3.151 \(\int \frac {x^2}{\sin ^{-1}(a+b x)^3} \, dx\)

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

[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 = 0.51, 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 = {4805, 4745, 4621, 4719, 4623, 3302, 4633, 4635, 4406, 12, 3299, 4641} \[ -\frac {a^2 \text {CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {a^2 (a+b x)}{2 b^3 \sin ^{-1}(a+b x)}-\frac {a^2 \sqrt {1-(a+b x)^2}}{2 b^3 \sin ^{-1}(a+b x)^2}-\frac {\text {CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {9 \text {CosIntegral}\left (3 \sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {2 a \text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b^3}+\frac {3 (a+b x)^3}{2 b^3 \sin ^{-1}(a+b x)}-\frac {2 a (a+b x)^2}{b^3 \sin ^{-1}(a+b x)}-\frac {\sqrt {1-(a+b x)^2} (a+b x)^2}{2 b^3 \sin ^{-1}(a+b x)^2}-\frac {a+b x}{b^3 \sin ^{-1}(a+b x)}+\frac {a \sqrt {1-(a+b x)^2} (a+b x)}{b^3 \sin ^{-1}(a+b x)^2}+\frac {a}{b^3 \sin ^{-1}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcSin[a + b*x]^3,x]

[Out]

-(a^2*Sqrt[1 - (a + b*x)^2])/(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 3299

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 3302

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 4406

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 4621

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 4623

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 4633

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 4635

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

Rule 4641

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

Rule 4719

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

Rule 4745

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 4805

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 {\operatorname {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 {\operatorname {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 {\operatorname {Subst}\left (\int \frac {x^2}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {x}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \operatorname {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 {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {3 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {(2 a) \operatorname {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 \operatorname {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 {\operatorname {Subst}\left (\int \frac {1}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b^3}-\frac {9 \operatorname {Subst}\left (\int \frac {x^2}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{2 b^3}+\frac {(4 a) \operatorname {Subst}\left (\int \frac {x}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b^3}-\frac {a^2 \operatorname {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 {\operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {9 \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b^3}+\frac {(4 a) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b^3}-\frac {a^2 \operatorname {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 \operatorname {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) \operatorname {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 \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {9 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b^3}+\frac {(2 a) \operatorname {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 = 0.58, size = 115, normalized size = 0.65 \[ \frac {\frac {4 b x \left (\left (2 a^2+5 a b x+3 b^2 x^2-2\right ) \sin ^{-1}(a+b x)-b x \sqrt {-a^2-2 a b x-b^2 x^2+1}\right )}{\sin ^{-1}(a+b x)^2}-\left (4 a^2+1\right ) \text {Ci}\left (\sin ^{-1}(a+b x)\right )+9 \text {Ci}\left (3 \sin ^{-1}(a+b x)\right )+16 a \text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{8 b^3} \]

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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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\arcsin \left (b x + a\right )^{3}}, x\right ) \]

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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giac [A]  time = 0.42, size = 272, normalized size = 1.55 \[ -\frac {a^{2} \operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{2 \, b^{3}} + \frac {{\left (b x + a\right )} a^{2}}{2 \, b^{3} \arcsin \left (b x + a\right )} + \frac {2 \, a \operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{b^{3}} + \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )}}{2 \, b^{3} \arcsin \left (b x + a\right )} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a}{b^{3} \arcsin \left (b x + a\right )} + \frac {9 \, \operatorname {Ci}\left (3 \, \arcsin \left (b x + a\right )\right )}{8 \, b^{3}} - \frac {\operatorname {Ci}\left (\arcsin \left (b x + a\right )\right )}{8 \, b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{b^{3} \arcsin \left (b x + a\right )^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} + \frac {b x + a}{2 \, b^{3} \arcsin \left (b x + a\right )} - \frac {a}{b^{3} \arcsin \left (b x + a\right )} + \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{2 \, b^{3} \arcsin \left (b x + a\right )^{2}} \]

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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maple [A]  time = 0.16, size = 215, normalized size = 1.22 \[ \frac {\frac {a \left (4 \Si \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}+2 \cos \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+\sin \left (2 \arcsin \left (b x +a \right )\right )\right )}{2 \arcsin \left (b x +a \right )^{2}}-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{8 \arcsin \left (b x +a \right )^{2}}+\frac {b x +a}{8 \arcsin \left (b x +a \right )}-\frac {\Ci \left (\arcsin \left (b x +a \right )\right )}{8}+\frac {\cos \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )^{2}}-\frac {3 \sin \left (3 \arcsin \left (b x +a \right )\right )}{8 \arcsin \left (b x +a \right )}+\frac {9 \Ci \left (3 \arcsin \left (b x +a \right )\right )}{8}-\frac {a^{2} \left (\Ci \left (\arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )^{2}-\left (b x +a \right ) \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \arcsin \left (b x +a \right )^{2}}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*sin(3*arcsin(b*x+a))/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 \[ -\frac {\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x^{2} + \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2} \int \frac {9 \, b^{2} x^{2} + 10 \, a b x + 2 \, a^{2} - 2}{\arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}\,{d x} - {\left (3 \, b^{2} x^{3} + 5 \, a b x^{2} + 2 \, {\left (a^{2} - 1\right )} x\right )} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}{2 \, b^{2} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2}} \]

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

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

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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