Optimal. Leaf size=74 \[ \frac {(d x)^{1+m} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac {2 b c (d x)^{3+m} \, _2F_1\left (1,\frac {3+m}{4};\frac {7+m}{4};c^2 x^4\right )}{d^3 (1+m) (3+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6049, 371}
\begin {gather*} \frac {(d x)^{m+1} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (m+1)}-\frac {2 b c (d x)^{m+3} \, _2F_1\left (1,\frac {m+3}{4};\frac {m+7}{4};c^2 x^4\right )}{d^3 (m+1) (m+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 6049
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac {(2 b c) \int \frac {x (d x)^{1+m}}{1-c^2 x^4} \, dx}{d (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac {(2 b c) \int \frac {(d x)^{2+m}}{1-c^2 x^4} \, dx}{d^2 (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d (1+m)}-\frac {2 b c (d x)^{3+m} \, _2F_1\left (1,\frac {3+m}{4};\frac {7+m}{4};c^2 x^4\right )}{d^3 (1+m) (3+m)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 64, normalized size = 0.86 \begin {gather*} -\frac {x (d x)^m \left (-\left ((3+m) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )\right )+2 b c x^2 \, _2F_1\left (1,\frac {3+m}{4};\frac {7+m}{4};c^2 x^4\right )\right )}{(1+m) (3+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \arctanh \left (c \,x^{2}\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (d x\right )^{m} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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