Optimal. Leaf size=42 \[ -\frac {1}{6 x^{3/2}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2 \sqrt {x}}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6098, 51, 63, 206} \[ -\frac {1}{6 x^{3/2}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2 \sqrt {x}}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 206
Rule 6098
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx &=-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{(1-x) x^{5/2}} \, dx\\ &=-\frac {1}{6 x^{3/2}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{(1-x) x^{3/2}} \, dx\\ &=-\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x}} \, dx\\ &=-\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 58, normalized size = 1.38 \[ -\frac {1}{6 x^{3/2}}-\frac {\coth ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2 \sqrt {x}}-\frac {1}{4} \log \left (1-\sqrt {x}\right )+\frac {1}{4} \log \left (\sqrt {x}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 38, normalized size = 0.90 \[ \frac {3 \, {\left (x^{2} - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) - 2 \, {\left (3 \, x + 1\right )} \sqrt {x}}{12 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 37, normalized size = 0.88 \[ -\frac {\mathrm {arccoth}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {1}{6 x^{\frac {3}{2}}}-\frac {1}{2 \sqrt {x}}-\frac {\ln \left (-1+\sqrt {x}\right )}{4}+\frac {\ln \left (1+\sqrt {x}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 36, normalized size = 0.86 \[ -\frac {3 \, x + 1}{6 \, x^{\frac {3}{2}}} - \frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{2 \, x^{2}} + \frac {1}{4} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{4} \, \log \left (\sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.49, size = 45, normalized size = 1.07 \[ \frac {\ln \left (1-\frac {1}{\sqrt {x}}\right )}{4\,x^2}-\frac {\frac {x}{2}+\frac {1}{6}}{x^{3/2}}-\frac {\ln \left (\frac {1}{\sqrt {x}}+1\right )}{4\,x^2}-\frac {\mathrm {atan}\left (\sqrt {x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 5.45, size = 160, normalized size = 3.81 \[ \frac {3 x^{\frac {7}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} - \frac {3 x^{\frac {5}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} - \frac {3 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} + \frac {3 \sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} - \frac {3 x^{3}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} + \frac {2 x^{2}}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} + \frac {x}{6 x^{\frac {7}{2}} - 6 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________