Optimal. Leaf size=52 \[ \frac {x^2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {2 x}{a^2}+\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{a^3}+\frac {x^3}{3} \]
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Rubi [A] time = 0.24, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6338, 6742, 266, 47, 63, 208} \[ \frac {x^2 \sqrt {\frac {1}{a^2 x^2}+1}}{a}+\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{a^3}+\frac {2 x}{a^2}+\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 6338
Rule 6742
Rubi steps
\begin {align*} \int e^{2 \text {csch}^{-1}(a x)} x^2 \, dx &=\int \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2 x^2 \, dx\\ &=\int \left (\frac {2}{a^2}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}} x}{a}+x^2\right ) \, dx\\ &=\frac {2 x}{a^2}+\frac {x^3}{3}+\frac {2 \int \sqrt {1+\frac {1}{a^2 x^2}} x \, dx}{a}\\ &=\frac {2 x}{a^2}+\frac {x^3}{3}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x^2} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=\frac {2 x}{a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{a}+\frac {x^3}{3}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a^3}\\ &=\frac {2 x}{a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{a}+\frac {x^3}{3}-\frac {\operatorname {Subst}\left (\int \frac {1}{-a^2+a^2 x^2} \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}\right )}{a}\\ &=\frac {2 x}{a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{a}+\frac {x^3}{3}+\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 1.10 \[ \frac {a x \left (a^2 x^2+3 a x \sqrt {\frac {1}{a^2 x^2}+1}+6\right )+3 \log \left (x \left (\sqrt {\frac {1}{a^2 x^2}+1}+1\right )\right )}{3 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.77, size = 72, normalized size = 1.38 \[ \frac {a^{3} x^{3} + 3 \, a^{2} x^{2} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 6 \, a x - 3 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 90, normalized size = 1.73 \[ \frac {x^{3}}{3}+\frac {2 x}{a^{2}}+\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (x \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right )\right )}{a^{3} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 89, normalized size = 1.71 \[ \frac {1}{3} \, x^{3} + \frac {\frac {2 \, \sqrt {\frac {1}{a^{2} x^{2}} + 1}}{a^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} - a^{2}} + \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{a^{2}} - \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a^{2}}}{2 \, a} + \frac {2 \, x}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 51, normalized size = 0.98 \[ \frac {2\,x}{a^2}+\frac {x^3}{3}+\frac {x^2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{a}-\frac {\mathrm {atan}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.33, size = 36, normalized size = 0.69 \[ \frac {x^{3}}{3} + \frac {x \sqrt {a^{2} x^{2} + 1}}{a^{2}} + \frac {2 x}{a^{2}} + \frac {\operatorname {asinh}{\left (a x \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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