Optimal. Leaf size=67 \[ -\frac {a+b \tanh ^{-1}\left (c x^n\right )}{x}-\frac {b c n x^{n-1} \, _2F_1\left (1,-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );c^2 x^{2 n}\right )}{1-n} \]
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Rubi [A] time = 0.03, antiderivative size = 67, 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 = {6097, 364} \[ -\frac {a+b \tanh ^{-1}\left (c x^n\right )}{x}-\frac {b c n x^{n-1} \, _2F_1\left (1,-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );c^2 x^{2 n}\right )}{1-n} \]
Antiderivative was successfully verified.
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Rule 364
Rule 6097
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^n\right )}{x^2} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^n\right )}{x}+(b c n) \int \frac {x^{-2+n}}{1-c^2 x^{2 n}} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (c x^n\right )}{x}-\frac {b c n x^{-1+n} \, _2F_1\left (1,-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );c^2 x^{2 n}\right )}{1-n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 66, normalized size = 0.99 \[ -\frac {a}{x}+\frac {b c n x^{n-1} \, _2F_1\left (1,\frac {n-1}{2 n};\frac {n-1}{2 n}+1;c^2 x^{2 n}\right )}{n-1}-\frac {b \tanh ^{-1}\left (c x^n\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arctanh \left (c \,x^{n}\right )}{x^{2}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (n \int \frac {1}{c x^{2} x^{n} + x^{2}}\,{d x} + n \int \frac {1}{c x^{2} x^{n} - x^{2}}\,{d x} + \frac {\log \left (c x^{n} + 1\right ) - \log \left (-c x^{n} + 1\right )}{x}\right )} b - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^n\right )}{x^2} \,d x \]
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 \[ \int \frac {a + b \operatorname {atanh}{\left (c x^{n} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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