Optimal. Leaf size=83 \[ \frac{8}{7} \left (\sqrt{\sqrt{x}+1}+2\right )^{7/2}-\frac{48}{5} \left (\sqrt{\sqrt{x}+1}+2\right )^{5/2}+\frac{88}{3} \left (\sqrt{\sqrt{x}+1}+2\right )^{3/2}-48 \sqrt{\sqrt{\sqrt{x}+1}+2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0465592, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {371, 1398, 772} \[ \frac{8}{7} \left (\sqrt{\sqrt{x}+1}+2\right )^{7/2}-\frac{48}{5} \left (\sqrt{\sqrt{x}+1}+2\right )^{5/2}+\frac{88}{3} \left (\sqrt{\sqrt{x}+1}+2\right )^{3/2}-48 \sqrt{\sqrt{\sqrt{x}+1}+2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 1398
Rule 772
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+\sqrt{1+\sqrt{x}}}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{2+\sqrt{1+x}}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{-1+x}{\sqrt{2+\sqrt{x}}} \, dx,x,1+\sqrt{x}\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x \left (-1+x^2\right )}{\sqrt{2+x}} \, dx,x,\sqrt{1+\sqrt{x}}\right )\\ &=4 \operatorname{Subst}\left (\int \left (-\frac{6}{\sqrt{2+x}}+11 \sqrt{2+x}-6 (2+x)^{3/2}+(2+x)^{5/2}\right ) \, dx,x,\sqrt{1+\sqrt{x}}\right )\\ &=-48 \sqrt{2+\sqrt{1+\sqrt{x}}}+\frac{88}{3} \left (2+\sqrt{1+\sqrt{x}}\right )^{3/2}-\frac{48}{5} \left (2+\sqrt{1+\sqrt{x}}\right )^{5/2}+\frac{8}{7} \left (2+\sqrt{1+\sqrt{x}}\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0204362, size = 58, normalized size = 0.7 \[ \frac{8}{105} \sqrt{\sqrt{\sqrt{x}+1}+2} \left (3 \sqrt{x} \left (5 \sqrt{\sqrt{x}+1}-12\right )+76 \sqrt{\sqrt{x}+1}-280\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0., size = 54, normalized size = 0.7 \begin{align*}{\frac{88}{3} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{3}{2}}}}-{\frac{48}{5} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{5}{2}}}}+{\frac{8}{7} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{7}{2}}}}-48\,\sqrt{2+\sqrt{1+\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01124, size = 72, normalized size = 0.87 \begin{align*} \frac{8}{7} \,{\left (\sqrt{\sqrt{x} + 1} + 2\right )}^{\frac{7}{2}} - \frac{48}{5} \,{\left (\sqrt{\sqrt{x} + 1} + 2\right )}^{\frac{5}{2}} + \frac{88}{3} \,{\left (\sqrt{\sqrt{x} + 1} + 2\right )}^{\frac{3}{2}} - 48 \, \sqrt{\sqrt{\sqrt{x} + 1} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.52566, size = 124, normalized size = 1.49 \begin{align*} \frac{8}{105} \,{\left ({\left (15 \, \sqrt{x} + 76\right )} \sqrt{\sqrt{x} + 1} - 36 \, \sqrt{x} - 280\right )} \sqrt{\sqrt{\sqrt{x} + 1} + 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{1}{\sqrt{\sqrt{\sqrt{x} + 1} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]