3.1.90 \(\int \frac {x}{\text {arcsinh}(a+b x)^3} \, dx\) [90]

3.1.90.1 Optimal result

Integrand size = 10, antiderivative size = 147 \[ \int \frac {x}{\text {arcsinh}(a+b x)^3} \, dx=\frac {a \sqrt {1+(a+b x)^2}}{2 b^2 \text {arcsinh}(a+b x)^2}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{2 b^2 \text {arcsinh}(a+b x)^2}-\frac {1}{2 b^2 \text {arcsinh}(a+b x)}+\frac {a (a+b x)}{2 b^2 \text {arcsinh}(a+b x)}-\frac {(a+b x)^2}{b^2 \text {arcsinh}(a+b x)}-\frac {a \text {Chi}(\text {arcsinh}(a+b x))}{2 b^2}+\frac {\text {Shi}(2 \text {arcsinh}(a+b x))}{b^2} \]

output

-1/2/b^2/arcsinh(b*x+a)+1/2*a*(b*x+a)/b^2/arcsinh(b*x+a)-(b*x+a)^2/b^2/arc 
sinh(b*x+a)-1/2*a*Chi(arcsinh(b*x+a))/b^2+Shi(2*arcsinh(b*x+a))/b^2+1/2*a* 
(1+(b*x+a)^2)^(1/2)/b^2/arcsinh(b*x+a)^2-1/2*(b*x+a)*(1+(b*x+a)^2)^(1/2)/b 
^2/arcsinh(b*x+a)^2
 

3.1.90.2 Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\text {arcsinh}(a+b x)^3} \, dx=-\frac {b x \sqrt {1+a^2+2 a b x+b^2 x^2}+\text {arcsinh}(a+b x)+a^2 \text {arcsinh}(a+b x)+3 a b x \text {arcsinh}(a+b x)+2 b^2 x^2 \text {arcsinh}(a+b x)+a \text {arcsinh}(a+b x)^2 \text {Chi}(\text {arcsinh}(a+b x))-2 \text {arcsinh}(a+b x)^2 \text {Shi}(2 \text {arcsinh}(a+b x))}{2 b^2 \text {arcsinh}(a+b x)^2} \]

input

Integrate[x/ArcSinh[a + b*x]^3,x]
 

output

-1/2*(b*x*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + ArcSinh[a + b*x] + a^2*ArcSi 
nh[a + b*x] + 3*a*b*x*ArcSinh[a + b*x] + 2*b^2*x^2*ArcSinh[a + b*x] + a*Ar 
cSinh[a + b*x]^2*CoshIntegral[ArcSinh[a + b*x]] - 2*ArcSinh[a + b*x]^2*Sin 
hIntegral[2*ArcSinh[a + b*x]])/(b^2*ArcSinh[a + b*x]^2)
 

3.1.90.3 Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6274, 25, 27, 6244, 2009}

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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6244

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {1}{2} a \text {Chi}(\text {arcsinh}(a+b x))-\text {Shi}(2 \text {arcsinh}(a+b x))+\frac {(a+b x)^2}{\text {arcsinh}(a+b x)}-\frac {a (a+b x)}{2 \text {arcsinh}(a+b x)}+\frac {\sqrt {(a+b x)^2+1} (a+b x)}{2 \text {arcsinh}(a+b x)^2}+\frac {1}{2 \text {arcsinh}(a+b x)}-\frac {a \sqrt {(a+b x)^2+1}}{2 \text {arcsinh}(a+b x)^2}}{b^2}\)

input

Int[x/ArcSinh[a + b*x]^3,x]
 

output

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

3.1.90.3.1 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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6244

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_S 
ymbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x] /; 
 FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -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]
 

3.1.90.4 Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4 \operatorname {arcsinh}\left (b x +a \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2 \,\operatorname {arcsinh}\left (b x +a \right )}+\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-\frac {a \left (\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}-\left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{2 \operatorname {arcsinh}\left (b x +a \right )^{2}}}{b^{2}}\)	\(107\)
default	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{4 \operatorname {arcsinh}\left (b x +a \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2 \,\operatorname {arcsinh}\left (b x +a \right )}+\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-\frac {a \left (\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}-\left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{2 \operatorname {arcsinh}\left (b x +a \right )^{2}}}{b^{2}}\)	\(107\)

input

int(x/arcsinh(b*x+a)^3,x,method=_RETURNVERBOSE)
 

output

1/b^2*(-1/4/arcsinh(b*x+a)^2*sinh(2*arcsinh(b*x+a))-1/2/arcsinh(b*x+a)*cos 
h(2*arcsinh(b*x+a))+Shi(2*arcsinh(b*x+a))-1/2*a*(Chi(arcsinh(b*x+a))*arcsi 
nh(b*x+a)^2-(b*x+a)*arcsinh(b*x+a)-(1+(b*x+a)^2)^(1/2))/arcsinh(b*x+a)^2)
 

3.1.90.5 Fricas [F]

\[ \int \frac {x}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]

input

integrate(x/arcsinh(b*x+a)^3,x, algorithm="fricas")
 

output

integral(x/arcsinh(b*x + a)^3, x)
 

3.1.90.6 Sympy [F]

\[ \int \frac {x}{\text {arcsinh}(a+b x)^3} \, dx=\int \frac {x}{\operatorname {asinh}^{3}{\left (a + b x \right )}}\, dx \]

input

integrate(x/asinh(b*x+a)**3,x)
 

output

Integral(x/asinh(a + b*x)**3, x)
 

3.1.90.7 Maxima [F]

\[ \int \frac {x}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]

input

integrate(x/arcsinh(b*x+a)^3,x, algorithm="maxima")
 

output

-1/2*(b^8*x^8 + 7*a*b^7*x^7 + 3*(7*a^2*b^6 + b^6)*x^6 + 5*(7*a^3*b^5 + 3*a 
*b^5)*x^5 + (35*a^4*b^4 + 30*a^2*b^4 + 3*b^4)*x^4 + 3*(7*a^5*b^3 + 10*a^3* 
b^3 + 3*a*b^3)*x^3 + (7*a^6*b^2 + 15*a^4*b^2 + 9*a^2*b^2 + b^2)*x^2 + (b^5 
*x^5 + 4*a*b^4*x^4 + (6*a^2*b^3 + b^3)*x^3 + 2*(2*a^3*b^2 + a*b^2)*x^2 + ( 
a^4*b + a^2*b)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (3*b^6*x^6 + 15*a* 
b^5*x^5 + 5*(6*a^2*b^4 + b^4)*x^4 + 15*(2*a^3*b^3 + a*b^3)*x^3 + (15*a^4*b 
^2 + 15*a^2*b^2 + 2*b^2)*x^2 + (3*a^5*b + 5*a^3*b + 2*a*b)*x)*(b^2*x^2 + 2 
*a*b*x + a^2 + 1) + (a^7*b + 3*a^5*b + 3*a^3*b + a*b)*x + (2*b^8*x^8 + 15* 
a*b^7*x^7 + a^8 + (49*a^2*b^6 + 6*b^6)*x^6 + 3*a^6 + (91*a^3*b^5 + 33*a*b^ 
5)*x^5 + 3*(35*a^4*b^4 + 25*a^2*b^4 + 2*b^4)*x^4 + 3*a^4 + (77*a^5*b^3 + 9 
0*a^3*b^3 + 21*a*b^3)*x^3 + (35*a^6*b^2 + 60*a^4*b^2 + 27*a^2*b^2 + 2*b^2) 
*x^2 + (2*b^5*x^5 + 9*a*b^4*x^4 + a^5 + 2*(8*a^2*b^3 + b^3)*x^3 + 2*a^3 + 
2*(7*a^3*b^2 + 3*a*b^2)*x^2 + 6*(a^4*b + a^2*b)*x + a)*(b^2*x^2 + 2*a*b*x 
+ a^2 + 1)^(3/2) + (6*b^6*x^6 + 33*a*b^5*x^5 + 3*a^6 + 5*(15*a^2*b^4 + 2*b 
^4)*x^4 + 7*a^4 + (90*a^3*b^3 + 37*a*b^3)*x^3 + (60*a^4*b^2 + 51*a^2*b^2 + 
 5*b^2)*x^2 + 5*a^2 + (21*a^5*b + 31*a^3*b + 10*a*b)*x + 1)*(b^2*x^2 + 2*a 
*b*x + a^2 + 1) + a^2 + 3*(3*a^7*b + 7*a^5*b + 5*a^3*b + a*b)*x + (6*b^7*x 
^7 + 39*a*b^6*x^6 + 3*a^7 + 2*(54*a^2*b^5 + 7*b^5)*x^5 + 8*a^5 + (165*a^3* 
b^4 + 64*a*b^4)*x^4 + (150*a^4*b^3 + 116*a^2*b^3 + 11*b^3)*x^3 + 7*a^3 + ( 
81*a^5*b^2 + 104*a^3*b^2 + 29*a*b^2)*x^2 + (24*a^6*b + 46*a^4*b + 25*a^...
 

3.1.90.8 Giac [F]

\[ \int \frac {x}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]

input

integrate(x/arcsinh(b*x+a)^3,x, algorithm="giac")
 

output

integrate(x/arcsinh(b*x + a)^3, x)
 

3.1.90.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\text {arcsinh}(a+b x)^3} \, dx=\int \frac {x}{{\mathrm {asinh}\left (a+b\,x\right )}^3} \,d x \]

input

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

output

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