Optimal. Leaf size=63 \[ \frac {1}{4} x^4 \csc ^{-1}(x)^2+\frac {x^2}{12}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {\log (x)}{3} \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5223, 3758, 4185, 4184, 3475} \[ \frac {x^2}{12}+\frac {1}{4} x^4 \csc ^{-1}(x)^2+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {\log (x)}{3} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3758
Rule 4184
Rule 4185
Rule 5223
Rubi steps
\begin {align*} \int x^3 \csc ^{-1}(x)^2 \, dx &=-\operatorname {Subst}\left (\int x^2 \cot (x) \csc ^4(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac {1}{4} x^4 \csc ^{-1}(x)^2-\frac {1}{2} \operatorname {Subst}\left (\int x \csc ^4(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac {x^2}{12}+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{4} x^4 \csc ^{-1}(x)^2-\frac {1}{3} \operatorname {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac {x^2}{12}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{4} x^4 \csc ^{-1}(x)^2-\frac {1}{3} \operatorname {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(x)\right )\\ &=\frac {x^2}{12}+\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)+\frac {1}{6} \sqrt {1-\frac {1}{x^2}} x^3 \csc ^{-1}(x)+\frac {1}{4} x^4 \csc ^{-1}(x)^2+\frac {\log (x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 42, normalized size = 0.67 \[ \frac {1}{12} \left (3 x^4 \csc ^{-1}(x)^2+x^2+2 \sqrt {1-\frac {1}{x^2}} \left (x^2+2\right ) x \csc ^{-1}(x)+4 \log (x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^3 \csc ^{-1}(x)^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.85, size = 35, normalized size = 0.56 \[ \frac {1}{4} \, x^{4} \operatorname {arccsc}\relax (x)^{2} + \frac {1}{6} \, {\left (x^{2} + 2\right )} \sqrt {x^{2} - 1} \operatorname {arccsc}\relax (x) + \frac {1}{12} \, x^{2} + \frac {1}{3} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.96, size = 106, normalized size = 1.68 \[ \frac {1}{4} \, x^{4} \arcsin \left (\frac {1}{x}\right )^{2} + \frac {1}{12} \, x^{2} {\left (\frac {2}{x^{2}} + 1\right )} + \frac {1}{48} \, {\left (x^{3} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{3} + 9 \, x {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )} - \frac {9 \, x^{2} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{2} + 1}{x^{3} {\left (\sqrt {-\frac {1}{x^{2}} + 1} - 1\right )}^{3}}\right )} \arcsin \left (\frac {1}{x}\right ) - \frac {1}{6} \, \log \left (\frac {1}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 56, normalized size = 0.89
method | result | size |
default | \(\frac {x^{4} \mathrm {arccsc}\relax (x )^{2}}{4}+\frac {x^{3} \mathrm {arccsc}\relax (x ) \sqrt {\frac {x^{2}-1}{x^{2}}}}{6}+\frac {x^{2}}{12}+\frac {\mathrm {arccsc}\relax (x ) \sqrt {\frac {x^{2}-1}{x^{2}}}\, x}{3}-\frac {\ln \left (\frac {1}{x}\right )}{3}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 95, normalized size = 1.51 \[ \frac {1}{4} \, x^{4} \operatorname {arccsc}\relax (x)^{2} + \frac {2 \, x^{4} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) + 2 \, x^{2} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) + {\left (x^{2} + 2 \, \log \left (x^{2}\right )\right )} \sqrt {x + 1} \sqrt {x - 1} - 4 \, \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right )}{12 \, \sqrt {x + 1} \sqrt {x - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,{\mathrm {asin}\left (\frac {1}{x}\right )}^2 \,d x \]
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 \[ \int x^{3} \operatorname {acsc}^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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