3.1.16 \(\int \frac {\sin ^{-1}(x)}{(1+x^2)^{3/2}} \, dx\) [16]

Optimal. Leaf size=22 \[ \frac {x \sin ^{-1}(x)}{\sqrt {1+x^2}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \]

[Out]

-1/2*arcsin(x^2)+x*arcsin(x)/(x^2+1)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {197, 4755, 281, 222} \begin {gather*} \frac {x \sin ^{-1}(x)}{\sqrt {x^2+1}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]/(1 + x^2)^(3/2),x]

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 222

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

Rule 281

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 4755

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \frac {x \sin ^{-1}(x)}{\sqrt {1+x^2}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 20.31, size = 71, normalized size = 3.23 \begin {gather*} \frac {x \text {ArcSin}\left [x\right ]}{\sqrt {1+x^2}}-\frac {\text {meijerg}\left [\left \{\left \{-\frac {1}{2},-\frac {1}{4},0,\frac {1}{4},\frac {1}{2},1\right \},\left \{\right \}\right \},\left \{\left \{-\frac {1}{4},\frac {1}{4}\right \},\left \{-\frac {1}{2},0,0,0\right \}\right \},\frac {\text {exp\_polar}\left [-2 I \text {Pi}\right ]}{x^4}\right ]}{8 \text {Pi}^{\frac {3}{2}}}+\frac {\frac {I}{8} \text {meijerg}\left [\left \{\left \{\frac {1}{4},\frac {3}{4}\right \},\left \{\frac {1}{2},\frac {1}{2},1,1\right \}\right \},\left \{\left \{0,\frac {1}{4},\frac {1}{2},\frac {3}{4},1,0\right \},\left \{\right \}\right \},\frac {1}{x^4}\right ]}{\text {Pi}^{\frac {3}{2}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[ArcSin[x]/(1 + x^2)^(3/2),x]')

[Out]

x ArcSin[x] / Sqrt[1 + x ^ 2] - meijerg[{{-1 / 2, -1 / 4, 0, 1 / 4, 1 / 2, 1}, {}}, {{-1 / 4, 1 / 4}, {-1 / 2,
0, 0, 0}}, exp_polar[-2 I Pi] / x ^ 4] / (8 Pi ^ (3 / 2)) + I / 8 meijerg[{{1 / 4, 3 / 4}, {1 / 2, 1 / 2, 1,
1}}, {{0, 1 / 4, 1 / 2, 3 / 4, 1, 0}, {}}, 1 / x ^ 4] / Pi ^ (3 / 2)

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\arcsin \left (x \right )}{\left (x^{2}+1\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.36, size = 18, normalized size = 0.82 \begin {gather*} \frac {x \arcsin \left (x\right )}{\sqrt {x^{2} + 1}} - \frac {1}{2} \, \arcsin \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arcsin(x)/sqrt(x^2 + 1) - 1/2*arcsin(x^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (18) = 36\).
time = 0.34, size = 56, normalized size = 2.55 \begin {gather*} \frac {2 \, \sqrt {x^{2} + 1} x \arcsin \left (x\right ) + {\left (x^{2} + 1\right )} \arctan \left (\frac {\sqrt {x^{2} + 1} \sqrt {-x^{2} + 1} x^{2}}{x^{4} - 1}\right )}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 19.11, size = 78, normalized size = 3.55 \begin {gather*} \frac {x \operatorname {asin}{\left (x \right )}}{\sqrt {x^{2} + 1}} + \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{x^{4}}} \right )}}{8 \pi ^{\frac {3}{2}}} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{4}}} \right )}}{8 \pi ^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*asin(x)/sqrt(x**2 + 1) + I*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), x**(-4))/
(8*pi**(3/2)) - meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), exp_polar(-2*I*pi)
/x**4)/(8*pi**(3/2))

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Giac [A]
time = 0.01, size = 32, normalized size = 1.45 \begin {gather*} \frac {2 x \sqrt {x^{2}+1} \arcsin x}{2 \left (x^{2}+1\right )}-\frac {2}{4} \arcsin \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)/(x^2+1)^(3/2),x)

[Out]

x*arcsin(x)/sqrt(x^2 + 1) - 1/2*arcsin(x^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\mathrm {asin}\left (x\right )}{{\left (x^2+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x)/(x^2 + 1)^(3/2),x)

[Out]

int(asin(x)/(x^2 + 1)^(3/2), x)

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