Optimal. Leaf size=75 \[ \frac {(d x)^{m+1} \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )}{d (m+1)}-\frac {2 b c d (d x)^{m-1} \, _2F_1\left (1,\frac {1-m}{4};\frac {5-m}{4};\frac {c^2}{x^4}\right )}{1-m^2} \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6097, 16, 339, 364} \[ \frac {(d x)^{m+1} \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )}{d (m+1)}-\frac {2 b c d (d x)^{m-1} \, _2F_1\left (1,\frac {1-m}{4};\frac {5-m}{4};\frac {c^2}{x^4}\right )}{1-m^2} \]
Antiderivative was successfully verified.
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Rule 16
Rule 339
Rule 364
Rule 6097
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right ) \, dx &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )}{d (1+m)}+\frac {(2 b c) \int \frac {(d x)^{1+m}}{\left (1-\frac {c^2}{x^4}\right ) x^3} \, dx}{d (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )}{d (1+m)}+\frac {\left (2 b c d^2\right ) \int \frac {(d x)^{-2+m}}{1-\frac {c^2}{x^4}} \, dx}{1+m}\\ &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )}{d (1+m)}-\frac {\left (2 b c d \left (\frac {1}{x}\right )^{-1+m} (d x)^{-1+m}\right ) \operatorname {Subst}\left (\int \frac {x^{-m}}{1-c^2 x^4} \, dx,x,\frac {1}{x}\right )}{1+m}\\ &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )}{d (1+m)}-\frac {2 b c d (d x)^{-1+m} \, _2F_1\left (1,\frac {1-m}{4};\frac {5-m}{4};\frac {c^2}{x^4}\right )}{1-m^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 0.91 \[ \frac {(d x)^m \left ((m-1) x^2 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )+2 b c \, _2F_1\left (1,\frac {1}{4}-\frac {m}{4};\frac {5}{4}-\frac {m}{4};\frac {c^2}{x^4}\right )\right )}{(m-1) (m+1) x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a\right )} \left (d x\right )^{m}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a\right )} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )\, dx \]
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 \[ \frac {1}{2} \, {\left (4 \, c d^{m} \int \frac {x^{2} x^{m}}{{\left (m + 1\right )} x^{4} - c^{2} {\left (m + 1\right )}}\,{d x} + \frac {d^{m} x x^{m} \log \left (x^{2} + c\right ) - d^{m} x x^{m} \log \left (x^{2} - c\right )}{m + 1}\right )} b + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \]
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 \[ \int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )\right ) \,d x \]
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 \[ \int \left (d x\right )^{m} \left (a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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