Optimal. Leaf size=51 \[ -2 \sqrt {1-x^2}+\tanh ^{-1}\left (\sqrt {1-x^2}\right )-x \sin ^{-1}(x) (1-\log (x))+\sqrt {1-x^2} \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {4715, 267, 2434,
272, 52, 65, 212} \begin {gather*} -x \text {ArcSin}(x)+x \text {ArcSin}(x) \log (x)-2 \sqrt {1-x^2}+\sqrt {1-x^2} \log (x)+\tanh ^{-1}\left (\sqrt {1-x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 212
Rule 267
Rule 272
Rule 2434
Rule 4715
Rubi steps
\begin {align*} \int \sin ^{-1}(x) \log (x) \, dx &=\sqrt {1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)-\int \left (\frac {\sqrt {1-x^2}}{x}+\sin ^{-1}(x)\right ) \, dx\\ &=\sqrt {1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)-\int \frac {\sqrt {1-x^2}}{x} \, dx-\int \sin ^{-1}(x) \, dx\\ &=-x \sin ^{-1}(x)+\sqrt {1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)-\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1-x}}{x} \, dx,x,x^2\right )+\int \frac {x}{\sqrt {1-x^2}} \, dx\\ &=-2 \sqrt {1-x^2}-x \sin ^{-1}(x)+\sqrt {1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=-2 \sqrt {1-x^2}-x \sin ^{-1}(x)+\sqrt {1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-2 \sqrt {1-x^2}-x \sin ^{-1}(x)+\tanh ^{-1}\left (\sqrt {1-x^2}\right )+\sqrt {1-x^2} \log (x)+x \sin ^{-1}(x) \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 52, normalized size = 1.02 \begin {gather*} -2 \sqrt {1-x^2}+x \sin ^{-1}(x) (-1+\log (x))+\left (-1+\sqrt {1-x^2}\right ) \log (x)+\log \left (1+\sqrt {1-x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(92\) vs.
\(2(45)=90\).
time = 0.03, size = 93, normalized size = 1.82
method | result | size |
default | \(\frac {2 \arcsin \left (x \right ) \tan \left (\frac {\arcsin \left (x \right )}{2}\right ) \ln \left (\frac {2 \tan \left (\frac {\arcsin \left (x \right )}{2}\right )}{1+\tan ^{2}\left (\frac {\arcsin \left (x \right )}{2}\right )}\right )-2 \left (\tan ^{2}\left (\frac {\arcsin \left (x \right )}{2}\right )\right ) \ln \left (\frac {2 \tan \left (\frac {\arcsin \left (x \right )}{2}\right )}{1+\tan ^{2}\left (\frac {\arcsin \left (x \right )}{2}\right )}\right )-2 \arcsin \left (x \right ) \tan \left (\frac {\arcsin \left (x \right )}{2}\right )-4}{1+\tan ^{2}\left (\frac {\arcsin \left (x \right )}{2}\right )}-\ln \left (1+\tan ^{2}\left (\frac {\arcsin \left (x \right )}{2}\right )\right )\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 4.01, size = 58, normalized size = 1.14 \begin {gather*} {\left (x \log \left (x\right ) - x\right )} \arcsin \left (x\right ) + \sqrt {-x^{2} + 1} \log \left (x\right ) - 2 \, \sqrt {-x^{2} + 1} + \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.66, size = 54, normalized size = 1.06 \begin {gather*} x \arcsin \left (x\right ) \log \left (x\right ) - x \arcsin \left (x\right ) + \sqrt {-x^{2} + 1} {\left (\log \left (x\right ) - 2\right )} + \frac {1}{2} \, \log \left (\sqrt {-x^{2} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {-x^{2} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 3.98, size = 102, normalized size = 2.00 \begin {gather*} x \log {\left (x \right )} \operatorname {asin}{\left (x \right )} - x \operatorname {asin}{\left (x \right )} + \sqrt {1 - x^{2}} \log {\left (x \right )} - \sqrt {1 - x^{2}} - \begin {cases} - \frac {x}{\sqrt {-1 + \frac {1}{x^{2}}}} - \operatorname {acosh}{\left (\frac {1}{x} \right )} + \frac {1}{x \sqrt {-1 + \frac {1}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {i x}{\sqrt {1 - \frac {1}{x^{2}}}} + i \operatorname {asin}{\left (\frac {1}{x} \right )} - \frac {i}{x \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 272 vs.
\(2 (42) = 84\).
time = 0.51, size = 272, normalized size = 5.33 \begin {gather*} x \arcsin \left (x\right ) \log \left (x\right ) + \sqrt {-x^{2} + 1} \log \left (x\right ) - \frac {2 \, x \arcsin \left (x\right )}{{\left (\sqrt {-x^{2} + 1} + 1\right )} {\left (\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} + \frac {x^{2} \log \left (\sqrt {-x^{2} + 1} + 1\right )}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2} {\left (\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} + \frac {\log \left (\sqrt {-x^{2} + 1} + 1\right )}{\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1} - \frac {x^{2} \log \left ({\left | x \right |}\right )}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2} {\left (\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} - \frac {\log \left ({\left | x \right |}\right )}{\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1} + \frac {2 \, x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2} {\left (\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1\right )}} - \frac {2}{\frac {x^{2}}{{\left (\sqrt {-x^{2} + 1} + 1\right )}^{2}} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {asin}\left (x\right )\,\ln \left (x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________