\(\int \frac {\arcsin (a+b x)}{x^5} \, dx\) [130]

Optimal result

Integrand size = 10, antiderivative size = 186 \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\arcsin (a+b x)}{4 x^4}-\frac {a \left (3+2 a^2\right ) b^4 \text {arctanh}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^{7/2}} \]

[Out]

-1/4*arcsin(b*x+a)/x^4-1/8*a*(2*a^2+3)*b^4*arctanh((1-a*(b*x+a))/(-a^2+1)^(1/2)/(1-(b*x+a)^2)^(1/2))/(-a^2+1)^
(7/2)-1/12*b*(1-(b*x+a)^2)^(1/2)/(-a^2+1)/x^3-5/24*a*b^2*(1-(b*x+a)^2)^(1/2)/(-a^2+1)^2/x^2-1/24*(11*a^2+4)*b^
3*(1-(b*x+a)^2)^(1/2)/(-a^2+1)^3/x

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4889, 4827, 759, 849, 821, 739, 212} \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=-\frac {a \left (2 a^2+3\right ) b^4 \text {arctanh}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^{7/2}}-\frac {\left (11 a^2+4\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {\arcsin (a+b x)}{4 x^4} \]

[In]

Int[ArcSin[a + b*x]/x^5,x]

[Out]

-1/12*(b*Sqrt[1 - (a + b*x)^2])/((1 - a^2)*x^3) - (5*a*b^2*Sqrt[1 - (a + b*x)^2])/(24*(1 - a^2)^2*x^2) - ((4 +
11*a^2)*b^3*Sqrt[1 - (a + b*x)^2])/(24*(1 - a^2)^3*x) - ArcSin[a + b*x]/(4*x^4) - (a*(3 + 2*a^2)*b^4*ArcTanh[
(1 - a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/(8*(1 - a^2)^(7/2))

Rule 212

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 759

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 4827

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^(
n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\arcsin (x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^5} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\arcsin (a+b x)}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right )^4 \sqrt {1-x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {\arcsin (a+b x)}{4 x^4}+\frac {b^2 \text {Subst}\left (\int \frac {\frac {3 a}{b}+\frac {2 x}{b}}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sqrt {1-x^2}} \, dx,x,a+b x\right )}{12 \left (1-a^2\right )} \\ & = -\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\arcsin (a+b x)}{4 x^4}-\frac {b^4 \text {Subst}\left (\int \frac {-\frac {2 \left (2+3 a^2\right )}{b^2}-\frac {5 a x}{b^2}}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1-x^2}} \, dx,x,a+b x\right )}{24 \left (1-a^2\right )^2} \\ & = -\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\arcsin (a+b x)}{4 x^4}+\frac {\left (a \left (3+2 a^2\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 \left (1-a^2\right )^3} \\ & = -\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\arcsin (a+b x)}{4 x^4}-\frac {\left (a \left (3+2 a^2\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{b^2}-\frac {a^2}{b^2}-x^2} \, dx,x,\frac {\frac {1}{b}-\frac {a (a+b x)}{b}}{\sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^3} \\ & = -\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\arcsin (a+b x)}{4 x^4}-\frac {a \left (3+2 a^2\right ) b^4 \text {arctanh}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.04 \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\frac {1}{8} \left (\frac {b \sqrt {1-a^2-2 a b x-b^2 x^2} \left (2+2 a^4+5 a b x-5 a^3 b x+4 b^2 x^2+a^2 \left (-4+11 b^2 x^2\right )\right )}{3 \left (-1+a^2\right )^3 x^3}-\frac {2 \arcsin (a+b x)}{x^4}+\frac {a \left (3+2 a^2\right ) b^4 \log (x)}{\left (1-a^2\right )^{7/2}}-\frac {a \left (3+2 a^2\right ) b^4 \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {1-a^2-2 a b x-b^2 x^2}\right )}{\left (1-a^2\right )^{7/2}}\right ) \]

[In]

Integrate[ArcSin[a + b*x]/x^5,x]

[Out]

((b*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(2 + 2*a^4 + 5*a*b*x - 5*a^3*b*x + 4*b^2*x^2 + a^2*(-4 + 11*b^2*x^2)))/(
3*(-1 + a^2)^3*x^3) - (2*ArcSin[a + b*x])/x^4 + (a*(3 + 2*a^2)*b^4*Log[x])/(1 - a^2)^(7/2) - (a*(3 + 2*a^2)*b^
4*Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]])/(1 - a^2)^(7/2))/8

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(393\) vs. \(2(164)=328\).

Time = 0.33 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.12

method	result	size
parts	\(-\frac {\arcsin \left (b x +a \right )}{4 x^{4}}+\frac {b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 \left (-a^{2}+1\right ) x^{3}}+\frac {5 a b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) x^{2}}+\frac {3 a b \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (-a^{2}+1\right )}+\frac {2 b^{2} \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) x}-\frac {a b \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (-a^{2}+1\right )}\right )}{4}\)	\(394\)
derivativedivides	\(b^{4} \left (-\frac {\arcsin \left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{12 \left (-a^{2}+1\right ) b^{3} x^{3}}+\frac {5 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b^{2} x^{2}}+\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (-a^{2}+1\right )}+\frac {-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}}{-6 a^{2}+6}\right )\)	\(408\)
default	\(b^{4} \left (-\frac {\arcsin \left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{12 \left (-a^{2}+1\right ) b^{3} x^{3}}+\frac {5 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b^{2} x^{2}}+\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (-a^{2}+1\right )}+\frac {-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}}{-6 a^{2}+6}\right )\)	\(408\)

[In]

int(arcsin(b*x+a)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*arcsin(b*x+a)/x^4+1/4*b*(-1/3/(-a^2+1)/x^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+5/3*a*b/(-a^2+1)*(-1/2/(-a^2+1)
/x^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3/2*a*b/(-a^2+1)*(-1/(-a^2+1)/x*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-a*b/(-a^2+1
)^(3/2)*ln((-2*a^2+2-2*a*b*x+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x))-1/2*b^2/(-a^2+1)^(3/2)*ln((-
2*a^2+2-2*a*b*x+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x))+2/3*b^2/(-a^2+1)*(-1/(-a^2+1)/x*(-b^2*x^2
-2*a*b*x-a^2+1)^(1/2)-a*b/(-a^2+1)^(3/2)*ln((-2*a^2+2-2*a*b*x+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))
/x)))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.60 \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\left [-\frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \sqrt {-a^{2} + 1} b^{4} x^{4} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 12 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \arcsin \left (b x + a\right ) - 2 \, {\left ({\left (11 \, a^{4} - 7 \, a^{2} - 4\right )} b^{3} x^{3} - 5 \, {\left (a^{5} - 2 \, a^{3} + a\right )} b^{2} x^{2} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{48 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}, -\frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \sqrt {a^{2} - 1} b^{4} x^{4} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + 6 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \arcsin \left (b x + a\right ) - {\left ({\left (11 \, a^{4} - 7 \, a^{2} - 4\right )} b^{3} x^{3} - 5 \, {\left (a^{5} - 2 \, a^{3} + a\right )} b^{2} x^{2} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{24 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}\right ] \]

[In]

integrate(arcsin(b*x+a)/x^5,x, algorithm="fricas")

[Out]

[-1/48*(3*(2*a^3 + 3*a)*sqrt(-a^2 + 1)*b^4*x^4*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x - 2*sqrt(-b^
2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 12*(a^8 - 4*a^6 + 6*a^4 - 4*a^
2 + 1)*arcsin(b*x + a) - 2*((11*a^4 - 7*a^2 - 4)*b^3*x^3 - 5*(a^5 - 2*a^3 + a)*b^2*x^2 + 2*(a^6 - 3*a^4 + 3*a^
2 - 1)*b*x)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*x^4), -1/24*(3*(2*a^3 + 3*a
)*sqrt(a^2 - 1)*b^4*x^4*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b
^2*x^2 + a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) + 6*(a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*arcsin(b*x + a) - ((11*a^4
- 7*a^2 - 4)*b^3*x^3 - 5*(a^5 - 2*a^3 + a)*b^2*x^2 + 2*(a^6 - 3*a^4 + 3*a^2 - 1)*b*x)*sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1))/((a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*x^4)]

Sympy [F]

\[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\int \frac {\operatorname {asin}{\left (a + b x \right )}}{x^{5}}\, dx \]

[In]

integrate(asin(b*x+a)/x**5,x)

[Out]

Integral(asin(a + b*x)/x**5, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(arcsin(b*x+a)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for mor
e details)Is

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1112 vs. \(2 (158) = 316\).

Time = 0.37 (sec) , antiderivative size = 1112, normalized size of antiderivative = 5.98 \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\text {Too large to display} \]

[In]

integrate(arcsin(b*x+a)/x^5,x, algorithm="giac")

[Out]

-1/12*b*(3*(2*a^3*b^4 + 3*a*b^4)*arctan(((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a/(b^2*x + a*b) - 1)/
sqrt(a^2 - 1))/((a^6*abs(b) - 3*a^4*abs(b) + 3*a^2*abs(b) - abs(b))*sqrt(a^2 - 1)) - (36*(sqrt(-b^2*x^2 - 2*a*
b*x - a^2 + 1)*abs(b) + b)^2*a^7*b^4/(b^2*x + a*b)^2 + 18*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^
7*b^4/(b^2*x + a*b)^4 + 18*a^7*b^4 - 81*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^6*b^4/(b^2*x + a*b)
- 108*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^6*b^4/(b^2*x + a*b)^3 - 27*(sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1)*abs(b) + b)^5*a^6*b^4/(b^2*x + a*b)^5 + 120*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^5*b
^4/(b^2*x + a*b)^2 + 81*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^5*b^4/(b^2*x + a*b)^4 - 5*a^5*b^4
+ 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^4*b^4/(b^2*x + a*b) - 42*(sqrt(-b^2*x^2 - 2*a*b*x - a^2
+ 1)*abs(b) + b)^3*a^4*b^4/(b^2*x + a*b)^3 + 18*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^5*a^4*b^4/(b^
2*x + a*b)^5 - 18*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^3*b^4/(b^2*x + a*b)^2 - 36*(sqrt(-b^2*x^
2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a^3*b^4/(b^2*x + a*b)^4 + 2*a^3*b^4 - 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 +
1)*abs(b) + b)*a^2*b^4/(b^2*x + a*b) + 8*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^2*b^4/(b^2*x + a*
b)^3 - 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^5*a^2*b^4/(b^2*x + a*b)^5 + 12*(sqrt(-b^2*x^2 - 2*a*b
*x - a^2 + 1)*abs(b) + b)^2*a*b^4/(b^2*x + a*b)^2 + 12*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^4*a*b^4
/(b^2*x + a*b)^4 - 8*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*b^4/(b^2*x + a*b)^3)/((a^9*abs(b) - 3*a
^7*abs(b) + 3*a^5*abs(b) - a^3*abs(b))*((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 +
a - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)/(b^2*x + a*b))^3)) - 1/4*arcsin(b*x + a)/x^4

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\int \frac {\mathrm {asin}\left (a+b\,x\right )}{x^5} \,d x \]

[In]

int(asin(a + b*x)/x^5,x)

[Out]

int(asin(a + b*x)/x^5, x)