\(\int \frac {1}{x^{3/2} (1+x^2)^3} \, dx\) [333]

Optimal result

Integrand size = 13, antiderivative size = 115 \[ \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx=-\frac {45}{16 \sqrt {x}}+\frac {1}{4 \sqrt {x} \left (1+x^2\right )^2}+\frac {9}{16 \sqrt {x} \left (1+x^2\right )}+\frac {45 \arctan \left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {45 \arctan \left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {45 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )}{32 \sqrt {2}} \] Output:

-45/16/x^(1/2)+1/4/x^(1/2)/(x^2+1)^2+9/16/x^(1/2)/(x^2+1)-45/64*arctan(-1+ 
2^(1/2)*x^(1/2))*2^(1/2)-45/64*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)+45/64*arc 
tanh(2^(1/2)*x^(1/2)/(1+x))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx=\frac {1}{64} \left (-\frac {4 \left (32+81 x^2+45 x^4\right )}{\sqrt {x} \left (1+x^2\right )^2}-45 \sqrt {2} \arctan \left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right )+45 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right )\right ) \] Input:

Integrate[1/(x^(3/2)*(1 + x^2)^3),x]
 

Output:

((-4*(32 + 81*x^2 + 45*x^4))/(Sqrt[x]*(1 + x^2)^2) - 45*Sqrt[2]*ArcTan[(-1 
 + x)/(Sqrt[2]*Sqrt[x])] + 45*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x])/(1 + x)])/ 
64
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.34, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {253, 253, 264, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {1}{x^{3/2} \left (x^2+1\right )^3} \, dx\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {9}{8} \int \frac {1}{x^{3/2} \left (x^2+1\right )^2}dx+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \int \frac {1}{x^{3/2} \left (x^2+1\right )}dx+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-\int \frac {\sqrt {x}}{x^2+1}dx-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \int \frac {x}{x^2+1}d\sqrt {x}-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \left (\frac {1}{2} \int \frac {x+1}{x^2+1}d\sqrt {x}-\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}\right )-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}+\frac {1}{2} \int \frac {1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )-\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}\right )-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x-1}d\left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-x-1}d\left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}\right )-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-x}{x^2+1}d\sqrt {x}\right )-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {x}+1\right )}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {x}}{x-\sqrt {2} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {x}+1}{x+\sqrt {2} \sqrt {x}+1}d\sqrt {x}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )\right )-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {9}{8} \left (\frac {5}{4} \left (-2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {x}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}-\frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{2 \sqrt {2}}\right )\right )-\frac {2}{\sqrt {x}}\right )+\frac {1}{2 \sqrt {x} \left (x^2+1\right )}\right )+\frac {1}{4 \sqrt {x} \left (x^2+1\right )^2}\)

Input:

Int[1/(x^(3/2)*(1 + x^2)^3),x]
 

Output:

1/(4*Sqrt[x]*(1 + x^2)^2) + (9*(1/(2*Sqrt[x]*(1 + x^2)) + (5*(-2/Sqrt[x] - 
 2*((-(ArcTan[1 - Sqrt[2]*Sqrt[x]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[x]]/ 
Sqrt[2])/2 + (Log[1 - Sqrt[2]*Sqrt[x] + x]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*S 
qrt[x] + x]/(2*Sqrt[2]))/2)))/4))/8
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70

method	result	size
risch	\(-\frac {45 x^{4}+81 x^{2}+32}{16 \sqrt {x}\, \left (x^{2}+1\right )^{2}}-\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\)	\(81\)
derivativedivides	\(-\frac {2}{\sqrt {x}}-\frac {2 \left (\frac {13 x^{\frac {7}{2}}}{32}+\frac {17 x^{\frac {3}{2}}}{32}\right )}{\left (x^{2}+1\right )^{2}}-\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\)	\(82\)
default	\(-\frac {2}{\sqrt {x}}-\frac {2 \left (\frac {13 x^{\frac {7}{2}}}{32}+\frac {17 x^{\frac {3}{2}}}{32}\right )}{\left (x^{2}+1\right )^{2}}-\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{128}\)	\(82\)
meijerg	\(-\frac {45 x^{4}+81 x^{2}+32}{16 \sqrt {x}\, \left (x^{2}+1\right )^{2}}-\frac {45 x^{\frac {3}{2}} \left (\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}-\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}\right )}{64}\)	\(155\)
trager	\(-\frac {45 x^{4}+81 x^{2}+32}{16 \sqrt {x}\, \left (x^{2}+1\right )^{2}}+\frac {45 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )-4 \sqrt {x}}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+x +1}\right )}{64}+\frac {45 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )-4 \sqrt {x}}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-x -1}\right )}{64}\)	\(204\)

Input:

int(1/x^(3/2)/(x^2+1)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/16*(45*x^4+81*x^2+32)/x^(1/2)/(x^2+1)^2-45/128*2^(1/2)*(ln((x-2^(1/2)*x 
^(1/2)+1)/(x+2^(1/2)*x^(1/2)+1))+2*arctan(1+2^(1/2)*x^(1/2))+2*arctan(-1+2 
^(1/2)*x^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx=-\frac {90 \, \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \left (\sqrt {2} \sqrt {x} + 1\right ) + 90 \, \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \left (\sqrt {2} \sqrt {x} - 1\right ) - 45 \, \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) + 45 \, \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + 8 \, {\left (45 \, x^{4} + 81 \, x^{2} + 32\right )} \sqrt {x}}{128 \, {\left (x^{5} + 2 \, x^{3} + x\right )}} \] Input:

integrate(1/x^(3/2)/(x^2+1)^3,x, algorithm="fricas")
 

Output:

-1/128*(90*sqrt(2)*(x^5 + 2*x^3 + x)*arctan(sqrt(2)*sqrt(x) + 1) + 90*sqrt 
(2)*(x^5 + 2*x^3 + x)*arctan(sqrt(2)*sqrt(x) - 1) - 45*sqrt(2)*(x^5 + 2*x^ 
3 + x)*log(sqrt(2)*sqrt(x) + x + 1) + 45*sqrt(2)*(x^5 + 2*x^3 + x)*log(-sq 
rt(2)*sqrt(x) + x + 1) + 8*(45*x^4 + 81*x^2 + 32)*sqrt(x))/(x^5 + 2*x^3 + 
x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (104) = 208\).

Time = 1.72 (sec) , antiderivative size = 653, normalized size of antiderivative = 5.68 \[ \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx=- \frac {45 \sqrt {2} x^{\frac {9}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} + \frac {45 \sqrt {2} x^{\frac {9}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} x^{\frac {9}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} x^{\frac {5}{2}} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} + \frac {90 \sqrt {2} x^{\frac {5}{2}} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {180 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {180 \sqrt {2} x^{\frac {5}{2}} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {45 \sqrt {2} \sqrt {x} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} + \frac {45 \sqrt {2} \sqrt {x} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} \sqrt {x} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {90 \sqrt {2} \sqrt {x} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {360 x^{4}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {648 x^{2}}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} - \frac {256}{128 x^{\frac {9}{2}} + 256 x^{\frac {5}{2}} + 128 \sqrt {x}} \] Input:

integrate(1/x**(3/2)/(x**2+1)**3,x)
 

Output:

-45*sqrt(2)*x**(9/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256 
*x**(5/2) + 128*sqrt(x)) + 45*sqrt(2)*x**(9/2)*log(4*sqrt(2)*sqrt(x) + 4*x 
 + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*sqrt(2)*x**(9/2)*at 
an(sqrt(2)*sqrt(x) - 1)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 90*s 
qrt(2)*x**(9/2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**(9/2) + 256*x**(5/2) + 1 
28*sqrt(x)) - 90*sqrt(2)*x**(5/2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x 
**(9/2) + 256*x**(5/2) + 128*sqrt(x)) + 90*sqrt(2)*x**(5/2)*log(4*sqrt(2)* 
sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 180*sqrt( 
2)*x**(5/2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**(9/2) + 256*x**(5/2) + 128*s 
qrt(x)) - 180*sqrt(2)*x**(5/2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**(9/2) + 2 
56*x**(5/2) + 128*sqrt(x)) - 45*sqrt(2)*sqrt(x)*log(-4*sqrt(2)*sqrt(x) + 4 
*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) + 45*sqrt(2)*sqrt(x)*l 
og(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x) 
) - 90*sqrt(2)*sqrt(x)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**(9/2) + 256*x**(5 
/2) + 128*sqrt(x)) - 90*sqrt(2)*sqrt(x)*atan(sqrt(2)*sqrt(x) + 1)/(128*x** 
(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 360*x**4/(128*x**(9/2) + 256*x**(5/2 
) + 128*sqrt(x)) - 648*x**2/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x)) - 
256/(128*x**(9/2) + 256*x**(5/2) + 128*sqrt(x))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx=-\frac {45}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \frac {45}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {45}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {45}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {45 \, x^{4} + 81 \, x^{2} + 32}{16 \, {\left (x^{\frac {9}{2}} + 2 \, x^{\frac {5}{2}} + \sqrt {x}\right )}} \] Input:

integrate(1/x^(3/2)/(x^2+1)^3,x, algorithm="maxima")
 

Output:

-45/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 45/64*sqrt(2)*a 
rctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 45/128*sqrt(2)*log(sqrt(2)*sqr 
t(x) + x + 1) - 45/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 1/16*(45*x^ 
4 + 81*x^2 + 32)/(x^(9/2) + 2*x^(5/2) + sqrt(x))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx=-\frac {45}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \frac {45}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {45}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {45}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {2}{\sqrt {x}} - \frac {13 \, x^{\frac {7}{2}} + 17 \, x^{\frac {3}{2}}}{16 \, {\left (x^{2} + 1\right )}^{2}} \] Input:

integrate(1/x^(3/2)/(x^2+1)^3,x, algorithm="giac")
 

Output:

-45/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - 45/64*sqrt(2)*a 
rctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 45/128*sqrt(2)*log(sqrt(2)*sqr 
t(x) + x + 1) - 45/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 2/sqrt(x) - 
 1/16*(13*x^(7/2) + 17*x^(3/2))/(x^2 + 1)^2
 

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx=-\frac {\frac {45\,x^4}{16}+\frac {81\,x^2}{16}+2}{\sqrt {x}+2\,x^{5/2}+x^{9/2}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {45}{64}+\frac {45}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {45}{64}-\frac {45}{64}{}\mathrm {i}\right ) \] Input:

int(1/(x^(3/2)*(x^2 + 1)^3),x)
 

Output:

- 2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 - 1i/2))*(45/64 - 45i/64) - 2^(1/2)*at 
an(2^(1/2)*x^(1/2)*(1/2 + 1i/2))*(45/64 + 45i/64) - ((81*x^2)/16 + (45*x^4 
)/16 + 2)/(x^(1/2) + 2*x^(5/2) + x^(9/2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^{3/2} \left (1+x^2\right )^3} \, dx=\frac {-90 \sqrt {x}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right ) x^{4}-180 \sqrt {x}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right ) x^{2}-90 \sqrt {x}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}-\sqrt {2}}{\sqrt {2}}\right )-90 \sqrt {x}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right ) x^{4}-180 \sqrt {x}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right ) x^{2}-90 \sqrt {x}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {x}+\sqrt {2}}{\sqrt {2}}\right )-45 \sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right ) x^{4}-90 \sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right ) x^{2}-45 \sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, \sqrt {2}+x +1\right )+45 \sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right ) x^{4}+90 \sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right ) x^{2}+45 \sqrt {x}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, \sqrt {2}+x +1\right )-360 x^{4}-648 x^{2}-256}{128 \sqrt {x}\, \left (x^{4}+2 x^{2}+1\right )} \] Input:

int(1/x^(3/2)/(x^2+1)^3,x)
 

Output:

( - 90*sqrt(x)*sqrt(2)*atan((2*sqrt(x) - sqrt(2))/sqrt(2))*x**4 - 180*sqrt 
(x)*sqrt(2)*atan((2*sqrt(x) - sqrt(2))/sqrt(2))*x**2 - 90*sqrt(x)*sqrt(2)* 
atan((2*sqrt(x) - sqrt(2))/sqrt(2)) - 90*sqrt(x)*sqrt(2)*atan((2*sqrt(x) + 
 sqrt(2))/sqrt(2))*x**4 - 180*sqrt(x)*sqrt(2)*atan((2*sqrt(x) + sqrt(2))/s 
qrt(2))*x**2 - 90*sqrt(x)*sqrt(2)*atan((2*sqrt(x) + sqrt(2))/sqrt(2)) - 45 
*sqrt(x)*sqrt(2)*log( - sqrt(x)*sqrt(2) + x + 1)*x**4 - 90*sqrt(x)*sqrt(2) 
*log( - sqrt(x)*sqrt(2) + x + 1)*x**2 - 45*sqrt(x)*sqrt(2)*log( - sqrt(x)* 
sqrt(2) + x + 1) + 45*sqrt(x)*sqrt(2)*log(sqrt(x)*sqrt(2) + x + 1)*x**4 + 
90*sqrt(x)*sqrt(2)*log(sqrt(x)*sqrt(2) + x + 1)*x**2 + 45*sqrt(x)*sqrt(2)* 
log(sqrt(x)*sqrt(2) + x + 1) - 360*x**4 - 648*x**2 - 256)/(128*sqrt(x)*(x* 
*4 + 2*x**2 + 1))