Optimal. Leaf size=72 \[ -\frac {a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3}-\frac {b c n x^{n-3} \, _2F_1\left (1,-\frac {3-n}{2 n};-\frac {3 (1-n)}{2 n};c^2 x^{2 n}\right )}{3 (3-n)} \]
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Rubi [A] time = 0.03, antiderivative size = 72, 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} \[ -\frac {a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3}-\frac {b c n x^{n-3} \, _2F_1\left (1,-\frac {3-n}{2 n};-\frac {3 (1-n)}{2 n};c^2 x^{2 n}\right )}{3 (3-n)} \]
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^4} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3}+\frac {1}{3} (b c n) \int \frac {x^{-4+n}}{1-c^2 x^{2 n}} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3}-\frac {b c n x^{-3+n} \, _2F_1\left (1,-\frac {3-n}{2 n};-\frac {3 (1-n)}{2 n};c^2 x^{2 n}\right )}{3 (3-n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 1.01 \[ -\frac {a}{3 x^3}+\frac {b c n x^{n-3} \, _2F_1\left (1,\frac {n-3}{2 n};\frac {n-3}{2 n}+1;c^2 x^{2 n}\right )}{3 (n-3)}-\frac {b \tanh ^{-1}\left (c x^n\right )}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{4}}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arctanh \left (c \,x^{n}\right )}{x^{4}}\, dx \]
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 \[ -\frac {1}{6} \, {\left (3 \, n \int \frac {1}{3 \, {\left (c x^{4} x^{n} + x^{4}\right )}}\,{d x} + 3 \, n \int \frac {1}{3 \, {\left (c x^{4} x^{n} - x^{4}\right )}}\,{d x} + \frac {\log \left (c x^{n} + 1\right ) - \log \left (-c x^{n} + 1\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \]
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 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^n\right )}{x^4} \,d x \]
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 \[ \int \frac {a + b \operatorname {atanh}{\left (c x^{n} \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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