Optimal. Leaf size=278 \[ \frac {2 \left (n^2+14 n+56\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a^2 (n+6) \left (n^2+6 n+8\right ) x}-\frac {\left (n^2+14 n+56\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a (n+4) (n+6)}+\frac {(n+8) x \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{n+6}-\frac {x \left (a-\frac {1}{x}\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a} \]
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Rubi [A] time = 0.27, 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 = {6176, 6181, 90, 79, 45, 37} \[ \frac {2 \left (n^2+14 n+56\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a^2 (n+6) \left (n^2+6 n+8\right ) x}-\frac {\left (n^2+14 n+56\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a (n+4) (n+6)}+\frac {(n+8) x \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{n+6}-\frac {x \left (a-\frac {1}{x}\right ) \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 79
Rule 90
Rule 6176
Rule 6181
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.08, size = 116, normalized size = 0.42 \[ \frac {2 c^2 (a x+1) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \left (2 n \left (3 a^2 x^2-10 a x+7\right )+8 \left (a^2 x^2-4 a x+7\right )+n^2 (a x-1)^2\right ) (c-a c x)^{n/2}}{a (n+2) (n+4) (n+6)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 104, normalized size = 0.37 \[ \frac {2 \left (a x +1\right ) \left (a^{2} n^{2} x^{2}+6 a^{2} n \,x^{2}+8 a^{2} x^{2}-2 a \,n^{2} x -20 a n x -32 a x +n^{2}+14 n +56\right ) {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )^{2+\frac {n}{2}}}{\left (a x -1\right )^{2} a \left (n^{3}+12 n^{2}+44 n +48\right )} \]
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 \[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {x^3\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+12\,n+16\right )}{n^3+12\,n^2+44\,n+48}+\frac {{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+28\,n+112\right )}{a^3\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {2\,x\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (n^2+6\,n-24\right )}{a^2\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {x^2\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+28\,n+48\right )}{a\,\left (n^3+12\,n^2+44\,n+48\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^2}-\frac {2\,x}{a}+x^2\right )} \]
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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