3.2.51 \(\int \frac {x^2}{\text {ArcSin}(a+b x)^3} \, dx\) [151]

Optimal. Leaf size=176 \[ -\frac {x^2 \sqrt {1-(a+b x)^2}}{2 b \text {ArcSin}(a+b x)^2}+\frac {a^2 (a+b x)}{2 b^3 \text {ArcSin}(a+b x)}-\frac {2 a (a+b x)^2}{b^3 \text {ArcSin}(a+b x)}+\frac {9 a+b x}{8 b^3 \text {ArcSin}(a+b x)}-\frac {\left (1+4 a^2\right ) \text {CosIntegral}(\text {ArcSin}(a+b x))}{8 b^3}+\frac {9 \text {CosIntegral}(3 \text {ArcSin}(a+b x))}{8 b^3}-\frac {3 \sin (3 \text {ArcSin}(a+b x))}{8 b^3 \text {ArcSin}(a+b x)}+\frac {2 a \text {Si}(2 \text {ArcSin}(a+b x))}{b^3} \]

[Out]

1/2*a^2*(b*x+a)/b^3/arcsin(b*x+a)-2*a*(b*x+a)^2/b^3/arcsin(b*x+a)+1/8*(b*x+9*a)/b^3/arcsin(b*x+a)-1/8*(4*a^2+1
)*Ci(arcsin(b*x+a))/b^3+9/8*Ci(3*arcsin(b*x+a))/b^3+2*a*Si(2*arcsin(b*x+a))/b^3-3/8*sin(3*arcsin(b*x+a))/b^3/a
rcsin(b*x+a)-1/2*x^2*(1-(b*x+a)^2)^(1/2)/b/arcsin(b*x+a)^2

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Rubi [A]
time = 0.34, antiderivative size = 263, normalized size of antiderivative = 1.49, number of steps used = 24, number of rules used = 12, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4889, 4829, 4717, 4807, 4719, 3383, 4729, 4731, 4491, 12, 3380, 4737} \begin {gather*} -\frac {a^2 \text {CosIntegral}(\text {ArcSin}(a+b x))}{2 b^3}+\frac {a^2 (a+b x)}{2 b^3 \text {ArcSin}(a+b x)}-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \text {ArcSin}(a+b x)^2}-\frac {\text {CosIntegral}(\text {ArcSin}(a+b x))}{8 b^3}+\frac {9 \text {CosIntegral}(3 \text {ArcSin}(a+b x))}{8 b^3}+\frac {2 a \text {Si}(2 \text {ArcSin}(a+b x))}{b^3}+\frac {3 (a+b x)^3}{2 b^3 \text {ArcSin}(a+b x)}-\frac {2 a (a+b x)^2}{b^3 \text {ArcSin}(a+b x)}-\frac {\sqrt {1-(a+b x)^2} (a+b x)^2}{2 b^3 \text {ArcSin}(a+b x)^2}-\frac {a+b x}{b^3 \text {ArcSin}(a+b x)}+\frac {a \sqrt {1-(a+b x)^2} (a+b x)}{b^3 \text {ArcSin}(a+b x)^2}+\frac {a}{b^3 \text {ArcSin}(a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/ArcSin[a + b*x]^3,x]

[Out]

-1/2*(a^2*Sqrt[1 - (a + b*x)^2])/(b^3*ArcSin[a + b*x]^2) + (a*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(b^3*ArcSin[a +
b*x]^2) - ((a + b*x)^2*Sqrt[1 - (a + b*x)^2])/(2*b^3*ArcSin[a + b*x]^2) + a/(b^3*ArcSin[a + b*x]) - (a + b*x)
/(b^3*ArcSin[a + b*x]) + (a^2*(a + b*x))/(2*b^3*ArcSin[a + b*x]) - (2*a*(a + b*x)^2)/(b^3*ArcSin[a + b*x]) + (
3*(a + b*x)^3)/(2*b^3*ArcSin[a + b*x]) - CosIntegral[ArcSin[a + b*x]]/(8*b^3) - (a^2*CosIntegral[ArcSin[a + b*
x]])/(2*b^3) + (9*CosIntegral[3*ArcSin[a + b*x]])/(8*b^3) + (2*a*SinIntegral[2*ArcSin[a + b*x]])/b^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4717

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/
(b*c*(n + 1))), x] + Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

Rule 4719

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 4729

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin
[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/
Sqrt[1 - c^2*x^2]), x], x] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2
]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4807

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Dist[f*(m/(b
*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rule 4829

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d + e
*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {x^2}{\sin ^{-1}(a+b x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2}{b^2 \sin ^{-1}(x)^3}-\frac {2 a x}{b^2 \sin ^{-1}(x)^3}+\frac {x^2}{b^2 \sin ^{-1}(x)^3}\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {x}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {3 \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b^3}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {(2 a) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {a^2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a}{b^3 \sin ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sin ^{-1}(a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \sin ^{-1}(a+b x)}-\frac {2 a (a+b x)^2}{b^3 \sin ^{-1}(a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \sin ^{-1}(a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b^3}-\frac {9 \text {Subst}\left (\int \frac {x^2}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{2 b^3}+\frac {(4 a) \text {Subst}\left (\int \frac {x}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b^3}-\frac {a^2 \text {Subst}\left (\int \frac {1}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{2 b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a}{b^3 \sin ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sin ^{-1}(a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \sin ^{-1}(a+b x)}-\frac {2 a (a+b x)^2}{b^3 \sin ^{-1}(a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \sin ^{-1}(a+b x)}+\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {9 \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {(4 a) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {a^2 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a}{b^3 \sin ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sin ^{-1}(a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \sin ^{-1}(a+b x)}-\frac {2 a (a+b x)^2}{b^3 \sin ^{-1}(a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \sin ^{-1}(a+b x)}+\frac {\text {Ci}\left (\sin ^{-1}(a+b x)\right )}{b^3}-\frac {a^2 \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b^3}-\frac {9 \text {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {(4 a) \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a}{b^3 \sin ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sin ^{-1}(a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \sin ^{-1}(a+b x)}-\frac {2 a (a+b x)^2}{b^3 \sin ^{-1}(a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \sin ^{-1}(a+b x)}+\frac {\text {Ci}\left (\sin ^{-1}(a+b x)\right )}{b^3}-\frac {a^2 \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b^3}-\frac {9 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {9 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {(2 a) \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1-(a+b x)^2}}{b^3 \sin ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}+\frac {a}{b^3 \sin ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sin ^{-1}(a+b x)}+\frac {a^2 (a+b x)}{2 b^3 \sin ^{-1}(a+b x)}-\frac {2 a (a+b x)^2}{b^3 \sin ^{-1}(a+b x)}+\frac {3 (a+b x)^3}{2 b^3 \sin ^{-1}(a+b x)}-\frac {\text {Ci}\left (\sin ^{-1}(a+b x)\right )}{8 b^3}-\frac {a^2 \text {Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {9 \text {Ci}\left (3 \sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {2 a \text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b^3}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 115, normalized size = 0.65 \begin {gather*} \frac {\frac {4 b x \left (-b x \sqrt {1-a^2-2 a b x-b^2 x^2}+\left (-2+2 a^2+5 a b x+3 b^2 x^2\right ) \text {ArcSin}(a+b x)\right )}{\text {ArcSin}(a+b x)^2}-\left (1+4 a^2\right ) \text {CosIntegral}(\text {ArcSin}(a+b x))+9 \text {CosIntegral}(3 \text {ArcSin}(a+b x))+16 a \text {Si}(2 \text {ArcSin}(a+b x))}{8 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcSin[a + b*x]^3,x]

[Out]

((4*b*x*(-(b*x*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]) + (-2 + 2*a^2 + 5*a*b*x + 3*b^2*x^2)*ArcSin[a + b*x]))/ArcSi
n[a + b*x]^2 - (1 + 4*a^2)*CosIntegral[ArcSin[a + b*x]] + 9*CosIntegral[3*ArcSin[a + b*x]] + 16*a*SinIntegral[
2*ArcSin[a + b*x]])/(8*b^3)

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Maple [A]
time = 0.20, size = 215, normalized size = 1.22

method	result	size
derivativedivides	\(\frac {\frac {a \left (4 \sinIntegral \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+2 \cos \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sin \left (2 \arcsin \left (b x +a \right )\right )\right )}{2 \arcsin \left (b x +a \right )^{2}}-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{8 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{8 \arcsin \left (b x +a \right )}-\frac {\cosineIntegral \left (\arcsin \left (b x +a \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )}+\frac {9 \cosineIntegral \left (3 \arcsin \left (b x +a \right )\right )}{8}-\frac {a^{2} \left (\cosineIntegral \left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}-\left (b x +a \right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \arcsin \left (b x +a \right )^{2}}}{b^{3}}\)	\(215\)
default	\(\frac {\frac {a \left (4 \sinIntegral \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+2 \cos \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sin \left (2 \arcsin \left (b x +a \right )\right )\right )}{2 \arcsin \left (b x +a \right )^{2}}-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{8 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{8 \arcsin \left (b x +a \right )}-\frac {\cosineIntegral \left (\arcsin \left (b x +a \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )}+\frac {9 \cosineIntegral \left (3 \arcsin \left (b x +a \right )\right )}{8}-\frac {a^{2} \left (\cosineIntegral \left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}-\left (b x +a \right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \arcsin \left (b x +a \right )^{2}}}{b^{3}}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arcsin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/b^3*(1/2*a*(4*Si(2*arcsin(b*x+a))*arcsin(b*x+a)^2+2*cos(2*arcsin(b*x+a))*arcsin(b*x+a)+sin(2*arcsin(b*x+a)))
/arcsin(b*x+a)^2-1/8/arcsin(b*x+a)^2*(1-(b*x+a)^2)^(1/2)+1/8*(b*x+a)/arcsin(b*x+a)-1/8*Ci(arcsin(b*x+a))+1/8/a
rcsin(b*x+a)^2*cos(3*arcsin(b*x+a))-3/8/arcsin(b*x+a)*sin(3*arcsin(b*x+a))+9/8*Ci(3*arcsin(b*x+a))-1/2*a^2*(Ci
(arcsin(b*x+a))*arcsin(b*x+a)^2-(b*x+a)*arcsin(b*x+a)+(1-(b*x+a)^2)^(1/2))/arcsin(b*x+a)^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*b*x^2 + arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^2*in
tegrate((9*b^2*x^2 + 10*a*b*x + 2*a^2 - 2)/arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)), x) - (3*b^2
*x^3 + 5*a*b*x^2 + 2*(a^2 - 1)*x)*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)))/(b^2*arctan2(b*x + a
, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^2)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(b*x+a)^3,x, algorithm="fricas")

[Out]

integral(x^2/arcsin(b*x + a)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\operatorname {asin}^{3}{\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/asin(b*x+a)**3,x)

[Out]

Integral(x**2/asin(a + b*x)**3, x)

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Giac [A]
time = 0.43, size = 272, normalized size = 1.55 \begin {gather*} -\frac {a^{2} \operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{2 \, b^{3}} + \frac {{\left (b x + a\right )} a^{2}}{2 \, b^{3} \arcsin \left (b x + a\right )} + \frac {2 \, a \operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{b^{3}} + \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )}}{2 \, b^{3} \arcsin \left (b x + a\right )} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a}{b^{3} \arcsin \left (b x + a\right )} + \frac {9 \, \operatorname {Ci}\left (3 \, \arcsin \left (b x + a\right )\right )}{8 \, b^{3}} - \frac {\operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{8 \, b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{b^{3} \arcsin \left (b x + a\right )^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} + \frac {b x + a}{2 \, b^{3} \arcsin \left (b x + a\right )} - \frac {a}{b^{3} \arcsin \left (b x + a\right )} + \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arcsin(b*x+a)^3,x, algorithm="giac")

[Out]

-1/2*a^2*cos_integral(arcsin(b*x + a))/b^3 + 1/2*(b*x + a)*a^2/(b^3*arcsin(b*x + a)) + 2*a*sin_integral(2*arcs
in(b*x + a))/b^3 + 3/2*((b*x + a)^2 - 1)*(b*x + a)/(b^3*arcsin(b*x + a)) - 2*((b*x + a)^2 - 1)*a/(b^3*arcsin(b
*x + a)) + 9/8*cos_integral(3*arcsin(b*x + a))/b^3 - 1/8*cos_integral(arcsin(b*x + a))/b^3 + sqrt(-(b*x + a)^2
+ 1)*(b*x + a)*a/(b^3*arcsin(b*x + a)^2) - 1/2*sqrt(-(b*x + a)^2 + 1)*a^2/(b^3*arcsin(b*x + a)^2) + 1/2*(b*x
+ a)/(b^3*arcsin(b*x + a)) - a/(b^3*arcsin(b*x + a)) + 1/2*(-(b*x + a)^2 + 1)^(3/2)/(b^3*arcsin(b*x + a)^2) -
1/2*sqrt(-(b*x + a)^2 + 1)/(b^3*arcsin(b*x + a)^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\mathrm {asin}\left (a+b\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/asin(a + b*x)^3,x)

[Out]

int(x^2/asin(a + b*x)^3, x)

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