Optimal. Leaf size=21 \[ \text {Int}\left (\frac {(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2},x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2} \, dx &=\int \frac {(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{b^{2} \operatorname {artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x^{2}\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (c^{2} d^{m} x^{4} - d^{m}\right )} x^{m}}{b^{2} c x \log \left (c x^{2} + 1\right ) - b^{2} c x \log \left (-c x^{2} + 1\right ) + 2 \, a b c x} + \int -\frac {{\left (c^{2} d^{m} {\left (m + 3\right )} x^{4} - d^{m} {\left (m - 1\right )}\right )} x^{m}}{b^{2} c x^{2} \log \left (c x^{2} + 1\right ) - b^{2} c x^{2} \log \left (-c x^{2} + 1\right ) + 2 \, a b c x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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