\(\int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx\) [53]

Optimal result

Integrand size = 22, antiderivative size = 67 \[ \int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{2} x \sqrt {1+x^2}-\frac {\text {arcsinh}(x)}{2}+\frac {2}{3} \arctan \left (\frac {1+x}{\sqrt {1+x^2}}\right )+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.57 \[ \int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx=\frac {1}{6} \left (-3 x \sqrt {1+x^2}-4 \arctan \left (\frac {1+x+2 x^2-(1+2 x) \sqrt {1+x^2}}{1-x+\sqrt {1+x^2}}\right )+4 \sqrt {2} \text {arctanh}\left (\frac {1-x+\sqrt {1+x^2}}{\sqrt {2}}\right )+3 \log \left (-x+\sqrt {1+x^2}\right )\right ) \] Input:

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

Output:

(-3*x*Sqrt[1 + x^2] - 4*ArcTan[(1 + x + 2*x^2 - (1 + 2*x)*Sqrt[1 + x^2])/( 
1 - x + Sqrt[1 + x^2])] + 4*Sqrt[2]*ArcTanh[(1 - x + Sqrt[1 + x^2])/Sqrt[2 
]] + 3*Log[-x + Sqrt[1 + x^2]])/6
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.92 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {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.

\(\displaystyle \int \frac {x^3 \sqrt {x^2+1}}{1-x^3} \, dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{3} (-1)^{2/3} \text {arcsinh}(x)+\frac {1}{3} \sqrt [3]{-1} \text {arcsinh}(x)-\frac {5 \text {arcsinh}(x)}{6}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )+\frac {1}{3} i \text {arctanh}\left (\frac {x+(-1)^{2/3}}{\sqrt {1-\sqrt [3]{-1}} \sqrt {x^2+1}}\right )-\frac {1}{3} \sqrt [3]{-1} \sqrt {1+(-1)^{2/3}} \text {arctanh}\left (\frac {2 \left ((-1)^{2/3} x+1\right )}{\left (-\sqrt {3}+i\right ) \sqrt {x^2+1}}\right )-\frac {1}{2} \sqrt {x^2+1} x-\frac {1}{3} (-1)^{2/3} \sqrt {x^2+1}+\frac {1}{3} \sqrt [3]{-1} \sqrt {x^2+1}-\frac {\sqrt {x^2+1}}{3}\)

Input:

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

Output:

-1/3*Sqrt[1 + x^2] + ((-1)^(1/3)*Sqrt[1 + x^2])/3 - ((-1)^(2/3)*Sqrt[1 + x 
^2])/3 - (x*Sqrt[1 + x^2])/2 - (5*ArcSinh[x])/6 + ((-1)^(1/3)*ArcSinh[x])/ 
3 - ((-1)^(2/3)*ArcSinh[x])/3 + (Sqrt[2]*ArcTanh[(1 + x)/(Sqrt[2]*Sqrt[1 + 
 x^2])])/3 + (I/3)*ArcTanh[((-1)^(2/3) + x)/(Sqrt[1 - (-1)^(1/3)]*Sqrt[1 + 
 x^2])] - ((-1)^(1/3)*Sqrt[1 + (-1)^(2/3)]*ArcTanh[(2*(1 + (-1)^(2/3)*x))/ 
((I - Sqrt[3])*Sqrt[1 + x^2])])/3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

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]
 

Maple [C] (verified)

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

Time = 0.38 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.66

method	result	size
trager	\(-\frac {\sqrt {x^{2}+1}\, x}{2}-\frac {\ln \left (x +\sqrt {x^{2}+1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+2 \sqrt {x^{2}+1}}{x -1}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -\sqrt {x^{2}+1}\, x -\sqrt {x^{2}+1}}{x^{2}+x +1}\right )}{3}\)	\(111\)
risch	\(-\frac {\sqrt {x^{2}+1}\, x}{2}-\frac {\operatorname {arcsinh}\left (x \right )}{2}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )}{3 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\)	\(161\)
default	\(-\frac {\sqrt {x^{2}+1}\, x}{2}-\frac {\operatorname {arcsinh}\left (x \right )}{2}-\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{3}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{3}+\frac {\sqrt {x^{2}+1}}{3}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )-\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\)	\(361\)

Input:

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

Output:

-1/2*(x^2+1)^(1/2)*x-1/2*ln(x+(x^2+1)^(1/2))+1/3*RootOf(_Z^2-2)*ln(-(RootO 
f(_Z^2-2)*x+RootOf(_Z^2-2)+2*(x^2+1)^(1/2))/(x-1))+1/3*RootOf(_Z^2+1)*ln(- 
(RootOf(_Z^2+1)*x-(x^2+1)^(1/2)*x-(x^2+1)^(1/2))/(x^2+x+1))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.30 \[ \int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{2} \, \sqrt {x^{2} + 1} x + \frac {1}{3} \, \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (x + 1\right )} + \sqrt {x^{2} + 1} {\left (\sqrt {2} + 2\right )} + x + 1}{x - 1}\right ) - \frac {2}{3} \, \arctan \left (-\frac {x^{2} - \sqrt {x^{2} + 1} {\left (x + 1\right )} + x + 1}{x}\right ) + \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \] Input:

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

Output:

-1/2*sqrt(x^2 + 1)*x + 1/3*sqrt(2)*log(-(sqrt(2)*(x + 1) + sqrt(x^2 + 1)*( 
sqrt(2) + 2) + x + 1)/(x - 1)) - 2/3*arctan(-(x^2 - sqrt(x^2 + 1)*(x + 1) 
+ x + 1)/x) + 1/2*log(-x + sqrt(x^2 + 1))
 

Sympy [F]

\[ \int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx=- \int \frac {x^{3} \sqrt {x^{2} + 1}}{x^{3} - 1}\, dx \] Input:

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

Output:

-Integral(x**3*sqrt(x**2 + 1)/(x**3 - 1), x)
 

Maxima [F]

\[ \int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx=\int { -\frac {\sqrt {x^{2} + 1} x^{3}}{x^{3} - 1} \,d x } \] Input:

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

Output:

-integrate(sqrt(x^2 + 1)*x^3/(x^3 - 1), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (50) = 100\).

Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.03 \[ \int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{3} \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} + 2 \right |}}\right ) + \frac {2}{3} \, \arctan \left (-\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 1}\right )}^{3} - \frac {3}{2} \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - \frac {3}{2} \, x + \frac {3}{2} \, \sqrt {x^{2} + 1} + \frac {1}{2}\right ) - \frac {2}{3} \, \arctan \left (-x + \sqrt {x^{2} + 1} - 2\right ) + \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \] Input:

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

Output:

-1/2*sqrt(x^2 + 1)*x - 1/3*sqrt(2)*log(abs(-2*x - 2*sqrt(2) + 2*sqrt(x^2 + 
 1) + 2)/abs(-2*x + 2*sqrt(2) + 2*sqrt(x^2 + 1) + 2)) + 2/3*arctan(-1/2*(x 
 - sqrt(x^2 + 1))^3 - 3/2*(x - sqrt(x^2 + 1))^2 - 3/2*x + 3/2*sqrt(x^2 + 1 
) + 1/2) - 2/3*arctan(-x + sqrt(x^2 + 1) - 2) + 1/2*log(-x + sqrt(x^2 + 1) 
)
 

Mupad [B] (verification not implemented)

Time = 22.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.66 \[ \int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx=-\frac {\mathrm {asinh}\left (x\right )}{2}-\frac {\sqrt {2}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{3}-\frac {x\,\sqrt {x^2+1}}{2}-\frac {\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}{3\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}-\frac {\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}}{3\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2} \] Input:

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

Output:

- asinh(x)/2 - (2^(1/2)*(log(x - 1) - log(x + 2^(1/2)*(x^2 + 1)^(1/2) + 1) 
))/3 - (x*(x^2 + 1)^(1/2))/2 - ((log(x - (3^(1/2)*1i)/2 + 1/2) - log((3^(1 
/2)/2 - 1i/2)*(x^2 + 1)^(1/2) - x/2 + (3^(1/2)*x*1i)/2 + 1))*(((3^(1/2)*1i 
)/2 - 1/2)^2 + 1)^(1/2))/(3*((3^(1/2)*1i)/2 - 1/2)^2) - ((log(x + (3^(1/2) 
*1i)/2 + 1/2) - log((3^(1/2)/2 + 1i/2)*(x^2 + 1)^(1/2) - x/2 - (3^(1/2)*x* 
1i)/2 + 1))*(((3^(1/2)*1i)/2 + 1/2)^2 + 1)^(1/2))/(3*((3^(1/2)*1i)/2 + 1/2 
)^2)
 

Reduce [F]

\[ \int \frac {x^3 \sqrt {1+x^2}}{1-x^3} \, dx=-\left (\int \frac {\sqrt {x^{2}+1}\, x^{3}}{x^{3}-1}d x \right ) \] Input:

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

Output:

 - int((sqrt(x**2 + 1)*x**3)/(x**3 - 1),x)