3.1.78 \(\int \frac {\text {arcsinh}(a+b x)^3}{x} \, dx\) [78]

3.1.78.1 Optimal result

Integrand size = 12, antiderivative size = 275 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x} \, dx=-\frac {1}{4} \text {arcsinh}(a+b x)^4+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+3 \text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+3 \text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )-6 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )-6 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \]

output

-1/4*arcsinh(b*x+a)^4+arcsinh(b*x+a)^3*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a 
-(a^2+1)^(1/2)))+arcsinh(b*x+a)^3*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2 
+1)^(1/2)))+3*arcsinh(b*x+a)^2*polylog(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a 
^2+1)^(1/2)))+3*arcsinh(b*x+a)^2*polylog(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+ 
(a^2+1)^(1/2)))-6*arcsinh(b*x+a)*polylog(3,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a- 
(a^2+1)^(1/2)))-6*arcsinh(b*x+a)*polylog(3,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+ 
(a^2+1)^(1/2)))+6*polylog(4,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2))) 
+6*polylog(4,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))
 

3.1.78.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.26 \[ \int \frac {\text {arcsinh}(a+b x)^3}{x} \, dx=-\frac {1}{4} \text {arcsinh}(a+b x)^4+\text {arcsinh}(a+b x)^3 \log \left (1+\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+\text {arcsinh}(a+b x)^3 \log \left (1+\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+3 \text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+3 \text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right )-6 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right )-6 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+6 \operatorname {PolyLog}\left (4,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+6 \operatorname {PolyLog}\left (4,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \]

input

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

output

-1/4*ArcSinh[a + b*x]^4 + ArcSinh[a + b*x]^3*Log[1 + E^ArcSinh[a + b*x]/(( 
-(a/b) - Sqrt[1 + a^2]/b)*b)] + ArcSinh[a + b*x]^3*Log[1 + E^ArcSinh[a + b 
*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b)] + 3*ArcSinh[a + b*x]^2*PolyLog[2, -(E^ 
ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b))] + 3*ArcSinh[a + b*x]^2*P 
olyLog[2, -(E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b))] - 6*ArcSin 
h[a + b*x]*PolyLog[3, -(E^ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b)) 
] - 6*ArcSinh[a + b*x]*PolyLog[3, -(E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + 
 a^2]/b)*b))] + 6*PolyLog[4, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*P 
olyLog[4, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]
 

3.1.78.3 Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6274, 25, 27, 6242, 6095, 2620, 3011, 7163, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6242

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

\(\Big \downarrow \) 6095

\(\displaystyle -\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)^3}{a-e^{\text {arcsinh}(a+b x)}-\sqrt {a^2+1}}d\text {arcsinh}(a+b x)-\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)^3}{a-e^{\text {arcsinh}(a+b x)}+\sqrt {a^2+1}}d\text {arcsinh}(a+b x)-\frac {1}{4} \text {arcsinh}(a+b x)^4\)

\(\Big \downarrow \) 2620

\(\displaystyle -3 \int \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-3 \int \text {arcsinh}(a+b x)^2 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{4} \text {arcsinh}(a+b x)^4\)

\(\Big \downarrow \) 3011

\(\displaystyle -3 \left (2 \int \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )\right )-3 \left (2 \int \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{4} \text {arcsinh}(a+b x)^4\)

\(\Big \downarrow \) 7163

\(\displaystyle -3 \left (2 \left (\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )-\int \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)\right )-\text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )\right )-3 \left (2 \left (\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )-\int \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)\right )-\text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{4} \text {arcsinh}(a+b x)^4\)

\(\Big \downarrow \) 2720

\(\displaystyle -3 \left (2 \left (\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )-\int e^{-\text {arcsinh}(a+b x)} \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )de^{\text {arcsinh}(a+b x)}\right )-\text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )\right )-3 \left (2 \left (\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )-\int e^{-\text {arcsinh}(a+b x)} \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )de^{\text {arcsinh}(a+b x)}\right )-\text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{4} \text {arcsinh}(a+b x)^4\)

\(\Big \downarrow \) 7143

\(\displaystyle -3 \left (2 \left (\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )-\operatorname {PolyLog}\left (4,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )\right )-\text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )\right )-3 \left (2 \left (\text {arcsinh}(a+b x) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )-\operatorname {PolyLog}\left (4,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )\right )-\text {arcsinh}(a+b x)^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x)^3 \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{4} \text {arcsinh}(a+b x)^4\)

input

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

output

-1/4*ArcSinh[a + b*x]^4 + ArcSinh[a + b*x]^3*Log[1 - E^ArcSinh[a + b*x]/(a 
 - Sqrt[1 + a^2])] + ArcSinh[a + b*x]^3*Log[1 - E^ArcSinh[a + b*x]/(a + Sq 
rt[1 + a^2])] - 3*(-(ArcSinh[a + b*x]^2*PolyLog[2, E^ArcSinh[a + b*x]/(a - 
 Sqrt[1 + a^2])]) + 2*(ArcSinh[a + b*x]*PolyLog[3, E^ArcSinh[a + b*x]/(a - 
 Sqrt[1 + a^2])] - PolyLog[4, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])) - 
3*(-(ArcSinh[a + b*x]^2*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] 
) + 2*(ArcSinh[a + b*x]*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] 
 - PolyLog[4, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]))
 

3.1.78.3.1 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 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 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 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6242

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo 
l] :> Subst[Int[(a + b*x)^n*(Cosh[x]/(c*d + e*Sinh[x])), x], x, ArcSinh[c*x 
]] /; FreeQ[{a, b, c, d, e}, x] && 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]
 

rule 7163

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

3.1.78.4 Maple [F]

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

input

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

output

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

3.1.78.5 Fricas [F]

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

input

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

output

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

3.1.78.6 Sympy [F]

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

input

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

output

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

3.1.78.7 Maxima [F]

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

input

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

output

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

3.1.78.8 Giac [F]

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

input

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

output

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

3.1.78.9 Mupad [F(-1)]

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

input

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

output

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