Optimal. Leaf size=32 \[ \frac {2 \sqrt {\left (x^2+1\right ) \left (\sqrt {x^2+1}+1\right )}}{\sqrt {x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6715, 1588} \[ \frac {2 \sqrt {\left (x^2+1\right ) \left (\sqrt {x^2+1}+1\right )}}{\sqrt {x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1588
Rule 6715
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+x^2+\left (1+x^2\right )^{3/2}}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x+(1+x)^{3/2}}} \, dx,x,x^2\right )\\ &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {x^2 (1+x)}} \, dx,x,\sqrt {1+x^2}\right )\\ &=\frac {2 \sqrt {\left (1+x^2\right ) \left (1+\sqrt {1+x^2}\right )}}{\sqrt {1+x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 37, normalized size = 1.16 \[ \frac {2 \left (x^2+\sqrt {x^2+1}+1\right )}{\sqrt {\left (x^2+1\right ) \left (\sqrt {x^2+1}+1\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 23, normalized size = 0.72 \[ \frac {2 \, \sqrt {x^{2} + {\left (x^{2} + 1\right )}^{\frac {3}{2}} + 1}}{\sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.06, size = 15, normalized size = 0.47 \[ 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} - 2 \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x^{2}+1+\left (x^{2}+1\right )^{\frac {3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x^{2} + {\left (x^{2} + 1\right )}^{\frac {3}{2}} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.59, size = 47, normalized size = 1.47 \[ \frac {2\,\left (x^2+1\right )\,\sqrt {\sqrt {x^2+1}+1}}{\left (\sqrt {\sqrt {x^2+1}+1}+1\right )\,\sqrt {{\left (x^2+1\right )}^{3/2}+x^2+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {\left (x^{2} + 1\right ) \left (\sqrt {x^{2} + 1} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________