Optimal. Leaf size=109 \[ -\frac {3 x^{-2+m}}{a \left (2+m-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}-\frac {3 x^{-2+m} \sqrt {\frac {1}{1+a x^3}} \sqrt {1+a x^3} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-2+m);\frac {4+m}{6};a^2 x^6\right )}{a \left (2+m-m^2\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6470, 30, 265,
371} \begin {gather*} -\frac {3 \sqrt {\frac {1}{a x^3+1}} \sqrt {a x^3+1} x^{m-2} \, _2F_1\left (\frac {1}{2},\frac {m-2}{6};\frac {m+4}{6};a^2 x^6\right )}{a \left (-m^2+m+2\right )}-\frac {3 x^{m-2}}{a \left (-m^2+m+2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^3\right )}}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 265
Rule 371
Rule 6470
Rubi steps
\begin {align*} \int e^{\text {sech}^{-1}\left (a x^3\right )} x^m \, dx &=\frac {e^{\text {sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}+\frac {3 \int x^{-3+m} \, dx}{a (1+m)}+\frac {\left (3 \sqrt {\frac {1}{1+a x^3}} \sqrt {1+a x^3}\right ) \int \frac {x^{-3+m}}{\sqrt {1-a x^3} \sqrt {1+a x^3}} \, dx}{a (1+m)}\\ &=-\frac {3 x^{-2+m}}{a \left (2+m-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}+\frac {\left (3 \sqrt {\frac {1}{1+a x^3}} \sqrt {1+a x^3}\right ) \int \frac {x^{-3+m}}{\sqrt {1-a^2 x^6}} \, dx}{a (1+m)}\\ &=-\frac {3 x^{-2+m}}{a \left (2+m-m^2\right )}+\frac {e^{\text {sech}^{-1}\left (a x^3\right )} x^{1+m}}{1+m}-\frac {3 x^{-2+m} \sqrt {\frac {1}{1+a x^3}} \sqrt {1+a x^3} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-2+m);\frac {4+m}{6};a^2 x^6\right )}{a \left (2+m-m^2\right )}\\ \end {align*}
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Mathematica [A]
time = 1.81, size = 214, normalized size = 1.96 \begin {gather*} \frac {2^{\frac {1+m}{3}} e^{-\frac {1}{3} (1+m) \text {sech}^{-1}\left (a x^3\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^3\right )}}{1+e^{2 \text {sech}^{-1}\left (a x^3\right )}}\right )^{\frac {1+m}{3}} \left (1+e^{2 \text {sech}^{-1}\left (a x^3\right )}\right )^{\frac {1+m}{3}} x^{1+m} \left (a x^3\right )^{\frac {1}{3} (-1-m)} \left (e^{\frac {1}{3} (4+m) \text {sech}^{-1}\left (a x^3\right )} (10+m) \, _2F_1\left (\frac {4+m}{6},\frac {4+m}{3};\frac {10+m}{6};-e^{2 \text {sech}^{-1}\left (a x^3\right )}\right )-e^{\frac {1}{3} (10+m) \text {sech}^{-1}\left (a x^3\right )} (4+m) \, _2F_1\left (\frac {4+m}{3},\frac {10+m}{6};\frac {16+m}{6};-e^{2 \text {sech}^{-1}\left (a x^3\right )}\right )\right )}{(4+m) (10+m)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (\frac {1}{a \,x^{3}}+\sqrt {\frac {1}{a \,x^{3}}-1}\, \sqrt {\frac {1}{a \,x^{3}}+1}\right ) x^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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*} \frac {\int \frac {x^{m}}{x^{3}}\, dx + \int a x^{m} \sqrt {-1 + \frac {1}{a x^{3}}} \sqrt {1 + \frac {1}{a x^{3}}}\, dx}{a} \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 x^m\,\left (\sqrt {\frac {1}{a\,x^3}-1}\,\sqrt {\frac {1}{a\,x^3}+1}+\frac {1}{a\,x^3}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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