\(\int \frac {\sqrt {x}}{\sqrt {1+x} (1+x^2)} \, dx\) [895]

Optimal result

Integrand size = 20, antiderivative size = 65 \[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx=-\frac {1}{2} (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt {1+x}}\right )-\frac {1}{2} (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt {1+x}}\right ) \] Output:

-1/2*(1-I)^(3/2)*arctanh((1-I)^(1/2)*x^(1/2)/(1+x)^(1/2))-1/2*(1+I)^(3/2)* 
arctanh((1+I)^(1/2)*x^(1/2)/(1+x)^(1/2))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx=-\text {RootSum}\left [16+32 \text {$\#$1}+16 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log \left (-2 x+2 \sqrt {x} \sqrt {1+x}+\text {$\#$1}\right ) \text {$\#$1}^2}{8+8 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \] Input:

Integrate[Sqrt[x]/(Sqrt[1 + x]*(1 + x^2)),x]
 

Output:

-RootSum[16 + 32*#1 + 16*#1^2 + #1^4 & , (Log[-2*x + 2*Sqrt[x]*Sqrt[1 + x] 
 + #1]*#1^2)/(8 + 8*#1 + #1^3) & ]
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {613, 104, 221}

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.

\(\displaystyle \int \frac {\sqrt {x}}{\sqrt {x+1} \left (x^2+1\right )} \, dx\)

\(\Big \downarrow \) 613

\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {x} (x+i) \sqrt {x+1}}dx-\frac {1}{2} \int \frac {1}{(i-x) \sqrt {x} \sqrt {x+1}}dx\)

\(\Big \downarrow \) 104

\(\displaystyle \int \frac {1}{\frac {(1-i) x}{x+1}+i}d\frac {\sqrt {x}}{\sqrt {x+1}}-\int \frac {1}{i-\frac {(1+i) x}{x+1}}d\frac {\sqrt {x}}{\sqrt {x+1}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {1}{2} (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt {x+1}}\right )-\frac {1}{2} (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt {x+1}}\right )\)

Input:

Int[Sqrt[x]/(Sqrt[1 + x]*(1 + x^2)),x]
 

Output:

-1/2*((1 - I)^(3/2)*ArcTanh[(Sqrt[1 - I]*Sqrt[x])/Sqrt[1 + x]]) - ((1 + I) 
^(3/2)*ArcTanh[(Sqrt[1 + I]*Sqrt[x])/Sqrt[1 + x]])/2
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 613

Int[Sqrt[(e_.)*(x_)]/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Sym 
bol] :> Simp[e/(2*b)   Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(Rt[-a/b, 2] + x)), x 
], x] - Simp[e/(2*b)   Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(Rt[-a/b, 2] - x)), x 
], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(45)=90\).

Time = 0.41 (sec) , antiderivative size = 305, normalized size of antiderivative = 4.69

method	result	size
default	\(\frac {\sqrt {\frac {\left (x +1\right ) x}{\left (-1+\sqrt {2}+x \right )^{2}}}\, \left (-1+\sqrt {2}+x \right ) \left (\sqrt {-2+2 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}}\, \sqrt {\frac {\left (3 \sqrt {2}-4\right ) x \left (4+3 \sqrt {2}\right ) \left (x +1\right )}{\left (-1+\sqrt {2}+x \right )^{2}}}\, \left (3+2 \sqrt {2}\right ) \left (\sqrt {2}+1-x \right ) \left (3 \sqrt {2}-4\right ) \left (-1+\sqrt {2}+x \right )}{4 \left (x +1\right ) x}\right ) \sqrt {1+\sqrt {2}}\, \sqrt {2}-2 \sqrt {-2+2 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}}\, \sqrt {\frac {\left (3 \sqrt {2}-4\right ) x \left (4+3 \sqrt {2}\right ) \left (x +1\right )}{\left (-1+\sqrt {2}+x \right )^{2}}}\, \left (3+2 \sqrt {2}\right ) \left (\sqrt {2}+1-x \right ) \left (3 \sqrt {2}-4\right ) \left (-1+\sqrt {2}+x \right )}{4 \left (x +1\right ) x}\right ) \sqrt {1+\sqrt {2}}+4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +1\right ) x}{\left (-1+\sqrt {2}+x \right )^{2}}}}{\sqrt {1+\sqrt {2}}}\right ) \sqrt {2}-6 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +1\right ) x}{\left (-1+\sqrt {2}+x \right )^{2}}}}{\sqrt {1+\sqrt {2}}}\right )\right ) \sqrt {2}}{4 \sqrt {x}\, \sqrt {x +1}\, \left (3 \sqrt {2}-4\right ) \sqrt {1+\sqrt {2}}}\)	\(305\)

Input:

int(x^(1/2)/(x+1)^(1/2)/(x^2+1),x,method=_RETURNVERBOSE)
 

Output:

1/4/x^(1/2)/(x+1)^(1/2)*((x+1)*x/(-1+2^(1/2)+x)^2)^(1/2)*(-1+2^(1/2)+x)*(( 
-2+2*2^(1/2))^(1/2)*arctan(1/4*(-2+2*2^(1/2))^(1/2)*((3*2^(1/2)-4)*x*(4+3* 
2^(1/2))*(x+1)/(-1+2^(1/2)+x)^2)^(1/2)*(3+2*2^(1/2))*(2^(1/2)+1-x)*(3*2^(1 
/2)-4)*(-1+2^(1/2)+x)/(x+1)/x)*(1+2^(1/2))^(1/2)*2^(1/2)-2*(-2+2*2^(1/2))^ 
(1/2)*arctan(1/4*(-2+2*2^(1/2))^(1/2)*((3*2^(1/2)-4)*x*(4+3*2^(1/2))*(x+1) 
/(-1+2^(1/2)+x)^2)^(1/2)*(3+2*2^(1/2))*(2^(1/2)+1-x)*(3*2^(1/2)-4)*(-1+2^( 
1/2)+x)/(x+1)/x)*(1+2^(1/2))^(1/2)+4*arctanh(2^(1/2)*((x+1)*x/(-1+2^(1/2)+ 
x)^2)^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)-6*arctanh(2^(1/2)*((x+1)*x/(-1+2^(1 
/2)+x)^2)^(1/2)/(1+2^(1/2))^(1/2)))*2^(1/2)/(3*2^(1/2)-4)/(1+2^(1/2))^(1/2 
)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (37) = 74\).

Time = 0.09 (sec) , antiderivative size = 290, normalized size of antiderivative = 4.46 \[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx=\frac {1}{2} \, \sqrt {\sqrt {2} + 1} \arctan \left ({\left (\sqrt {x + 1} \sqrt {x} {\left (\sqrt {2} + 2\right )} - \sqrt {2} {\left (x + 1\right )} + {\left (2 \, \sqrt {x + 1} \sqrt {x} {\left (\sqrt {2} + 1\right )} - \sqrt {2} {\left (2 \, x + 1\right )} - 2 \, x - 1\right )} \sqrt {\sqrt {2} - 1} - 2 \, x\right )} \sqrt {\sqrt {2} + 1}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \arctan \left (-{\left (\sqrt {x + 1} \sqrt {x} {\left (\sqrt {2} + 2\right )} - \sqrt {2} {\left (x + 1\right )} - {\left (2 \, \sqrt {x + 1} \sqrt {x} {\left (\sqrt {2} + 1\right )} - \sqrt {2} {\left (2 \, x + 1\right )} - 2 \, x - 1\right )} \sqrt {\sqrt {2} - 1} - 2 \, x\right )} \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} - 1} \log \left (-2 \, \sqrt {x + 1} x^{\frac {3}{2}} + 2 \, x^{2} + {\left (\sqrt {2} {\left (x - 1\right )} - \sqrt {2} \sqrt {x + 1} \sqrt {x} - 2\right )} \sqrt {\sqrt {2} - 1} + x + \sqrt {2} + 1\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} - 1} \log \left (-2 \, \sqrt {x + 1} x^{\frac {3}{2}} + 2 \, x^{2} - {\left (\sqrt {2} {\left (x - 1\right )} - \sqrt {2} \sqrt {x + 1} \sqrt {x} - 2\right )} \sqrt {\sqrt {2} - 1} + x + \sqrt {2} + 1\right ) \] Input:

integrate(x^(1/2)/(1+x)^(1/2)/(x^2+1),x, algorithm="fricas")
 

Output:

1/2*sqrt(sqrt(2) + 1)*arctan((sqrt(x + 1)*sqrt(x)*(sqrt(2) + 2) - sqrt(2)* 
(x + 1) + (2*sqrt(x + 1)*sqrt(x)*(sqrt(2) + 1) - sqrt(2)*(2*x + 1) - 2*x - 
 1)*sqrt(sqrt(2) - 1) - 2*x)*sqrt(sqrt(2) + 1)) - 1/2*sqrt(sqrt(2) + 1)*ar 
ctan(-(sqrt(x + 1)*sqrt(x)*(sqrt(2) + 2) - sqrt(2)*(x + 1) - (2*sqrt(x + 1 
)*sqrt(x)*(sqrt(2) + 1) - sqrt(2)*(2*x + 1) - 2*x - 1)*sqrt(sqrt(2) - 1) - 
 2*x)*sqrt(sqrt(2) + 1)) + 1/4*sqrt(sqrt(2) - 1)*log(-2*sqrt(x + 1)*x^(3/2 
) + 2*x^2 + (sqrt(2)*(x - 1) - sqrt(2)*sqrt(x + 1)*sqrt(x) - 2)*sqrt(sqrt( 
2) - 1) + x + sqrt(2) + 1) - 1/4*sqrt(sqrt(2) - 1)*log(-2*sqrt(x + 1)*x^(3 
/2) + 2*x^2 - (sqrt(2)*(x - 1) - sqrt(2)*sqrt(x + 1)*sqrt(x) - 2)*sqrt(sqr 
t(2) - 1) + x + sqrt(2) + 1)
 

Sympy [F]

\[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx=\int \frac {\sqrt {x}}{\sqrt {x + 1} \left (x^{2} + 1\right )}\, dx \] Input:

integrate(x**(1/2)/(1+x)**(1/2)/(x**2+1),x)
 

Output:

Integral(sqrt(x)/(sqrt(x + 1)*(x**2 + 1)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx=\int { \frac {\sqrt {x}}{{\left (x^{2} + 1\right )} \sqrt {x + 1}} \,d x } \] Input:

integrate(x^(1/2)/(1+x)^(1/2)/(x^2+1),x, algorithm="maxima")
 

Output:

integrate(sqrt(x)/((x^2 + 1)*sqrt(x + 1)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (37) = 74\).

Time = 0.54 (sec) , antiderivative size = 375, normalized size of antiderivative = 5.77 \[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx=\frac {1}{4} \, {\left (\sqrt {2 \, \sqrt {2} + 2} + \sqrt {2 \, \sqrt {2} - 2}\right )} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {-\frac {1}{x + 1} + 1}\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, {\left (\sqrt {2 \, \sqrt {2} + 2} + \sqrt {2 \, \sqrt {2} - 2}\right )} \arctan \left (-\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {-\frac {1}{x + 1} + 1}\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, {\left (\sqrt {2 \, \sqrt {2} + 2} - \sqrt {2 \, \sqrt {2} - 2}\right )} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {-\frac {1}{x + 1} + 1} + \sqrt {\frac {1}{2}} - \frac {1}{x + 1} + 1\right ) + \frac {1}{8} \, {\left (\sqrt {2 \, \sqrt {2} + 2} - \sqrt {2 \, \sqrt {2} - 2}\right )} \log \left (-\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {-\frac {1}{x + 1} + 1} + \sqrt {\frac {1}{2}} - \frac {1}{x + 1} + 1\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (-\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + \sqrt {\frac {1}{2}} + 1\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (-\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + \sqrt {\frac {1}{2}} + 1\right ) \] Input:

integrate(x^(1/2)/(1+x)^(1/2)/(x^2+1),x, algorithm="giac")
 

Output:

1/4*(sqrt(2*sqrt(2) + 2) + sqrt(2*sqrt(2) - 2))*arctan(2*(1/2)^(3/4)*((1/2 
)^(1/4)*sqrt(sqrt(2) + 2) + 2*sqrt(-1/(x + 1) + 1))/sqrt(-sqrt(2) + 2)) + 
1/4*(sqrt(2*sqrt(2) + 2) + sqrt(2*sqrt(2) - 2))*arctan(-2*(1/2)^(3/4)*((1/ 
2)^(1/4)*sqrt(sqrt(2) + 2) - 2*sqrt(-1/(x + 1) + 1))/sqrt(-sqrt(2) + 2)) - 
 1/8*(sqrt(2*sqrt(2) + 2) - sqrt(2*sqrt(2) - 2))*log((1/2)^(1/4)*sqrt(sqrt 
(2) + 2)*sqrt(-1/(x + 1) + 1) + sqrt(1/2) - 1/(x + 1) + 1) + 1/8*(sqrt(2*s 
qrt(2) + 2) - sqrt(2*sqrt(2) - 2))*log(-(1/2)^(1/4)*sqrt(sqrt(2) + 2)*sqrt 
(-1/(x + 1) + 1) + sqrt(1/2) - 1/(x + 1) + 1) - 1/4*sqrt(2*sqrt(2) + 2)*ar 
ctan(2*(1/2)^(3/4)*((1/2)^(1/4)*sqrt(sqrt(2) + 2) + 2)/sqrt(-sqrt(2) + 2)) 
 - 1/4*sqrt(2*sqrt(2) + 2)*arctan(-2*(1/2)^(3/4)*((1/2)^(1/4)*sqrt(sqrt(2) 
 + 2) - 2)/sqrt(-sqrt(2) + 2)) - 1/8*sqrt(2*sqrt(2) - 2)*log((1/2)^(1/4)*s 
qrt(sqrt(2) + 2) + sqrt(1/2) + 1) + 1/8*sqrt(2*sqrt(2) - 2)*log(-(1/2)^(1/ 
4)*sqrt(sqrt(2) + 2) + sqrt(1/2) + 1)
 

Mupad [B] (verification not implemented)

Time = 13.84 (sec) , antiderivative size = 1610, normalized size of antiderivative = 24.77 \[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx=\text {Too large to display} \] Input:

int(x^(1/2)/((x^2 + 1)*(x + 1)^(1/2)),x)
 

Output:

- atan(((((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((2845 
4158336*x^(1/2))/((x + 1)^(1/2) - 1) + ((- 2^(1/2)/16 - 1/16)^(1/2) - (2^( 
1/2)/16 - 1/16)^(1/2))*(((112742891520*x^(1/2))/((x + 1)^(1/2) - 1) - ((53 
1502202880*x)/((x + 1)^(1/2) - 1)^2 - 241591910400)*((- 2^(1/2)/16 - 1/16) 
^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/ 
2)/16 - 1/16)^(1/2)) - (12079595520*x)/((x + 1)^(1/2) - 1)^2 + 68451041280 
))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + (1355599052 
8*x)/((x + 1)^(1/2) - 1)^2 + 9529458688) + (3556769792*x^(1/2))/((x + 1)^( 
1/2) - 1))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*1i - 
(((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*((13555990528*x 
)/((x + 1)^(1/2) - 1)^2 - ((28454158336*x^(1/2))/((x + 1)^(1/2) - 1) + ((- 
 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((112742891520*x^( 
1/2))/((x + 1)^(1/2) - 1) + ((531502202880*x)/((x + 1)^(1/2) - 1)^2 - 2415 
91910400)*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)))*((- 2 
^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + (12079595520*x)/((x 
 + 1)^(1/2) - 1)^2 - 68451041280))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2) 
/16 - 1/16)^(1/2)) + 9529458688) - (3556769792*x^(1/2))/((x + 1)^(1/2) - 1 
))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*1i)/((((- 2^( 
1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((28454158336*x^(1/2)) 
/((x + 1)^(1/2) - 1) + ((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/1...
 

Reduce [F]

\[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx=\int \frac {\sqrt {x}}{\sqrt {x +1}\, x^{2}+\sqrt {x +1}}d x \] Input:

int(x^(1/2)/(1+x)^(1/2)/(x^2+1),x)
 

Output:

int(sqrt(x)/(sqrt(x + 1)*x**2 + sqrt(x + 1)),x)