Optimal. Leaf size=278 \[ -\frac {\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a (4+n) (6+n)}+\frac {2 \left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a^2 (6+n) \left (8+6 n+n^2\right ) x}+\frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a} \]
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Rubi [A]
time = 0.18, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 92,
80, 47, 37} \begin {gather*} \frac {2 \left (n^2+14 n+56\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a^2 (n+6) \left (n^2+6 n+8\right ) x}-\frac {\left (n^2+14 n+56\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a (n+4) (n+6)}+\frac {(n+8) x \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{n+6}-\frac {x \left (a-\frac {1}{x}\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 80
Rule 92
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx &=\left (\left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} x^{-2-\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{2+\frac {n}{2}} x^{2+\frac {n}{2}} \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (\frac {1}{x}\right )^{2+\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-4-\frac {n}{2}} \left (1-\frac {x}{a}\right )^2 \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a}+\left (a \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (\frac {1}{x}\right )^{2+\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-4-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \left (-\frac {8+n}{2 a}+\frac {(4+n) x}{2 a^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a}+\frac {\left (\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (\frac {1}{x}\right )^{2+\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-3-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{2 a (6+n)}\\ &=-\frac {\left (56+14 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}} (c-a c x)^{\frac {4+n}{2}}}{a (4+n) (6+n)}+\frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a}-\frac {\left (\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (\frac {1}{x}\right )^{2+\frac {n}{2}} (c-a c x)^{2+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{a^2 (4+n) (6+n)}\\ &=-\frac {\left (56+14 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}} (c-a c x)^{\frac {4+n}{2}}}{a (4+n) (6+n)}+\frac {2 \left (56+14 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}} (c-a c x)^{\frac {4+n}{2}}}{a^2 (2+n) (4+n) (6+n) x}+\frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 116, normalized size = 0.42 \begin {gather*} \frac {2 c^2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} (1+a x) (c-a c x)^{n/2} \left (n^2 (-1+a x)^2+8 \left (7-4 a x+a^2 x^2\right )+2 n \left (7-10 a x+3 a^2 x^2\right )\right )}{a (2+n) (4+n) (6+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 104, normalized size = 0.37
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (a^{2} n^{2} x^{2}+6 a^{2} n \,x^{2}+8 a^{2} x^{2}-2 a \,n^{2} x -20 a n x -32 a x +n^{2}+14 n +56\right ) {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )^{2+\frac {n}{2}}}{\left (a x -1\right )^{2} a \left (n^{3}+12 n^{2}+44 n +48\right )}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 122, normalized size = 0.44 \begin {gather*} \frac {2 \, {\left ({\left (n^{2} + 6 \, n + 8\right )} a^{3} \left (-c\right )^{\frac {1}{2} \, n} c^{2} x^{3} - {\left (n^{2} + 14 \, n + 24\right )} a^{2} \left (-c\right )^{\frac {1}{2} \, n} c^{2} x^{2} - {\left (n^{2} + 6 \, n - 24\right )} a \left (-c\right )^{\frac {1}{2} \, n} c^{2} x + {\left (n^{2} + 14 \, n + 56\right )} \left (-c\right )^{\frac {1}{2} \, n} c^{2}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{{\left (n^{3} + 12 \, n^{2} + 44 \, n + 48\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 185, normalized size = 0.67 \begin {gather*} \frac {2 \, {\left ({\left (a^{3} n^{2} + 6 \, a^{3} n + 8 \, a^{3}\right )} x^{3} - {\left (a^{2} n^{2} + 14 \, a^{2} n + 24 \, a^{2}\right )} x^{2} + n^{2} - {\left (a n^{2} + 6 \, a n - 24 \, a\right )} x + 14 \, n + 56\right )} {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a n^{3} + 12 \, a n^{2} + {\left (a^{3} n^{3} + 12 \, a^{3} n^{2} + 44 \, a^{3} n + 48 \, a^{3}\right )} x^{2} + 44 \, a n - 2 \, {\left (a^{2} n^{3} + 12 \, a^{2} n^{2} + 44 \, a^{2} n + 48 \, a^{2}\right )} x + 48 \, a} \end {gather*}
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 \begin {gather*} \begin {cases} c^{\frac {n}{2} + 2} x e^{\frac {i \pi n}{2}} & \text {for}\: a = 0 \\- \frac {\int \frac {1}{a x e^{6 \operatorname {acoth}{\left (a x \right )}} - e^{6 \operatorname {acoth}{\left (a x \right )}}}\, dx}{c} & \text {for}\: n = -6 \\\int e^{- 4 \operatorname {acoth}{\left (a x \right )}}\, dx & \text {for}\: n = -4 \\- c \left (\int a x e^{- 2 \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{- 2 \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) & \text {for}\: n = -2 \\\frac {2 a^{3} c^{2} n^{2} x^{3} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} + \frac {12 a^{3} c^{2} n x^{3} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} + \frac {16 a^{3} c^{2} x^{3} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} - \frac {2 a^{2} c^{2} n^{2} x^{2} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} - \frac {28 a^{2} c^{2} n x^{2} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} - \frac {48 a^{2} c^{2} x^{2} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} - \frac {2 a c^{2} n^{2} x \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} - \frac {12 a c^{2} n x \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} + \frac {48 a c^{2} x \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} + \frac {2 c^{2} n^{2} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} + \frac {28 c^{2} n \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} + \frac {112 c^{2} \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n^{3} + 12 a n^{2} + 44 a n + 48 a} & \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 = 1.79, size = 223, normalized size = 0.80 \begin {gather*} \frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {x^3\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+12\,n+16\right )}{n^3+12\,n^2+44\,n+48}+\frac {{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+28\,n+112\right )}{a^3\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {2\,x\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (n^2+6\,n-24\right )}{a^2\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {x^2\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+28\,n+48\right )}{a\,\left (n^3+12\,n^2+44\,n+48\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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