Optimal. Leaf size=136 \[ \frac{3 (1-x)}{16 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{1-x}{8 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{5/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{3 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{16 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0332407, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6345, 12, 51, 63, 206} \[ \frac{3 (1-x)}{16 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{3/2}}+\frac{1-x}{8 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} x^{5/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{3 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{16 \sqrt{\frac{1}{\sqrt{x}}-1} \sqrt{\frac{1}{\sqrt{x}}+1} \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6345
Rule 12
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{x^3} \, dx &=-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{\sqrt{1-x} \int \frac{1}{2 \sqrt{1-x} x^3} \, dx}{2 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x} x^3} \, dx}{4 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{8 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{\left (3 \sqrt{1-x}\right ) \int \frac{1}{\sqrt{1-x} x^2} \, dx}{16 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{8 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{3 (1-x)}{16 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{\left (3 \sqrt{1-x}\right ) \int \frac{1}{\sqrt{1-x} x} \, dx}{32 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{8 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{3 (1-x)}{16 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{\left (3 \sqrt{1-x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x}\right )}{16 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ &=\frac{1-x}{8 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{5/2}}+\frac{3 (1-x)}{16 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} x^{3/2}}-\frac{\text{sech}^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{3 \sqrt{1-x} \tanh ^{-1}\left (\sqrt{1-x}\right )}{16 \sqrt{-1+\frac{1}{\sqrt{x}}} \sqrt{1+\frac{1}{\sqrt{x}}} \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.112036, size = 125, normalized size = 0.92 \[ \frac{1}{16} \left (\frac{\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}} \left (3 x^{3/2}+3 x+2 \sqrt{x}+2\right )}{x^2}-\frac{8 \text{sech}^{-1}\left (\sqrt{x}\right )}{x^2}+3 \log \left (\sqrt{x} \sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+\sqrt{\frac{1-\sqrt{x}}{\sqrt{x}+1}}+1\right )-\frac{3 \log (x)}{2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.133, size = 79, normalized size = 0.6 \begin{align*} -{\frac{1}{2\,{x}^{2}}{\rm arcsech} \left (\sqrt{x}\right )}+{\frac{1}{16}\sqrt{-{ \left ( -1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}}\sqrt{{ \left ( 1+\sqrt{x} \right ){\frac{1}{\sqrt{x}}}}} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{1-x}}} \right ){x}^{2}+3\,\sqrt{1-x}x+2\,\sqrt{1-x} \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{1-x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.993, size = 124, normalized size = 0.91 \begin{align*} -\frac{3 \, x^{\frac{3}{2}}{\left (\frac{1}{x} - 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x} \sqrt{\frac{1}{x} - 1}}{16 \,{\left (x^{2}{\left (\frac{1}{x} - 1\right )}^{2} - 2 \, x{\left (\frac{1}{x} - 1\right )} + 1\right )}} - \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{2 \, x^{2}} + \frac{3}{32} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} + 1\right ) - \frac{3}{32} \, \log \left (\sqrt{x} \sqrt{\frac{1}{x} - 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.88476, size = 134, normalized size = 0.99 \begin{align*} \frac{{\left (3 \, x + 2\right )} \sqrt{x} \sqrt{-\frac{x - 1}{x}} +{\left (3 \, x^{2} - 8\right )} \log \left (\frac{x \sqrt{-\frac{x - 1}{x}} + \sqrt{x}}{x}\right )}{16 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asech}{\left (\sqrt{x} \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsech}\left (\sqrt{x}\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]