Optimal. Leaf size=36 \[ a \log (x)-\frac {b \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b \text {Li}_2\left (c x^n\right )}{2 n} \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6095, 5912} \[ -\frac {b \text {PolyLog}\left (2,-c x^n\right )}{2 n}+\frac {b \text {PolyLog}\left (2,c x^n\right )}{2 n}+a \log (x) \]
Antiderivative was successfully verified.
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Rule 5912
Rule 6095
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^n\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^n\right )}{n}\\ &=a \log (x)-\frac {b \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b \text {Li}_2\left (c x^n\right )}{2 n}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 39, normalized size = 1.08 \[ \frac {b c x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};c^2 x^{2 n}\right )}{n}+a \log (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 141, normalized size = 3.92 \[ -\frac {b n \log \left (c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right ) + 1\right ) \log \relax (x) - b n \log \left (-c \cosh \left (n \log \relax (x)\right ) - c \sinh \left (n \log \relax (x)\right ) + 1\right ) \log \relax (x) - b n \log \relax (x) \log \left (-\frac {c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right ) + 1}{c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right ) - 1}\right ) - 2 \, a n \log \relax (x) - b {\rm Li}_2\left (c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right )\right ) + b {\rm Li}_2\left (-c \cosh \left (n \log \relax (x)\right ) - c \sinh \left (n \log \relax (x)\right )\right )}{2 \, n} \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 76, normalized size = 2.11 \[ \frac {a \ln \left (c \,x^{n}\right )}{n}+\frac {b \ln \left (c \,x^{n}\right ) \arctanh \left (c \,x^{n}\right )}{n}-\frac {b \dilog \left (c \,x^{n}\right )}{2 n}-\frac {b \dilog \left (c \,x^{n}+1\right )}{2 n}-\frac {b \ln \left (c \,x^{n}\right ) \ln \left (c \,x^{n}+1\right )}{2 n} \]
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 \[ \frac {1}{2} \, {\left (n \int \frac {\log \relax (x)}{c x x^{n} + x}\,{d x} + n \int \frac {\log \relax (x)}{c x x^{n} - x}\,{d x} + \log \left (c x^{n} + 1\right ) \log \relax (x) - \log \left (-c x^{n} + 1\right ) \log \relax (x)\right )} b + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^n\right )}{x} \,d x \]
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 \[ \int \frac {a + b \operatorname {atanh}{\left (c x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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