44\,n+48\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {3\,x}{a^2}-\frac {1}{a^3}+x^3-\frac {3\,x^2}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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