Optimal. Leaf size=42 \[ -\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^{n+1}}{b c (n+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2512, 2302, 30} \[ -\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^{n+1}}{b c (n+1)} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rule 2512
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^n}{1-c^2 x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a+b \log (x))^n}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {\operatorname {Subst}\left (\int x^n \, dx,x,a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c}\\ &=-\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^{1+n}}{b c (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 1.00 \[ -\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^{n+1}}{b c (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 56, normalized size = 1.33 \[ -\frac {{\left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )} {\left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{b c n + b c} \]
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 \[ \int -\frac {{\left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{c^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )+a \right )^{n}}{-c^{2} x^{2}+1}\, dx \]
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 \[ -\int \frac {{\left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{n}}{c^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {{\left (a+b\,\ln \left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^n}{c^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 133.77, size = 94, normalized size = 2.24 \[ \begin {cases} - \frac {a^{n} \operatorname {atan}{\left (\frac {x}{\sqrt {- \frac {1}{c^{2}}}} \right )}}{c^{2} \sqrt {- \frac {1}{c^{2}}}} & \text {for}\: b = 0 \\a^{n} x & \text {for}\: c = 0 \\- \frac {\begin {cases} \frac {\left (a + b \log {\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b \log {\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} \right )} & \text {otherwise} \end {cases}}{b c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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