3.821 \(\int \frac{\sqrt{x}-\sqrt{x^3}}{x-x^3} \, dx\)

Optimal. Leaf size=52 \[ \frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.183135, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {1593, 6725, 329, 212, 206, 203, 15, 298} \[ \frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[x] - Sqrt[x^3])/(x - x^3),x]

[Out]

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 212

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

Rule 206

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

Rule 203

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 298

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

Rubi steps

\begin{align*} \int \frac{\sqrt{x}-\sqrt{x^3}}{x-x^3} \, dx &=\int \frac{\sqrt{x}-\sqrt{x^3}}{x \left (1-x^2\right )} \, dx\\ &=\int \left (-\frac{1}{\sqrt{x} \left (-1+x^2\right )}+\frac{\sqrt{x^3}}{x \left (-1+x^2\right )}\right ) \, dx\\ &=-\int \frac{1}{\sqrt{x} \left (-1+x^2\right )} \, dx+\int \frac{\sqrt{x^3}}{x \left (-1+x^2\right )} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\sqrt{x}\right )\right )+\frac{\sqrt{x^3} \int \frac{\sqrt{x}}{-1+x^2} \, dx}{x^{3/2}}\\ &=\frac{\left (2 \sqrt{x^3}\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\sqrt{x}\right )}{x^{3/2}}+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x^3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )}{x^{3/2}}+\frac{\sqrt{x^3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )}{x^{3/2}}\\ &=\tan ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tanh ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0603503, size = 49, normalized size = 0.94 \[ \frac{\left (x^{3/2}+\sqrt{x^3}\right ) \tan ^{-1}\left (\sqrt{x}\right )+\left (x^{3/2}-\sqrt{x^3}\right ) \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[x] - Sqrt[x^3])/(x - x^3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 41, normalized size = 0.8 \begin{align*}{\it Artanh} \left ( \sqrt{x} \right ) +\arctan \left ( \sqrt{x} \right ) +{\frac{1}{2}\sqrt{{x}^{3}} \left ( \ln \left ( -1+\sqrt{x} \right ) -\ln \left ( 1+\sqrt{x} \right ) +2\,\arctan \left ( \sqrt{x} \right ) \right ){x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \arctan \left (\sqrt{x}\right ) - \int \frac{\sqrt{x}}{2 \,{\left (x + 1\right )}}\,{d x} + \int \frac{1}{4 \,{\left (\sqrt{x} + 1\right )}}\,{d x} + \int \frac{1}{4 \,{\left (\sqrt{x} - 1\right )}}\,{d x} + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x)) - integrate(1/2*sqrt(x)/(x + 1), x) + integrate(1/4/(sqrt(x) + 1), x) + integrate(1/4/(sqrt(x)
- 1), x) + 1/2*log(sqrt(x) + 1) - 1/2*log(sqrt(x) - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.9677, size = 26, normalized size = 0.5 \begin{align*} 2 \, \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(x))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sqrt{x}}{x^{3} - x}\, dx - \int - \frac{\sqrt{x^{3}}}{x^{3} - x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sqrt(x)/(x**3 - x), x) - Integral(-sqrt(x**3)/(x**3 - x), x)

________________________________________________________________________________________

Giac [A]  time = 1.10667, size = 8, normalized size = 0.15 \begin{align*} 2 \, \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(x))