Integrand size = 10, antiderivative size = 167 \[ \int \frac {\text {arcsinh}(a+b x)}{x^5} \, dx=-\frac {b \sqrt {1+(a+b x)^2}}{12 \left (1+a^2\right ) x^3}+\frac {5 a b^2 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^2 x^2}+\frac {\left (4-11 a^2\right ) b^3 \sqrt {1+(a+b x)^2}}{24 \left (1+a^2\right )^3 x}-\frac {\text {arcsinh}(a+b x)}{4 x^4}-\frac {a \left (3-2 a^2\right ) b^4 \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{8 \left (1+a^2\right )^{7/2}} \]
-1/4*arcsinh(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 = 0.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.07 \[ \int \frac {\text {arcsinh}(a+b x)}{x^5} \, dx=\frac {1}{8} \left (-\frac {b \sqrt {1+a^2+2 a b x+b^2 x^2} \left (2+2 a^4-5 a b x-5 a^3 b x-4 b^2 x^2+a^2 \left (4+11 b^2 x^2\right )\right )}{3 \left (1+a^2\right )^3 x^3}-\frac {2 \text {arcsinh}(a+b x)}{x^4}-\frac {a \left (-3+2 a^2\right ) b^4 \log (x)}{\left (1+a^2\right )^{7/2}}+\frac {a \left (-3+2 a^2\right ) b^4 \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{\left (1+a^2\right )^{7/2}}\right ) \]
(-1/3*(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)))/((1 + a^2)^3*x^3) - (2*ArcSinh[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 = 0.41 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6274, 25, 27, 6243, 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.
| \(\displaystyle \int \frac {\text {arcsinh}(a+b x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {\text {arcsinh}(a+b x)}{x^5}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {\text {arcsinh}(a+b x)}{x^5}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b^4 \int -\frac {\text {arcsinh}(a+b x)}{b^5 x^5}d(a+b x)\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle -b^4 \left (\frac {\text {arcsinh}(a+b x)}{4 b^4 x^4}-\frac {1}{4} \int \frac {1}{b^4 x^4 \sqrt {(a+b x)^2+1}}d(a+b x)\right )\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\int \frac {3 a+2 (a+b x)}{b^3 x^3 \sqrt {(a+b x)^2+1}}d(a+b x)}{3 \left (a^2+1\right )}+\frac {\sqrt {(a+b x)^2+1}}{3 \left (a^2+1\right ) b^3 x^3}\right )+\frac {\text {arcsinh}(a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\sqrt {(a+b x)^2+1}}{3 \left (a^2+1\right ) b^3 x^3}-\frac {\int -\frac {3 a+2 (a+b x)}{b^3 x^3 \sqrt {(a+b x)^2+1}}d(a+b x)}{3 \left (a^2+1\right )}\right )+\frac {\text {arcsinh}(a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\sqrt {(a+b x)^2+1}}{3 \left (a^2+1\right ) b^3 x^3}-\frac {\frac {5 a \sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}-\frac {\int \frac {2 \left (2-3 a^2\right )-5 a (a+b x)}{b^2 x^2 \sqrt {(a+b x)^2+1}}d(a+b x)}{2 \left (a^2+1\right )}}{3 \left (a^2+1\right )}\right )+\frac {\text {arcsinh}(a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\sqrt {(a+b x)^2+1}}{3 \left (a^2+1\right ) b^3 x^3}-\frac {\frac {5 a \sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}-\frac {\frac {3 a \left (3-2 a^2\right ) \int -\frac {1}{b x \sqrt {(a+b x)^2+1}}d(a+b x)}{a^2+1}-\frac {\left (4-11 a^2\right ) \sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}}{2 \left (a^2+1\right )}}{3 \left (a^2+1\right )}\right )+\frac {\text {arcsinh}(a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\sqrt {(a+b x)^2+1}}{3 \left (a^2+1\right ) b^3 x^3}-\frac {\frac {5 a \sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}-\frac {-\frac {3 a \left (3-2 a^2\right ) \int \frac {1}{a^2-\frac {(-a (a+b x)-1)^2}{(a+b x)^2+1}+1}d\frac {-a (a+b x)-1}{\sqrt {(a+b x)^2+1}}}{a^2+1}-\frac {\left (4-11 a^2\right ) \sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}}{2 \left (a^2+1\right )}}{3 \left (a^2+1\right )}\right )+\frac {\text {arcsinh}(a+b x)}{4 b^4 x^4}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -b^4 \left (\frac {1}{4} \left (\frac {\sqrt {(a+b x)^2+1}}{3 \left (a^2+1\right ) b^3 x^3}-\frac {\frac {5 a \sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}-\frac {-\frac {3 a \left (3-2 a^2\right ) \text {arctanh}\left (\frac {-a (a+b x)-1}{\sqrt {a^2+1} \sqrt {(a+b x)^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {\left (4-11 a^2\right ) \sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}}{2 \left (a^2+1\right )}}{3 \left (a^2+1\right )}\right )+\frac {\text {arcsinh}(a+b x)}{4 b^4 x^4}\right )\) |
-(b^4*(ArcSinh[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 - 1 1*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))
3.1.66.3.1 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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[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_.) + ArcSinh[(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* ArcSinh[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. \(344\) vs. \(2(145)=290\).
Time = 0.02 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.07
| method | result | size |
| parts | \(-\frac {\operatorname {arcsinh}\left (b x +a \right )}{4 x^{4}}+\frac {b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{3 \left (a^{2}+1\right ) x^{3}}-\frac {5 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {3 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (a^{2}+1\right )}-\frac {2 b^{2} \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{3 \left (a^{2}+1\right )}\right )}{4}\) | \(345\) |
| derivativedivides | \(b^{4} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{12 \left (a^{2}+1\right ) b^{3} x^{3}}-\frac {5 a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {3 a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (a^{2}+1\right )}-\frac {-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}}{6 \left (a^{2}+1\right )}\right )\) | \(359\) |
| default | \(b^{4} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{4 b^{4} x^{4}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{12 \left (a^{2}+1\right ) b^{3} x^{3}}-\frac {5 a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {3 a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{12 \left (a^{2}+1\right )}-\frac {-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}}{6 \left (a^{2}+1\right )}\right )\) | \(359\) |
-1/4*arcsinh(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)))
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (145) = 290\).
Time = 0.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.05 \[ \int \frac {\text {arcsinh}(a+b x)}{x^5} \, dx=\frac {3 \, {\left (2 \, a^{3} - 3 \, a\right )} \sqrt {a^{2} + 1} b^{4} x^{4} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + \sqrt {a^{2} + 1} a + 1\right )} + {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - {\left (11 \, a^{4} + 7 \, a^{2} - 4\right )} b^{4} x^{4} + 6 \, {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} x^{4} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 6 \, {\left (a^{8} + 4 \, a^{6} - {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} x^{4} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left ({\left (11 \, a^{4} + 7 \, a^{2} - 4\right )} b^{3} x^{3} - 5 \, {\left (a^{5} + 2 \, a^{3} + a\right )} b^{2} x^{2} + 2 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{24 \, {\left (a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1\right )} x^{4}} \]
1/24*(3*(2*a^3 - 3*a)*sqrt(a^2 + 1)*b^4*x^4*log(-(a^2*b*x + a^3 + sqrt(b^2 *x^2 + 2*a*b*x + a^2 + 1)*(a^2 + sqrt(a^2 + 1)*a + 1) + (a*b*x + a^2 + 1)* sqrt(a^2 + 1) + a)/x) - (11*a^4 + 7*a^2 - 4)*b^4*x^4 + 6*(a^8 + 4*a^6 + 6* a^4 + 4*a^2 + 1)*x^4*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 6 *(a^8 + 4*a^6 - (a^8 + 4*a^6 + 6*a^4 + 4*a^2 + 1)*x^4 + 6*a^4 + 4*a^2 + 1) *log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - ((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)*s qrt(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)
\[ \int \frac {\text {arcsinh}(a+b x)}{x^5} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x^{5}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (145) = 290\).
Time = 0.19 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.14 \[ \int \frac {\text {arcsinh}(a+b x)}{x^5} \, dx=\frac {1}{24} \, {\left (\frac {15 \, a^{3} b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {7}{2}}} - \frac {9 \, a b^{3} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {5}{2}}} - \frac {15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} b^{2}}{{\left (a^{2} + 1\right )}^{3} x} + \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}}{{\left (a^{2} + 1\right )}^{2} x} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b}{{\left (a^{2} + 1\right )}^{2} x^{2}} - \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (a^{2} + 1\right )} x^{3}}\right )} b - \frac {\operatorname {arsinh}\left (b x + a\right )}{4 \, x^{4}} \]
1/24*(15*a^3*b^3*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x )) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^ 2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(7/2) - 9*a*b^3*arcsinh(2*a*b*x/(s qrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^ 2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(5/2) - 15*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*b^2/((a^2 + 1)^3*x) + 4*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2/((a^2 + 1)^2*x) + 5*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*b/((a^2 + 1)^2*x^2) - 2*sqrt(b^2*x^2 + 2*a*b*x + a ^2 + 1)/((a^2 + 1)*x^3))*b - 1/4*arcsinh(b*x + a)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (145) = 290\).
Time = 0.36 (sec) , antiderivative size = 709, normalized size of antiderivative = 4.25 \[ \int \frac {\text {arcsinh}(a+b x)}{x^5} \, dx=-\frac {1}{24} \, b {\left (\frac {3 \, {\left (2 \, a^{3} b^{3} - 3 \, a b^{3}\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} \sqrt {a^{2} + 1}} - \frac {2 \, {\left (6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a^{3} b^{3} - 16 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{5} b^{3} + 42 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{7} b^{3} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{6} b^{2} {\left | b \right |} + 20 \, a^{8} b^{2} {\left | b \right |} - 9 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{5} a b^{3} + 8 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{3} b^{3} + 93 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{5} b^{3} + 36 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{4} b^{2} {\left | b \right |} + 56 \, a^{6} b^{2} {\left | b \right |} + 24 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a b^{3} + 60 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{3} b^{3} + 36 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{2} b^{2} {\left | b \right |} + 48 \, a^{4} b^{2} {\left | b \right |} + 9 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a b^{3} + 12 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} b^{2} {\left | b \right |} + 8 \, a^{2} b^{2} {\left | b \right |} - 4 \, b^{2} {\left | b \right |}\right )}}{{\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )}^{3}}\right )} - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}{4 \, x^{4}} \]
-1/24*b*(3*(2*a^3*b^3 - 3*a*b^3)*log(abs(-2*x*abs(b) + 2*sqrt(b^2*x^2 + 2* a*b*x + a^2 + 1) - 2*sqrt(a^2 + 1))/abs(-2*x*abs(b) + 2*sqrt(b^2*x^2 + 2*a *b*x + a^2 + 1) + 2*sqrt(a^2 + 1)))/((a^6 + 3*a^4 + 3*a^2 + 1)*sqrt(a^2 + 1)) - 2*(6*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^5*a^3*b^3 - 16*( x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*a^5*b^3 + 42*(x*abs(b) - s qrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^7*b^3 + 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a^6*b^2*abs(b) + 20*a^8*b^2*abs(b) - 9*(x*abs(b) - s qrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^5*a*b^3 + 8*(x*abs(b) - sqrt(b^2*x^2 + 2 *a*b*x + a^2 + 1))^3*a^3*b^3 + 93*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^5*b^3 + 36*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a^4*b ^2*abs(b) + 56*a^6*b^2*abs(b) + 24*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^ 2 + 1))^3*a*b^3 + 60*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^3*b^ 3 + 36*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a^2*b^2*abs(b) + 4 8*a^4*b^2*abs(b) + 9*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a*b^3 + 12*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2*b^2*abs(b) + 8*a^2*b ^2*abs(b) - 4*b^2*abs(b))/((a^6 + 3*a^4 + 3*a^2 + 1)*((x*abs(b) - sqrt(b^2 *x^2 + 2*a*b*x + a^2 + 1))^2 - a^2 - 1)^3)) - 1/4*log(b*x + a + sqrt((b*x + a)^2 + 1))/x^4
Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{x^5} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x^5} \,d x \]