Optimal. Leaf size=15 \[ \text {Int}\left (\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2,x\right ) \]
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Rubi [A] time = 0.01, 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 = {} \[ \int \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx &=\int \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx\\ \end {align*}
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Mathematica [A] time = 1.97, size = 0, normalized size = 0.00 \[ \int \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \operatorname {artanh}\left (c x^{n}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x^{n}\right ) + a^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (a +b \arctanh \left (c \,x^{n}\right )\right )^{2}\, dx \]
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 \[ \frac {1}{4} \, b^{2} x \log \left (-c x^{n} + 1\right )^{2} + a^{2} x - \int -\frac {{\left (b^{2} c x^{n} - b^{2}\right )} \log \left (c x^{n} + 1\right )^{2} + 4 \, {\left (a b c x^{n} - a b\right )} \log \left (c x^{n} + 1\right ) + 2 \, {\left (2 \, a b - {\left (b^{2} c n + 2 \, a b c\right )} x^{n} - {\left (b^{2} c x^{n} - b^{2}\right )} \log \left (c x^{n} + 1\right )\right )} \log \left (-c x^{n} + 1\right )}{4 \, {\left (c x^{n} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.07 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atanh}{\left (c x^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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