Optimal. Leaf size=75 \[ -\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)^3}{3 a^3}+\frac {\left (4 \sqrt {\frac {1-a x}{a x+1}}+3\right ) (a x+1)^2}{6 a^3}-\frac {x}{a^2} \]
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Rubi [A] time = 0.51, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6337, 1804, 1814, 12, 261} \[ -\frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)^3}{3 a^3}+\frac {\left (4 \sqrt {\frac {1-a x}{a x+1}}+3\right ) (a x+1)^2}{6 a^3}-\frac {x}{a^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 261
Rule 1804
Rule 1814
Rule 6337
Rubi steps
\begin {align*} \int e^{-\text {sech}^{-1}(a x)} x^2 \, dx &=\int \frac {x^2}{\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}} \, dx\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {(-1+x)^3 x (1+x)}{\left (1+x^2\right )^4} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^3}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3}{3 a^3}-\frac {2 \operatorname {Subst}\left (\int \frac {-4+6 x+12 x^2-6 x^3}{\left (1+x^2\right )^3} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{3 a^3}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3}{3 a^3}+\frac {(1+a x)^2 \left (3+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^3}+\frac {\operatorname {Subst}\left (\int \frac {24 x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^3}\\ &=-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3}{3 a^3}+\frac {(1+a x)^2 \left (3+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^3}+\frac {4 \operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )}{a^3}\\ &=-\frac {x}{a^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^3}{3 a^3}+\frac {(1+a x)^2 \left (3+4 \sqrt {\frac {1-a x}{1+a x}}\right )}{6 a^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 48, normalized size = 0.64 \[ \frac {3 a^2 x^2-2 (a x-1) \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2}{6 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 54, normalized size = 0.72 \[ \frac {3 \, a x^{2} - 2 \, {\left (a^{2} x^{3} - x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{6 \, a^{2}} \]
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 \[ \int \frac {x^{2}}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 269, normalized size = 3.59 \[ -\frac {\left (a x -1\right ) \left (3 a^{6} x^{6} \left (\frac {a x +1}{a x}\right )^{\frac {5}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {3}{2}}+3 x^{4} \ln \left (a^{2} x^{2}\right ) \left (\frac {a x +1}{a x}\right )^{\frac {5}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {3}{2}} a^{4}+3 \ln \left (a^{2} x^{2}\right ) \sqrt {-\frac {a x -1}{a x}}\, \left (\frac {a x +1}{a x}\right )^{\frac {5}{2}} x^{4} a^{4}-2 x^{7} a^{7}-3 x^{3} \ln \left (a^{2} x^{2}\right ) \left (\frac {a x +1}{a x}\right )^{\frac {5}{2}} \sqrt {-\frac {a x -1}{a x}}\, a^{3}-2 x^{6} a^{6}+6 x^{5} a^{5}+6 x^{4} a^{4}-6 x^{3} a^{3}-6 a^{2} x^{2}+2 a x +2\right )}{6 x^{5} a^{8} \left (\frac {a x +1}{a x}\right )^{\frac {5}{2}} \left (-\frac {a x -1}{a x}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {x^{2}}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.06, size = 57, normalized size = 0.76 \[ \frac {x^2}{2\,a}+\frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {x}{3\,a^2}+\frac {1}{3\,a^3}-\frac {x^3}{3}-\frac {x^2}{3\,a}\right )}{\sqrt {\frac {1}{a\,x}+1}} \]
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 \[ a \int \frac {x^{3}}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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