3.1.65 \(\int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx\) [65]

3.1.65.1 Optimal result

Integrand size = 10, antiderivative size = 129 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=-\frac {b \sqrt {1+(a+b x)^2}}{6 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right )^2 x}-\frac {\text {arcsinh}(a+b x)}{3 x^3}+\frac {\left (1-2 a^2\right ) b^3 \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{6 \left (1+a^2\right )^{5/2}} \]

output

-1/3*arcsinh(b*x+a)/x^3+1/6*(-2*a^2+1)*b^3*arctanh((1+a*(b*x+a))/(a^2+1)^( 
1/2)/(1+(b*x+a)^2)^(1/2))/(a^2+1)^(5/2)-1/6*b*(1+(b*x+a)^2)^(1/2)/(a^2+1)/ 
x^2+1/2*a*b^2*(1+(b*x+a)^2)^(1/2)/(a^2+1)^2/x
 

3.1.65.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.16 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\frac {-\sqrt {1+a^2} b x \left (1+a^2-3 a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}-2 \left (1+a^2\right )^{5/2} \text {arcsinh}(a+b x)+\left (-1+2 a^2\right ) b^3 x^3 \log (x)+\left (1-2 a^2\right ) b^3 x^3 \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{6 \left (1+a^2\right )^{5/2} x^3} \]

input

Integrate[ArcSinh[a + b*x]/x^4,x]
 

output

(-(Sqrt[1 + a^2]*b*x*(1 + a^2 - 3*a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] 
) - 2*(1 + a^2)^(5/2)*ArcSinh[a + b*x] + (-1 + 2*a^2)*b^3*x^3*Log[x] + (1 
- 2*a^2)*b^3*x^3*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b* 
x + b^2*x^2]])/(6*(1 + a^2)^(5/2)*x^3)
 

3.1.65.3 Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6274, 27, 6243, 498, 25, 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.

\(\displaystyle \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \frac {\text {arcsinh}(a+b x)}{x^4}d(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle b^3 \int \frac {\text {arcsinh}(a+b x)}{b^4 x^4}d(a+b x)\)

\(\Big \downarrow \) 6243

\(\displaystyle b^3 \left (-\frac {1}{3} \int -\frac {1}{b^3 x^3 \sqrt {(a+b x)^2+1}}d(a+b x)-\frac {\text {arcsinh}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 498

\(\displaystyle b^3 \left (\frac {1}{3} \left (\frac {\int -\frac {3 a+b x}{b^2 x^2 \sqrt {(a+b x)^2+1}}d(a+b x)}{2 \left (a^2+1\right )}-\frac {\sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}\right )-\frac {\text {arcsinh}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle b^3 \left (\frac {1}{3} \left (-\frac {\int \frac {3 a+b x}{b^2 x^2 \sqrt {(a+b x)^2+1}}d(a+b x)}{2 \left (a^2+1\right )}-\frac {\sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}\right )-\frac {\text {arcsinh}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 679

\(\displaystyle b^3 \left (\frac {1}{3} \left (-\frac {-\frac {\left (1-2 a^2\right ) \int -\frac {1}{b x \sqrt {(a+b x)^2+1}}d(a+b x)}{a^2+1}-\frac {3 a \sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}}{2 \left (a^2+1\right )}-\frac {\sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}\right )-\frac {\text {arcsinh}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 488

\(\displaystyle b^3 \left (\frac {1}{3} \left (-\frac {\frac {\left (1-2 a^2\right ) \int \frac {1}{a^2-\frac {(-a (a+b x)-1)^2}{(a+b x)^2+1}+1}d\frac {-a (a+b x)-1}{\sqrt {(a+b x)^2+1}}}{a^2+1}-\frac {3 a \sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}}{2 \left (a^2+1\right )}-\frac {\sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}\right )-\frac {\text {arcsinh}(a+b x)}{3 b^3 x^3}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle b^3 \left (\frac {1}{3} \left (-\frac {\frac {\left (1-2 a^2\right ) \text {arctanh}\left (\frac {-a (a+b x)-1}{\sqrt {a^2+1} \sqrt {(a+b x)^2+1}}\right )}{\left (a^2+1\right )^{3/2}}-\frac {3 a \sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}}{2 \left (a^2+1\right )}-\frac {\sqrt {(a+b x)^2+1}}{2 \left (a^2+1\right ) b^2 x^2}\right )-\frac {\text {arcsinh}(a+b x)}{3 b^3 x^3}\right )\)

input

Int[ArcSinh[a + b*x]/x^4,x]
 

output

b^3*(-1/3*ArcSinh[a + b*x]/(b^3*x^3) + (-1/2*Sqrt[1 + (a + b*x)^2]/((1 + a 
^2)*b^2*x^2) - ((-3*a*Sqrt[1 + (a + b*x)^2])/((1 + a^2)*b*x) + ((1 - 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)
 

3.1.65.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 488

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]
 

rule 498

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])
 

rule 679

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]
 

rule 6243

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]
 

rule 6274

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]
 

3.1.65.4 Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.57

method	result	size
parts	\(-\frac {\operatorname {arcsinh}\left (b x +a \right )}{3 x^{3}}+\frac {b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {3 a b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{3}\)	\(203\)
derivativedivides	\(b^{3} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{3 b^{3} x^{3}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{6 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\)	\(213\)
default	\(b^{3} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{3 b^{3} x^{3}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 \left (a^{2}+1\right ) b^{2} x^{2}}-\frac {a \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}+\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{6 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\)	\(213\)

input

int(arcsinh(b*x+a)/x^4,x,method=_RETURNVERBOSE)
 

output

-1/3*arcsinh(b*x+a)/x^3+1/3*b*(-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))
 

3.1.65.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (111) = 222\).

Time = 0.29 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.21 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\frac {{\left (2 \, a^{2} - 1\right )} \sqrt {a^{2} + 1} b^{3} x^{3} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) + 3 \, {\left (a^{3} + a\right )} b^{3} x^{3} + 2 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 2 \, {\left (a^{6} + 3 \, a^{4} - {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3} + 3 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (3 \, {\left (a^{3} + a\right )} b^{2} x^{2} - {\left (a^{4} + 2 \, a^{2} + 1\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{6 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \]

input

integrate(arcsinh(b*x+a)/x^4,x, algorithm="fricas")
 

output

1/6*((2*a^2 - 1)*sqrt(a^2 + 1)*b^3*x^3*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) + 3*(a^3 + a)*b^3*x^3 + 2*(a^6 + 3*a^4 + 3*a^2 + 1)*x^3*l 
og(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 2*(a^6 + 3*a^4 - (a^6 + 
 3*a^4 + 3*a^2 + 1)*x^3 + 3*a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 + 1)) + (3*(a^3 + a)*b^2*x^2 - (a^4 + 2*a^2 + 1)*b*x)*sqrt(b^2*x^2 + 
 2*a*b*x + a^2 + 1))/((a^6 + 3*a^4 + 3*a^2 + 1)*x^3)
 

3.1.65.6 Sympy [F]

\[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x^{4}}\, dx \]

input

integrate(asinh(b*x+a)/x**4,x)
 

output

Integral(asinh(a + b*x)/x**4, x)
 

3.1.65.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (111) = 222\).

Time = 0.18 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.20 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=-\frac {1}{6} \, {\left (\frac {3 \, a^{2} b^{2} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {5}{2}}} - \frac {b^{2} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b}{{\left (a^{2} + 1\right )}^{2} x} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (a^{2} + 1\right )} x^{2}}\right )} b - \frac {\operatorname {arsinh}\left (b x + a\right )}{3 \, x^{3}} \]

input

integrate(arcsinh(b*x+a)/x^4,x, algorithm="maxima")
 

output

-1/6*(3*a^2*b^2*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)^(5/2) - b^2*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)^ 
(3/2) - 3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*b/((a^2 + 1)^2*x) + sqrt(b^2 
*x^2 + 2*a*b*x + a^2 + 1)/((a^2 + 1)*x^2))*b - 1/3*arcsinh(b*x + a)/x^3
 

3.1.65.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (111) = 222\).

Time = 0.38 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.95 \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\frac {1}{6} \, b {\left (\frac {{\left (2 \, a^{2} b^{2} - b^{2}\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} + 1\right )} \sqrt {a^{2} + 1}} - \frac {2 \, {\left (2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{2} b^{2} - 6 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{4} b^{2} - 4 \, a^{5} b {\left | b \right |} - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} b^{2} - 7 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{2} b^{2} - 8 \, a^{3} b {\left | b \right |} - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b^{2} - 4 \, a b {\left | b \right |}\right )}}{{\left (a^{4} + 2 \, a^{2} + 1\right )} {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )}^{2}}\right )} - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}{3 \, x^{3}} \]

input

integrate(arcsinh(b*x+a)/x^4,x, algorithm="giac")
 

output

1/6*b*((2*a^2*b^2 - b^2)*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^4 + 2*a^2 + 1)*sqrt(a^2 + 1)) - 2*(2*(x*ab 
s(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*a^2*b^2 - 6*(x*abs(b) - sqrt(b 
^2*x^2 + 2*a*b*x + a^2 + 1))*a^4*b^2 - 4*a^5*b*abs(b) - (x*abs(b) - sqrt(b 
^2*x^2 + 2*a*b*x + a^2 + 1))^3*b^2 - 7*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 + 1))*a^2*b^2 - 8*a^3*b*abs(b) - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 + 1))*b^2 - 4*a*b*abs(b))/((a^4 + 2*a^2 + 1)*((x*abs(b) - sqrt(b^2*x 
^2 + 2*a*b*x + a^2 + 1))^2 - a^2 - 1)^2)) - 1/3*log(b*x + a + sqrt((b*x + 
a)^2 + 1))/x^3
 

3.1.65.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x^4} \,d x \]

input

int(asinh(a + b*x)/x^4,x)
 

output

int(asinh(a + b*x)/x^4, x)