Optimal. Leaf size=106 \[ \frac {3+2 x^4}{12 x \sqrt {x^2}}-\frac {5 \sqrt {-1+x^2} \csc ^{-1}(x)}{2 x^2}-\frac {5 \left (-1+x^2\right )^{3/2} \csc ^{-1}(x)}{3 x^2}+\frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{3 x^2}-\frac {5 x \csc ^{-1}(x)^2}{4 \sqrt {x^2}}-\frac {7 x \log (x)}{3 \sqrt {x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 133, normalized size of antiderivative = 1.25, number of steps
used = 11, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {5351, 4785,
4741, 4737, 30, 14, 272, 45} \begin {gather*} \frac {x \sqrt {x^2}}{6}-\frac {7 \sqrt {x^2} \log (x)}{3 x}+\frac {1}{3} \left (x^2\right )^{3/2} \left (1-\frac {1}{x^2}\right )^{5/2} \csc ^{-1}(x)-\frac {5}{3} \sqrt {x^2} \left (1-\frac {1}{x^2}\right )^{3/2} \csc ^{-1}(x)-\frac {5 \sqrt {1-\frac {1}{x^2}} \csc ^{-1}(x)}{2 \sqrt {x^2}}-\frac {5 \sqrt {x^2} \csc ^{-1}(x)^2}{4 x}+\frac {\sqrt {x^2}}{4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 45
Rule 272
Rule 4737
Rule 4741
Rule 4785
Rule 5351
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right )^{5/2} \csc ^{-1}(x)}{x^3} \, dx &=-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {\left (1-x^2\right )^{5/2} \sin ^{-1}(x)}{x^4} \, dx,x,\frac {1}{x}\right )}{x}\\ &=\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^3} \, dx,x,\frac {1}{x}\right )}{3 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \sin ^{-1}(x)}{x^2} \, dx,x,\frac {1}{x}\right )}{3 x}\\ &=-\frac {5}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \csc ^{-1}(x)+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {(1-x)^2}{x^2} \, dx,x,\frac {1}{x^2}\right )}{6 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,\frac {1}{x}\right )}{3 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x) \, dx,x,\frac {1}{x}\right )}{x}\\ &=-\frac {5 \sqrt {1-\frac {1}{x^2}} \csc ^{-1}(x)}{2 \sqrt {x^2}}-\frac {5}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \csc ^{-1}(x)+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac {\sqrt {x^2} \text {Subst}\left (\int \left (1+\frac {1}{x^2}-\frac {2}{x}\right ) \, dx,x,\frac {1}{x^2}\right )}{6 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,\frac {1}{x}\right )}{3 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int x \, dx,x,\frac {1}{x}\right )}{2 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{2 x}\\ &=\frac {\sqrt {x^2}}{4 x^3}+\frac {x \sqrt {x^2}}{6}-\frac {5 \sqrt {1-\frac {1}{x^2}} \csc ^{-1}(x)}{2 \sqrt {x^2}}-\frac {5}{3} \left (1-\frac {1}{x^2}\right )^{3/2} \sqrt {x^2} \csc ^{-1}(x)+\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{5/2} \left (x^2\right )^{3/2} \csc ^{-1}(x)-\frac {5 \sqrt {x^2} \csc ^{-1}(x)^2}{4 x}-\frac {7 \sqrt {x^2} \log (x)}{3 x}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 86, normalized size = 0.81 \begin {gather*} \frac {\sqrt {-1+x^2} \left (4 x^2-30 \csc ^{-1}(x)^2-3 \cos \left (2 \csc ^{-1}(x)\right )+48 \log \left (\frac {1}{x}\right )-8 \log (x)+\csc ^{-1}(x) \left (8 \sqrt {1-\frac {1}{x^2}} x \left (-7+x^2\right )-6 \sin \left (2 \csc ^{-1}(x)\right )\right )\right )}{24 \sqrt {1-\frac {1}{x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.41, size = 305, normalized size = 2.88
method | result | size |
default | \(-\frac {5 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \mathrm {arccsc}\left (x \right )^{2}}{4 \sqrt {x^{2}-1}}+\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -2 x^{2}+2\right ) \left (2 \,\mathrm {arccsc}\left (x \right )+i\right )}{16 x^{2} \sqrt {x^{2}-1}}-\frac {\left (i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-2 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +2 x^{2}-2\right ) \left (-i+2 \,\mathrm {arccsc}\left (x \right )\right )}{16 x^{2} \sqrt {x^{2}-1}}-\frac {14 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \,\mathrm {arccsc}\left (x \right )}{3 \sqrt {x^{2}-1}}+\frac {\left (x^{4}+7 i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -8 x^{2}+7\right ) \left (2 \,\mathrm {arccsc}\left (x \right ) x^{4}+\sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-30 \,\mathrm {arccsc}\left (x \right ) x^{2}-7 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x -7 i+126 \,\mathrm {arccsc}\left (x \right )\right )}{6 \sqrt {x^{2}-1}\, \left (x^{4}-15 x^{2}+63\right )}+\frac {7 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \ln \left (\left (\frac {i}{x}+\sqrt {1-\frac {1}{x^{2}}}\right )^{2}-1\right )}{3 \sqrt {x^{2}-1}}\) | \(305\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.99, size = 51, normalized size = 0.48 \begin {gather*} \frac {2 \, x^{4} - 15 \, x^{2} \operatorname {arccsc}\left (x\right )^{2} - 28 \, x^{2} \log \left (x\right ) + 2 \, {\left (2 \, x^{4} - 14 \, x^{2} - 3\right )} \sqrt {x^{2} - 1} \operatorname {arccsc}\left (x\right ) + 3}{12 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {asin}\left (\frac {1}{x}\right )\,{\left (x^2-1\right )}^{5/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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