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]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [B]  time = 0.95, size = 45, normalized size = 1.80 \[ -\frac {x^{4} + \sqrt {x^{4} + 1} x^{2} + {\left (x^{4} + 1\right )} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) + 1}{2 \, {\left (x^{4} + 1\right )}} \]

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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giac [A]  time = 0.18, size = 29, normalized size = 1.16 \[ -\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \]

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))

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maple [A]  time = 0.01, size = 20, normalized size = 0.80 \[ -\frac {x^{2}}{2 \sqrt {x^{4}+1}}+\frac {\arcsinh \left (x^{2}\right )}{2} \]

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

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maxima [B]  time = 1.30, size = 45, normalized size = 1.80 \[ -\frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^5}{{\left (x^4+1\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^4 + 1)^(3/2),x)

[Out]

int(x^5/(x^4 + 1)^(3/2), x)

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sympy [A]  time = 2.30, size = 19, normalized size = 0.76 \[ - \frac {x^{2}}{2 \sqrt {x^{4} + 1}} + \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} \]

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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