Integrand size = 40, antiderivative size = 40 \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\text {Int}\left (\frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx \\ \end{align*}
Not integrable
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx \]
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Not integrable
Time = 0.65 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90
\[\int \frac {\left (a +b \,\operatorname {arccoth}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )^{n}}{-c^{2} x^{2}+1}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{c^{2} x^{2} - 1} \,d x } \]
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Not integrable
Time = 8.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=- \int \frac {\left (a + b \operatorname {acoth}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}\right )^{n}}{c^{2} x^{2} - 1}\, dx \]
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Not integrable
Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{c^{2} x^{2} - 1} \,d x } \]
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Not integrable
Time = 0.73 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \operatorname {arcoth}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{c^{2} x^{2} - 1} \,d x } \]
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Not integrable
Time = 4.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \coth ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx=-\int \frac {{\left (a+b\,\mathrm {acoth}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^n}{c^2\,x^2-1} \,d x \]
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