Optimal. Leaf size=72 \[ \frac {(d x)^{m+1} \left (a+b \tanh ^{-1}(c x)\right )}{d (m+1)}-\frac {b c (d x)^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{d^2 (m+1) (m+2)} \]
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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 = {5916, 364} \[ \frac {(d x)^{m+1} \left (a+b \tanh ^{-1}(c x)\right )}{d (m+1)}-\frac {b c (d x)^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{d^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 5916
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}(c x)\right )}{d (1+m)}-\frac {(b c) \int \frac {(d x)^{1+m}}{1-c^2 x^2} \, dx}{d (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}(c x)\right )}{d (1+m)}-\frac {b c (d x)^{2+m} \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{d^2 (1+m) (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 59, normalized size = 0.82 \[ -\frac {x (d x)^m \left (b c x \, _2F_1\left (1,\frac {m}{2}+1;\frac {m}{2}+2;c^2 x^2\right )-(m+2) \left (a+b \tanh ^{-1}(c x)\right )\right )}{(m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {artanh}\left (c x\right ) + a\right )} \left (d x\right )^{m}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.47, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctanh \left (c x \right )\right )\, dx \]
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 \[ \frac {1}{2} \, {\left (2 \, c d^{m} \int \frac {x x^{m}}{c^{2} {\left (m + 1\right )} x^{2} - m - 1}\,{d x} + \frac {d^{m} x x^{m} \log \left (c x + 1\right ) - d^{m} x x^{m} \log \left (-c x + 1\right )}{m + 1}\right )} b + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \]
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 \[ \int \left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d\,x\right )}^m \,d x \]
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 \[ \int \left (d x\right )^{m} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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