3.943 \(\int \frac{x^5}{(1+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right )-\frac{x^2}{2 \sqrt{x^4+1}} \]

[Out]

-x^2/(2*Sqrt[1 + x^4]) + ArcSinh[x^2]/2

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Rubi [A]  time = 0.0089134, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 288, 215} \[ \frac{1}{2} \sinh ^{-1}\left (x^2\right )-\frac{x^2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(1 + x^4)^(3/2),x]

[Out]

-x^2/(2*Sqrt[1 + x^4]) + ArcSinh[x^2]/2

Rule 275

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 215

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

Rubi steps

\begin{align*} \int \frac{x^5}{\left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (1+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{x^2}{2 \sqrt{1+x^4}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^2\right )\\ &=-\frac{x^2}{2 \sqrt{1+x^4}}+\frac{1}{2} \sinh ^{-1}\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0126422, size = 23, normalized size = 0.92 \[ \frac{1}{2} \left (\sinh ^{-1}\left (x^2\right )-\frac{x^2}{\sqrt{x^4+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 + x^4)^(3/2),x]

[Out]

(-(x^2/Sqrt[1 + x^4]) + ArcSinh[x^2])/2

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Maple [A]  time = 0.008, size = 20, normalized size = 0.8 \begin{align*}{\frac{{\it Arcsinh} \left ({x}^{2} \right ) }{2}}-{\frac{{x}^{2}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^4+1)^(3/2),x)

[Out]

1/2*arcsinh(x^2)-1/2*x^2/(x^4+1)^(1/2)

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Maxima [B]  time = 0.997293, size = 61, normalized size = 2.44 \begin{align*} -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} + \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

-1/2*x^2/sqrt(x^4 + 1) + 1/4*log(sqrt(x^4 + 1)/x^2 + 1) - 1/4*log(sqrt(x^4 + 1)/x^2 - 1)

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Fricas [B]  time = 1.52817, size = 113, normalized size = 4.52 \begin{align*} -\frac{x^{4} + \sqrt{x^{4} + 1} x^{2} +{\left (x^{4} + 1\right )} \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) + 1}{2 \,{\left (x^{4} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/2*(x^4 + sqrt(x^4 + 1)*x^2 + (x^4 + 1)*log(-x^2 + sqrt(x^4 + 1)) + 1)/(x^4 + 1)

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Sympy [A]  time = 1.4812, size = 19, normalized size = 0.76 \begin{align*} - \frac{x^{2}}{2 \sqrt{x^{4} + 1}} + \frac{\operatorname{asinh}{\left (x^{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**4+1)**(3/2),x)

[Out]

-x**2/(2*sqrt(x**4 + 1)) + asinh(x**2)/2

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Giac [A]  time = 1.17991, size = 39, normalized size = 1.56 \begin{align*} -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

-1/2*x^2/sqrt(x^4 + 1) - 1/2*log(-x^2 + sqrt(x^4 + 1))