Integrand size = 10, antiderivative size = 186 \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\arcsin (a+b x)}{4 x^4}-\frac {a \left (3+2 a^2\right ) b^4 \text {arctanh}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^{7/2}} \]
-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
Time = 0.34 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.04 \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\frac {1}{8} \left (\frac {b \sqrt {1-a^2-2 a b x-b^2 x^2} \left (2+2 a^4+5 a b x-5 a^3 b x+4 b^2 x^2+a^2 \left (-4+11 b^2 x^2\right )\right )}{3 \left (-1+a^2\right )^3 x^3}-\frac {2 \arcsin (a+b x)}{x^4}+\frac {a \left (3+2 a^2\right ) b^4 \log (x)}{\left (1-a^2\right )^{7/2}}-\frac {a \left (3+2 a^2\right ) b^4 \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {1-a^2-2 a b x-b^2 x^2}\right )}{\left (1-a^2\right )^{7/2}}\right ) \]
((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
Time = 0.40 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5304, 25, 27, 5242, 498, 25, 688, 679, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arcsin (a+b x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {\arcsin (a+b x)}{x^5}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {\arcsin (a+b x)}{x^5}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b^4 \int -\frac {\arcsin (a+b x)}{b^5 x^5}d(a+b x)\) |
| \(\Big \downarrow \) 5242 |
| \(\displaystyle -b^4 \left (\frac {\arcsin (a+b x)}{4 b^4 x^4}-\frac {1}{4} \int \frac {1}{b^4 x^4 \sqrt {1-(a+b x)^2}}d(a+b x)\right )\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\sqrt {1-(a+b x)^2}}{3 \left (1-a^2\right ) b^3 x^3}-\frac {\int \frac {3 a+2 (a+b x)}{b^3 x^3 \sqrt {1-(a+b x)^2}}d(a+b x)}{3 \left (1-a^2\right )}\right )+\frac {\arcsin (a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\int -\frac {3 a+2 (a+b x)}{b^3 x^3 \sqrt {1-(a+b x)^2}}d(a+b x)}{3 \left (1-a^2\right )}+\frac {\sqrt {1-(a+b x)^2}}{3 \left (1-a^2\right ) b^3 x^3}\right )+\frac {\arcsin (a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\frac {5 a \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}-\frac {\int \frac {2 \left (3 a^2+2\right )+5 a (a+b x)}{b^2 x^2 \sqrt {1-(a+b x)^2}}d(a+b x)}{2 \left (1-a^2\right )}}{3 \left (1-a^2\right )}+\frac {\sqrt {1-(a+b x)^2}}{3 \left (1-a^2\right ) b^3 x^3}\right )+\frac {\arcsin (a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\frac {5 a \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}-\frac {-\frac {3 a \left (2 a^2+3\right ) \int -\frac {1}{b x \sqrt {1-(a+b x)^2}}d(a+b x)}{1-a^2}-\frac {\left (11 a^2+4\right ) \sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}}{2 \left (1-a^2\right )}}{3 \left (1-a^2\right )}+\frac {\sqrt {1-(a+b x)^2}}{3 \left (1-a^2\right ) b^3 x^3}\right )+\frac {\arcsin (a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\frac {5 a \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}-\frac {\frac {3 a \left (2 a^2+3\right ) \int \frac {1}{-a^2-\frac {(a (a+b x)-1)^2}{1-(a+b x)^2}+1}d\frac {a (a+b x)-1}{\sqrt {1-(a+b x)^2}}}{1-a^2}-\frac {\left (11 a^2+4\right ) \sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}}{2 \left (1-a^2\right )}}{3 \left (1-a^2\right )}+\frac {\sqrt {1-(a+b x)^2}}{3 \left (1-a^2\right ) b^3 x^3}\right )+\frac {\arcsin (a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\frac {5 a \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) b^2 x^2}-\frac {\frac {3 a \left (2 a^2+3\right ) \text {arctanh}\left (\frac {a (a+b x)-1}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{\left (1-a^2\right )^{3/2}}-\frac {\left (11 a^2+4\right ) \sqrt {1-(a+b x)^2}}{\left (1-a^2\right ) b x}}{2 \left (1-a^2\right )}}{3 \left (1-a^2\right )}+\frac {\sqrt {1-(a+b x)^2}}{3 \left (1-a^2\right ) b^3 x^3}\right )+\frac {\arcsin (a+b x)}{4 b^4 x^4}\right )\) |
-(b^4*(ArcSin[a + b*x]/(4*b^4*x^4) + (Sqrt[1 - (a + b*x)^2]/(3*(1 - a^2)*b ^3*x^3) + ((5*a*Sqrt[1 - (a + b*x)^2])/(2*(1 - a^2)*b^2*x^2) - (-(((4 + 11 *a^2)*Sqrt[1 - (a + b*x)^2])/((1 - a^2)*b*x)) + (3*a*(3 + 2*a^2)*ArcTanh[( -1 + a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/(1 - a^2)^(3/2)) /(2*(1 - a^2)))/(3*(1 - a^2)))/4))
3.2.30.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
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] + Simp[(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] && EqQ[Simplify[m + 2*p + 3], 0]
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] + Simp[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] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Simp[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]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
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 | \(b^{4} \left (-\frac {\arcsin \left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{12 \left (-a^{2}+1\right ) b^{3} x^{3}}+\frac {5 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b^{2} x^{2}}+\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (-a^{2}+1\right )}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (-a^{2}+1\right )}+\frac {-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\left (-a^{2}+1\right )^{\frac {3}{2}}}}{-6 a^{2}+6}\right )\) | \(408\) |
-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)))
Time = 0.34 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.60 \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\left [-\frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \sqrt {-a^{2} + 1} b^{4} x^{4} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 12 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \arcsin \left (b x + a\right ) - 2 \, {\left ({\left (11 \, a^{4} - 7 \, a^{2} - 4\right )} b^{3} x^{3} - 5 \, {\left (a^{5} - 2 \, a^{3} + a\right )} b^{2} x^{2} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{48 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}, -\frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \sqrt {a^{2} - 1} b^{4} x^{4} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + 6 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \arcsin \left (b x + a\right ) - {\left ({\left (11 \, a^{4} - 7 \, a^{2} - 4\right )} b^{3} x^{3} - 5 \, {\left (a^{5} - 2 \, a^{3} + a\right )} b^{2} x^{2} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{24 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}\right ] \]
[-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)*sq rt(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)]
\[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\int \frac {\operatorname {asin}{\left (a + b x \right )}}{x^{5}}\, dx \]
Exception generated. \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more details)Is
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 \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\text {Too large to display} \]
-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*ab s(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) - 10 8*(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)*ab s(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*(sq rt(-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 + 1 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a*b^4/(b^2*x + a*b)...
Timed out. \[ \int \frac {\arcsin (a+b x)}{x^5} \, dx=\int \frac {\mathrm {asin}\left (a+b\,x\right )}{x^5} \,d x \]