Optimal. Leaf size=109 \[ -\frac {3 \sqrt {\frac {1}{a x^3+1}} \sqrt {a x^3+1} x^{m-2} \, _2F_1\left (\frac {1}{2},\frac {m-2}{6};\frac {m+4}{6};a^2 x^6\right )}{a \left (-m^2+m+2\right )}-\frac {3 x^{m-2}}{a \left (-m^2+m+2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^3\right )}}{m+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, 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 = {6335, 30, 259, 364} \[ -\frac {3 \sqrt {\frac {1}{a x^3+1}} \sqrt {a x^3+1} x^{m-2} \, _2F_1\left (\frac {1}{2},\frac {m-2}{6};\frac {m+4}{6};a^2 x^6\right )}{a \left (-m^2+m+2\right )}-\frac {3 x^{m-2}}{a \left (-m^2+m+2\right )}+\frac {x^{m+1} e^{\text {sech}^{-1}\left (a x^3\right )}}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 259
Rule 364
Rule 6335
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*}
________________________________________________________________________________________
Mathematica [A] time = 2.52, size = 159, normalized size = 1.46 \[ \frac {2^{\frac {m+1}{3}} x^{m+1} \left (a x^3\right )^{\frac {1}{3} (-m-1)} e^{\text {sech}^{-1}\left (a x^3\right )} \left (\frac {e^{\text {sech}^{-1}\left (a x^3\right )}}{e^{2 \text {sech}^{-1}\left (a x^3\right )}+1}\right )^{\frac {m+1}{3}} \left ((m+10) \, _2F_1\left (1,\frac {2-m}{6};\frac {m+10}{6};-e^{2 \text {sech}^{-1}\left (a x^3\right )}\right )-(m+4) e^{2 \text {sech}^{-1}\left (a x^3\right )} \, _2F_1\left (1,\frac {8-m}{6};\frac {m+16}{6};-e^{2 \text {sech}^{-1}\left (a x^3\right )}\right )\right )}{(m+4) (m+10)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 2.37, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{3} x^{m} \sqrt {\frac {a x^{3} + 1}{a x^{3}}} \sqrt {-\frac {a x^{3} - 1}{a x^{3}}} + x^{m}}{a x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} {\left (\sqrt {\frac {1}{a x^{3}} + 1} \sqrt {\frac {1}{a x^{3}} - 1} + \frac {1}{a x^{3}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (\frac {1}{x^{3} a}+\sqrt {\frac {1}{x^{3} a}-1}\, \sqrt {\frac {1}{x^{3} a}+1}\right ) x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,\left (\sqrt {\frac {1}{a\,x^3}-1}\,\sqrt {\frac {1}{a\,x^3}+1}+\frac {1}{a\,x^3}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{m}}{x^{3}}\, dx + \int a x^{m} \sqrt {-1 + \frac {1}{a x^{3}}} \sqrt {1 + \frac {1}{a x^{3}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________