3.1.79 \(\int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx\) [79]

3.1.79.1 Optimal result

Integrand size = 12, antiderivative size = 268 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=-\frac {\text {arcsinh}(a+b x)^3}{x}-\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {3 b \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {6 b \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}+\frac {6 b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}}-\frac {6 b \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2}} \]

output

-arcsinh(b*x+a)^3/x-3*b*arcsinh(b*x+a)^2*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/ 
(a-(a^2+1)^(1/2)))/(a^2+1)^(1/2)+3*b*arcsinh(b*x+a)^2*ln(1-(b*x+a+(1+(b*x+ 
a)^2)^(1/2))/(a+(a^2+1)^(1/2)))/(a^2+1)^(1/2)-6*b*arcsinh(b*x+a)*polylog(2 
,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))/(a^2+1)^(1/2)+6*b*arcsinh( 
b*x+a)*polylog(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))/(a^2+1)^(1 
/2)+6*b*polylog(3,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))/(a^2+1)^( 
1/2)-6*b*polylog(3,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))/(a^2+1)^ 
(1/2)
 

3.1.79.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.97 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=-\frac {\sqrt {1+a^2} \text {arcsinh}(a+b x)^3-3 b x \text {arcsinh}(a+b x)^2 \log \left (\frac {a+\sqrt {1+a^2}-e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+3 b x \text {arcsinh}(a+b x)^2 \log \left (\frac {-a+\sqrt {1+a^2}+e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )+6 b x \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )-6 b x \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-6 b x \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+6 b x \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\sqrt {1+a^2} x} \]

input

Integrate[ArcSinh[a + b*x]^3/x^2,x]
 

output

-((Sqrt[1 + a^2]*ArcSinh[a + b*x]^3 - 3*b*x*ArcSinh[a + b*x]^2*Log[(a + Sq 
rt[1 + a^2] - E^ArcSinh[a + b*x])/(a + Sqrt[1 + a^2])] + 3*b*x*ArcSinh[a + 
 b*x]^2*Log[(-a + Sqrt[1 + a^2] + E^ArcSinh[a + b*x])/(-a + Sqrt[1 + a^2]) 
] + 6*b*x*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2 
])] - 6*b*x*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a 
^2])] - 6*b*x*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*b*x*P 
olyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(Sqrt[1 + a^2]*x))
 

3.1.79.3 Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6274, 27, 6243, 6258, 3042, 3803, 2694, 27, 2620, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6243

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

\(\Big \downarrow \) 6258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

\(\displaystyle b \left (-6 \int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)^2}{2 e^{\text {arcsinh}(a+b x)} a-e^{2 \text {arcsinh}(a+b x)}+1}d\text {arcsinh}(a+b x)-\frac {\text {arcsinh}(a+b x)^3}{b x}\right )\)

\(\Big \downarrow \) 2694

\(\displaystyle b \left (-6 \left (\frac {\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)^2}{2 \left (a-e^{\text {arcsinh}(a+b x)}+\sqrt {a^2+1}\right )}d\text {arcsinh}(a+b x)}{\sqrt {a^2+1}}-\frac {\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)^2}{2 \left (a-e^{\text {arcsinh}(a+b x)}-\sqrt {a^2+1}\right )}d\text {arcsinh}(a+b x)}{\sqrt {a^2+1}}\right )-\frac {\text {arcsinh}(a+b x)^3}{b x}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle b \left (-6 \left (\frac {\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)^2}{a-e^{\text {arcsinh}(a+b x)}+\sqrt {a^2+1}}d\text {arcsinh}(a+b x)}{2 \sqrt {a^2+1}}-\frac {\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)^2}{a-e^{\text {arcsinh}(a+b x)}-\sqrt {a^2+1}}d\text {arcsinh}(a+b x)}{2 \sqrt {a^2+1}}\right )-\frac {\text {arcsinh}(a+b x)^3}{b x}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle b \left (-6 \left (\frac {2 \int \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {2 \int \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )-\frac {\text {arcsinh}(a+b x)^3}{b x}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle b \left (-6 \left (\frac {2 \left (\int \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )\right )-\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {2 \left (\int \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )\right )-\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )-\frac {\text {arcsinh}(a+b x)^3}{b x}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle b \left (-6 \left (\frac {2 \left (\int e^{-\text {arcsinh}(a+b x)} \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )de^{\text {arcsinh}(a+b x)}-\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )\right )-\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {2 \left (\int e^{-\text {arcsinh}(a+b x)} \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )de^{\text {arcsinh}(a+b x)}-\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )\right )-\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )-\frac {\text {arcsinh}(a+b x)^3}{b x}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle b \left (-6 \left (\frac {2 \left (\operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )-\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )\right )-\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{2 \sqrt {a^2+1}}-\frac {2 \left (\operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )-\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )\right )-\text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{2 \sqrt {a^2+1}}\right )-\frac {\text {arcsinh}(a+b x)^3}{b x}\right )\)

input

Int[ArcSinh[a + b*x]^3/x^2,x]
 

output

b*(-(ArcSinh[a + b*x]^3/(b*x)) - 6*(-1/2*(-(ArcSinh[a + b*x]^2*Log[1 - E^A 
rcSinh[a + b*x]/(a - Sqrt[1 + a^2])]) + 2*(-(ArcSinh[a + b*x]*PolyLog[2, E 
^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])]) + PolyLog[3, E^ArcSinh[a + b*x]/(a 
 - Sqrt[1 + a^2])]))/Sqrt[1 + a^2] + (-(ArcSinh[a + b*x]^2*Log[1 - E^ArcSi 
nh[a + b*x]/(a + Sqrt[1 + a^2])]) + 2*(-(ArcSinh[a + b*x]*PolyLog[2, E^Arc 
Sinh[a + b*x]/(a + Sqrt[1 + a^2])]) + PolyLog[3, E^ArcSinh[a + b*x]/(a + S 
qrt[1 + a^2])]))/(2*Sqrt[1 + a^2])))
 

3.1.79.3.1 Defintions of rubi rules used

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* 
(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( 
-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; 
FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 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 6258

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/S 
qrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[I 
nt[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b 
, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (Gt 
Q[m, 0] || IGtQ[n, 0])
 

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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

3.1.79.4 Maple [F]

\[\int \frac {\operatorname {arcsinh}\left (b x +a \right )^{3}}{x^{2}}d x\]

input

int(arcsinh(b*x+a)^3/x^2,x)
 

output

int(arcsinh(b*x+a)^3/x^2,x)
 

3.1.79.5 Fricas [F]

\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

input

integrate(arcsinh(b*x+a)^3/x^2,x, algorithm="fricas")
 

output

integral(arcsinh(b*x + a)^3/x^2, x)
 

3.1.79.6 Sympy [F]

\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]

input

integrate(asinh(b*x+a)**3/x**2,x)
 

output

Integral(asinh(a + b*x)**3/x**2, x)
 

3.1.79.7 Maxima [F]

\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

input

integrate(arcsinh(b*x+a)^3/x^2,x, algorithm="maxima")
 

output

-log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3/x + integrate(3*(b^3*x 
^2 + 2*a*b^2*x + a^2*b + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*x + a*b) + 
 b)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2/(b^3*x^4 + 3*a*b^2* 
x^3 + (3*a^2*b + b)*x^2 + (a^3 + a)*x + (b^2*x^3 + 2*a*b*x^2 + (a^2 + 1)*x 
)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)), x)
 

3.1.79.8 Giac [F]

\[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{3}}{x^{2}} \,d x } \]

input

integrate(arcsinh(b*x+a)^3/x^2,x, algorithm="giac")
 

output

integrate(arcsinh(b*x + a)^3/x^2, x)
 

3.1.79.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^3}{x^2} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^3}{x^2} \,d x \]

input

int(asinh(a + b*x)^3/x^2,x)
 

output

int(asinh(a + b*x)^3/x^2, x)