Optimal. Leaf size=21 \[ \text {Int}\left ((d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^3,x\right ) \]
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Rubi [A] time = 0.02, 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 = {} \[ \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^3 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^3 \, dx &=\int (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^3 \, dx\\ \end {align*}
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Mathematica [A] time = 8.42, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^3 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} b^{3} \operatorname {artanh}\left (c x^{n}\right )^{3} + 3 \, \left (d x\right )^{m} a b^{2} \operatorname {artanh}\left (c x^{n}\right )^{2} + 3 \, \left (d x\right )^{m} a^{2} b \operatorname {artanh}\left (c x^{n}\right ) + \left (d x\right )^{m} a^{3}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )}^{3} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctanh \left (c \,x^{n}\right )\right )^{3}\, dx \]
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 \[ -\frac {b^{3} d^{m} x x^{m} \log \left (-c x^{n} + 1\right )^{3}}{8 \, {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} + \int \frac {{\left (b^{3} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - b^{3} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )^{3} + 6 \, {\left (a b^{2} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - a b^{2} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )^{2} - 3 \, {\left (2 \, a b^{2} d^{m} {\left (m + 1\right )} x^{m} - {\left (2 \, a b^{2} c d^{m} {\left (m + 1\right )} + b^{3} c d^{m} n\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - {\left (b^{3} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - b^{3} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )\right )} \log \left (-c x^{n} + 1\right )^{2} + 12 \, {\left (a^{2} b c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - a^{2} b d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right ) - 3 \, {\left (4 \, a^{2} b c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - 4 \, a^{2} b d^{m} {\left (m + 1\right )} x^{m} + {\left (b^{3} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - b^{3} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )^{2} + 4 \, {\left (a b^{2} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - a b^{2} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )\right )} \log \left (-c x^{n} + 1\right )}{8 \, {\left (c {\left (m + 1\right )} x^{n} - m - 1\right )}}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right )}^3 \,d x \]
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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