Integrand size = 33, antiderivative size = 147 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}-\frac {7}{4} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )-16 \text {RootSum}\left [25+20 \text {$\#$1}-18 \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}}{5-9 \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )-2 \text {RootSum}\left [-1+8 \text {$\#$1}-6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}}{2-3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 - 2*RootSum[-1 + 8*#1 - 6*#1^2 + #1^4 & , ( -Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + Sqr t[x + Sqrt[1 + x]] - #1]*#1)/(2 - 3*#1 + #1^3) & ]
Time = 1.00 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.51, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2003, 7267, 25, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {x+1} \left (x^2+1\right )}{\left (x^2-1\right ) \sqrt {x+\sqrt {x+1}}} \, dx\) |
\(\Big \downarrow \) 2003 |
\(\displaystyle \int \frac {x^2+1}{(x-1) \sqrt {x+1} \sqrt {x+\sqrt {x+1}}}dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int -\frac {x^2+1}{(1-x) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {x^2+1}{(1-x) \sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -2 \int \left (\frac {2}{(1-x) \sqrt {x+\sqrt {x+1}}}-\frac {x+1}{\sqrt {x+\sqrt {x+1}}}\right )d\sqrt {x+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\arctan \left (\frac {\left (4-\sqrt {2}\right ) \sqrt {x+1}+2 \left (1+\sqrt {2}\right )}{2 \sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {7}{8} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-\frac {\text {arctanh}\left (\frac {\left (4+\sqrt {2}\right ) \sqrt {x+1}+2 \left (1-\sqrt {2}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {x+1}}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {1}{2} \sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{4} \sqrt {x+\sqrt {x+1}}\right )\) |
2*((-3*Sqrt[x + Sqrt[1 + x]])/4 + (Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/2 + ArcTan[(2*(1 + Sqrt[2]) + (4 - Sqrt[2])*Sqrt[1 + x])/(2*Sqrt[2*(-1 + Sqrt[ 2])]*Sqrt[x + Sqrt[1 + x]])]/Sqrt[2*(-1 + Sqrt[2])] + (7*ArcTanh[(1 + 2*Sq rt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/8 - ArcTanh[(2*(1 - Sqrt[2]) + (4 + Sqrt[2])*Sqrt[1 + x])/(2*Sqrt[2*(1 + Sqrt[2])]*Sqrt[x + Sqrt[1 + x]])]/Sq rt[2*(1 + Sqrt[2])])
3.21.56.3.1 Defintions of rubi rules used
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.23 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.48
method | result | size |
derivativedivides | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {2+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )}{2 \sqrt {1+\sqrt {2}}\, \sqrt {\left (\sqrt {1+x}-\sqrt {2}\right )^{2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )+1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}-\frac {\sqrt {2}\, \arctan \left (\frac {2-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}\, \sqrt {\left (\sqrt {1+x}+\sqrt {2}\right )^{2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )+1-\sqrt {2}}}\right )}{\sqrt {\sqrt {2}-1}}\) | \(217\) |
default | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {2+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )}{2 \sqrt {1+\sqrt {2}}\, \sqrt {\left (\sqrt {1+x}-\sqrt {2}\right )^{2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )+1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}-\frac {\sqrt {2}\, \arctan \left (\frac {2-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}\, \sqrt {\left (\sqrt {1+x}+\sqrt {2}\right )^{2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )+1-\sqrt {2}}}\right )}{\sqrt {\sqrt {2}-1}}\) | \(217\) |
(1+x)^(1/2)*(x+(1+x)^(1/2))^(1/2)-3/2*(x+(1+x)^(1/2))^(1/2)+7/4*ln(1/2+(1+ x)^(1/2)+(x+(1+x)^(1/2))^(1/2))-2^(1/2)/(1+2^(1/2))^(1/2)*arctanh(1/2*(2+2 *2^(1/2)+(1+2*2^(1/2))*((1+x)^(1/2)-2^(1/2)))/(1+2^(1/2))^(1/2)/(((1+x)^(1 /2)-2^(1/2))^2+(1+2*2^(1/2))*((1+x)^(1/2)-2^(1/2))+1+2^(1/2))^(1/2))-2^(1/ 2)/(2^(1/2)-1)^(1/2)*arctan(1/2*(2-2*2^(1/2)+(1-2*2^(1/2))*((1+x)^(1/2)+2^ (1/2)))/(2^(1/2)-1)^(1/2)/(((1+x)^(1/2)+2^(1/2))^2+(1-2*2^(1/2))*((1+x)^(1 /2)+2^(1/2))+1-2^(1/2))^(1/2))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 4.26 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.84 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (-\frac {2 \, {\left (\sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 3\right )} + 2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} + 4\right )} + 6 \, x + 6\right )} \sqrt {\sqrt {2} - 1} + 4 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + 2\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x - 1}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 3\right )} + 2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} + 4\right )} + 6 \, x + 6\right )} \sqrt {\sqrt {2} - 1} - 4 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + 2\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x - 1}\right ) + \frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} - \frac {1}{4} \, \sqrt {-8 \, \sqrt {2} - 8} \log \left (\frac {2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} - 4\right )} \sqrt {-8 \, \sqrt {2} - 8} + 8 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {2} - 2\right )} \sqrt {x + \sqrt {x + 1}} + {\left (\sqrt {2} {\left (5 \, x + 3\right )} - 6 \, x - 6\right )} \sqrt {-8 \, \sqrt {2} - 8}}{x - 1}\right ) + \frac {1}{4} \, \sqrt {-8 \, \sqrt {2} - 8} \log \left (-\frac {2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} - 4\right )} \sqrt {-8 \, \sqrt {2} - 8} - 8 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {2} - 2\right )} \sqrt {x + \sqrt {x + 1}} + {\left (\sqrt {2} {\left (5 \, x + 3\right )} - 6 \, x - 6\right )} \sqrt {-8 \, \sqrt {2} - 8}}{x - 1}\right ) + \frac {7}{8} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \]
-1/2*sqrt(2)*sqrt(sqrt(2) - 1)*log(-2*(sqrt(2)*(sqrt(2)*(5*x + 3) + 2*sqrt (x + 1)*(3*sqrt(2) + 4) + 6*x + 6)*sqrt(sqrt(2) - 1) + 4*(sqrt(x + 1)*(sqr t(2) + 1) + sqrt(2) + 2)*sqrt(x + sqrt(x + 1)))/(x - 1)) + 1/2*sqrt(2)*sqr t(sqrt(2) - 1)*log(2*(sqrt(2)*(sqrt(2)*(5*x + 3) + 2*sqrt(x + 1)*(3*sqrt(2 ) + 4) + 6*x + 6)*sqrt(sqrt(2) - 1) - 4*(sqrt(x + 1)*(sqrt(2) + 1) + sqrt( 2) + 2)*sqrt(x + sqrt(x + 1)))/(x - 1)) + 1/2*sqrt(x + sqrt(x + 1))*(2*sqr t(x + 1) - 3) - 1/4*sqrt(-8*sqrt(2) - 8)*log((2*sqrt(x + 1)*(3*sqrt(2) - 4 )*sqrt(-8*sqrt(2) - 8) + 8*(sqrt(x + 1)*(sqrt(2) - 1) + sqrt(2) - 2)*sqrt( x + sqrt(x + 1)) + (sqrt(2)*(5*x + 3) - 6*x - 6)*sqrt(-8*sqrt(2) - 8))/(x - 1)) + 1/4*sqrt(-8*sqrt(2) - 8)*log(-(2*sqrt(x + 1)*(3*sqrt(2) - 4)*sqrt( -8*sqrt(2) - 8) - 8*(sqrt(x + 1)*(sqrt(2) - 1) + sqrt(2) - 2)*sqrt(x + sqr t(x + 1)) + (sqrt(2)*(5*x + 3) - 6*x - 6)*sqrt(-8*sqrt(2) - 8))/(x - 1)) + 7/8*log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) + 8*x + 8*sqrt(x + 1) + 5)
Not integrable
Time = 15.80 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x^{2} + 1}{\left (x - 1\right ) \sqrt {x + 1} \sqrt {x + \sqrt {x + 1}}}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {{\left (x^{2} + 1\right )} \sqrt {x + 1}}{{\left (x^{2} - 1\right )} \sqrt {x + \sqrt {x + 1}}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.76 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + 4 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \arctan \left (\frac {\sqrt {2} - \sqrt {x + \sqrt {x + 1}} + \sqrt {x + 1}}{\sqrt {\sqrt {2} - 1}}\right ) - 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left ({\left | 10 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + 20 \, \sqrt {2} + 20 \, \sqrt {x + \sqrt {x + 1}} - 20 \, \sqrt {x + 1} + 20 \, \sqrt {2 \, \sqrt {2} - 2} \right |}\right ) + 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left ({\left | -2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + 4 \, \sqrt {2} + 4 \, \sqrt {x + \sqrt {x + 1}} - 4 \, \sqrt {x + 1} - 4 \, \sqrt {2 \, \sqrt {2} - 2} \right |}\right ) - \frac {7}{4} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \]
1/2*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) - 3) + 4*sqrt(1/2*sqrt(2) + 1/2)* arctan((sqrt(2) - sqrt(x + sqrt(x + 1)) + sqrt(x + 1))/sqrt(sqrt(2) - 1)) - 2*sqrt(1/2*sqrt(2) - 1/2)*log(abs(10*sqrt(2)*sqrt(2*sqrt(2) - 2) + 20*sq rt(2) + 20*sqrt(x + sqrt(x + 1)) - 20*sqrt(x + 1) + 20*sqrt(2*sqrt(2) - 2) )) + 2*sqrt(1/2*sqrt(2) - 1/2)*log(abs(-2*sqrt(2)*sqrt(2*sqrt(2) - 2) + 4* sqrt(2) + 4*sqrt(x + sqrt(x + 1)) - 4*sqrt(x + 1) - 4*sqrt(2*sqrt(2) - 2)) ) - 7/4*log(-2*sqrt(x + sqrt(x + 1)) + 2*sqrt(x + 1) + 1)
Not integrable
Time = 6.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {\left (x^2+1\right )\,\sqrt {x+1}}{\sqrt {x+\sqrt {x+1}}\,\left (x^2-1\right )} \,d x \]