Optimal. Leaf size=74 \[ \frac {24 \sqrt {x^2-1}}{x}+\frac {\sqrt {x^2-1} \csc ^{-1}(x)^4}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}-\frac {12 \sqrt {x^2-1} \csc ^{-1}(x)^2}{x}+\frac {24 \csc ^{-1}(x)}{\sqrt {x^2}} \]
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Rubi [A] time = 0.18, antiderivative size = 101, normalized size of antiderivative = 1.36, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5243, 4677, 4619, 261} \[ \frac {24 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2}}{x}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}-\frac {12 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^2}{x}+\frac {24 \csc ^{-1}(x)}{\sqrt {x^2}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 4677
Rule 5243
Rubi steps
\begin {align*} \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx &=-\frac {\sqrt {x^2} \operatorname {Subst}\left (\int \frac {x \sin ^{-1}(x)^4}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{x}\\ &=\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}-\frac {\left (4 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \sin ^{-1}(x)^3 \, dx,x,\frac {1}{x}\right )}{x}\\ &=-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}+\frac {\left (12 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x \sin ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{x}\\ &=-\frac {12 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^2}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}+\frac {\left (24 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \sin ^{-1}(x) \, dx,x,\frac {1}{x}\right )}{x}\\ &=\frac {24 \csc ^{-1}(x)}{\sqrt {x^2}}-\frac {12 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^2}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}-\frac {\left (24 \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{x}\\ &=\frac {24 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2}}{x}+\frac {24 \csc ^{-1}(x)}{\sqrt {x^2}}-\frac {12 \sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^2}{x}-\frac {4 \csc ^{-1}(x)^3}{\sqrt {x^2}}+\frac {\sqrt {1-\frac {1}{x^2}} \sqrt {x^2} \csc ^{-1}(x)^4}{x}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 76, normalized size = 1.03 \[ \frac {24 \left (x^2-1\right )+\left (x^2-1\right ) \csc ^{-1}(x)^4-4 \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)^3-12 \left (x^2-1\right ) \csc ^{-1}(x)^2+24 \sqrt {1-\frac {1}{x^2}} x \csc ^{-1}(x)}{x \sqrt {x^2-1}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{-1}(x)^4}{x^2 \sqrt {-1+x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.19, size = 37, normalized size = 0.50 \[ -\frac {4 \, \operatorname {arccsc}\relax (x)^{3} - {\left (\operatorname {arccsc}\relax (x)^{4} - 12 \, \operatorname {arccsc}\relax (x)^{2} + 24\right )} \sqrt {x^{2} - 1} - 24 \, \operatorname {arccsc}\relax (x)}{x} \]
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 \[ \int \frac {\operatorname {arccsc}\relax (x)^{4}}{\sqrt {x^{2} - 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.66, size = 330, normalized size = 4.46
method | result | size |
default | \(\frac {\left (i x^{2}-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -2 i\right ) \mathrm {arccsc}\relax (x )^{3}}{\sqrt {x^{2}-1}\, x}+\frac {\left (x^{2}-2+2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right ) \mathrm {arccsc}\relax (x )^{4}}{4 \sqrt {x^{2}-1}\, x}-\frac {6 \left (i x^{2}-2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -2 i\right ) \mathrm {arccsc}\relax (x )}{\sqrt {x^{2}-1}\, x}-\frac {3 \left (x^{2}-2+2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \right ) \mathrm {arccsc}\relax (x )^{2}}{\sqrt {x^{2}-1}\, x}+\frac {6 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-24 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -18 x^{2}+24}{\sqrt {x^{2}-1}\, \left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -1\right ) x}+\frac {\left (-i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +x^{2}-1\right ) \left (3 \mathrm {arccsc}\relax (x )^{4}-4 i \mathrm {arccsc}\relax (x )^{3}-36 \mathrm {arccsc}\relax (x )^{2}+24 i \mathrm {arccsc}\relax (x )+72\right )}{4 \sqrt {x^{2}-1}\, x}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +1\right ) \left (\mathrm {arccsc}\relax (x )^{4}+4 i \mathrm {arccsc}\relax (x )^{3}-12 \mathrm {arccsc}\relax (x )^{2}-24 i \mathrm {arccsc}\relax (x )+24\right )}{4 \sqrt {x^{2}-1}\, x}\) | \(330\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 58, normalized size = 0.78 \[ \frac {\sqrt {x^{2} - 1} \operatorname {arccsc}\relax (x)^{4}}{x} - 12 \, \sqrt {-\frac {1}{x^{2}} + 1} \operatorname {arccsc}\relax (x)^{2} - \frac {4 \, \operatorname {arccsc}\relax (x)^{3}}{x} + 24 \, \sqrt {-\frac {1}{x^{2}} + 1} + \frac {24 \, \operatorname {arccsc}\relax (x)}{x} \]
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 \[ \int \frac {{\mathrm {asin}\left (\frac {1}{x}\right )}^4}{x^2\,\sqrt {x^2-1}} \,d x \]
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 \[ \int \frac {\operatorname {acsc}^{4}{\relax (x )}}{x^{2} \sqrt {\left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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