Optimal. Leaf size=104 \[ \frac {\left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^3 (4+n)}-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^3 (2+n) (4+n)} \]
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Rubi [A]
time = 0.10, 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 = {6310, 6315, 80,
37} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 80
Rule 6310
Rule 6315
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 {\text {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) \text {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 = 0.16, size = 64, normalized size = 0.62 \begin {gather*} \frac {e^{n \coth ^{-1}(a x)} (3+n-a x) \left (\cosh \left (3 \coth ^{-1}(a x)\right )+\sinh \left (3 \coth ^{-1}(a x)\right )\right )}{a^2 c^3 (2+n) (4+n) \sqrt {1-\frac {1}{a^2 x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 46, normalized size = 0.44
method | result | size |
gosper | \(-\frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (a x -n -3\right ) \left (a x +1\right )}{\left (a x -1\right )^{2} c^{3} \left (n^{2}+6 n +8\right ) a}\) | \(46\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 128, normalized size = 1.23 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 52.64, size = 1112, normalized size = 10.69 \begin {gather*} \begin {cases} \frac {x e^{\frac {i \pi n}{2}}}{c^{3}} & \text {for}\: a = 0 \\\frac {a^{2} x^{2} \operatorname {acoth}{\left (a x \right )}}{2 a^{3} c^{3} x^{2} e^{4 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{4 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{4 \operatorname {acoth}{\left (a x \right )}}} + \frac {2 a x \operatorname {acoth}{\left (a x \right )}}{2 a^{3} c^{3} x^{2} e^{4 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{4 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{4 \operatorname {acoth}{\left (a x \right )}}} - \frac {a x}{2 a^{3} c^{3} x^{2} e^{4 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{4 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{4 \operatorname {acoth}{\left (a x \right )}}} + \frac {\operatorname {acoth}{\left (a x \right )}}{2 a^{3} c^{3} x^{2} e^{4 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{4 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{4 \operatorname {acoth}{\left (a x \right )}}} - \frac {1}{2 a^{3} c^{3} x^{2} e^{4 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{4 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{4 \operatorname {acoth}{\left (a x \right )}}} & \text {for}\: n = -4 \\- \frac {a^{2} x^{2} \operatorname {acoth}{\left (a x \right )}}{2 a^{3} c^{3} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{2 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{2 \operatorname {acoth}{\left (a x \right )}}} + \frac {a x}{2 a^{3} c^{3} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{2 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{2 \operatorname {acoth}{\left (a x \right )}}} + \frac {\operatorname {acoth}{\left (a x \right )}}{2 a^{3} c^{3} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{2 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{2 \operatorname {acoth}{\left (a x \right )}}} + \frac {1}{2 a^{3} c^{3} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a^{2} c^{3} x e^{2 \operatorname {acoth}{\left (a x \right )}} + 2 a c^{3} e^{2 \operatorname {acoth}{\left (a x \right )}}} & \text {for}\: n = -2 \\- \frac {a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} + \frac {a n x e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} + \frac {2 a x e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} + \frac {n e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} + \frac {3 e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{3} n^{2} x^{2} + 6 a^{3} c^{3} n x^{2} + 8 a^{3} c^{3} x^{2} - 2 a^{2} c^{3} n^{2} x - 12 a^{2} c^{3} n x - 16 a^{2} c^{3} x + a c^{3} n^{2} + 6 a c^{3} n + 8 a c^{3}} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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