3.1.3 \(\int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx\) [3]

3.1.3.1 Optimal result

Integrand size = 14, antiderivative size = 348 \[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=-\frac {\text {arcsinh}(c x)^4}{4 e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \]

output

-1/4*arcsinh(c*x)^4/e+arcsinh(c*x)^3*ln(1+e*(c*x+(c^2*x^2+1)^(1/2))/(c*d-( 
c^2*d^2+e^2)^(1/2)))/e+arcsinh(c*x)^3*ln(1+e*(c*x+(c^2*x^2+1)^(1/2))/(c*d+ 
(c^2*d^2+e^2)^(1/2)))/e+3*arcsinh(c*x)^2*polylog(2,-e*(c*x+(c^2*x^2+1)^(1/ 
2))/(c*d-(c^2*d^2+e^2)^(1/2)))/e+3*arcsinh(c*x)^2*polylog(2,-e*(c*x+(c^2*x 
^2+1)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)))/e-6*arcsinh(c*x)*polylog(3,-e*(c*x 
+(c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2+e^2)^(1/2)))/e-6*arcsinh(c*x)*polylog(3, 
-e*(c*x+(c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)))/e+6*polylog(4,-e*(c* 
x+(c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2+e^2)^(1/2)))/e+6*polylog(4,-e*(c*x+(c^2 
*x^2+1)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)))/e
 

3.1.3.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.93 \[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=\frac {-\text {arcsinh}(c x)^4+4 \text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )+4 \text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )+12 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )+12 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )-24 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-24 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )+24 \operatorname {PolyLog}\left (4,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )+24 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{4 e} \]

input

Integrate[ArcSinh[c*x]^3/(d + e*x),x]
 

output

(-ArcSinh[c*x]^4 + 4*ArcSinh[c*x]^3*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt 
[c^2*d^2 + e^2])] + 4*ArcSinh[c*x]^3*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqr 
t[c^2*d^2 + e^2])] + 12*ArcSinh[c*x]^2*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c* 
d) + Sqrt[c^2*d^2 + e^2])] + 12*ArcSinh[c*x]^2*PolyLog[2, -((e*E^ArcSinh[c 
*x])/(c*d + Sqrt[c^2*d^2 + e^2]))] - 24*ArcSinh[c*x]*PolyLog[3, (e*E^ArcSi 
nh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] - 24*ArcSinh[c*x]*PolyLog[3, -((e 
*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))] + 24*PolyLog[4, (e*E^ArcSin 
h[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] + 24*PolyLog[4, -((e*E^ArcSinh[c*x 
])/(c*d + Sqrt[c^2*d^2 + e^2]))])/(4*e)
 

3.1.3.3 Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {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.

\(\displaystyle \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx\)

\(\Big \downarrow \) 6242

\(\displaystyle \int \frac {\sqrt {c^2 x^2+1} \text {arcsinh}(c x)^3}{c d+c e x}d\text {arcsinh}(c x)\)

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

Int[ArcSinh[c*x]^3/(d + e*x),x]
 

output

-1/4*ArcSinh[c*x]^4/e + (ArcSinh[c*x]^3*Log[1 + (e*E^ArcSinh[c*x])/(c*d - 
Sqrt[c^2*d^2 + e^2])])/e + (ArcSinh[c*x]^3*Log[1 + (e*E^ArcSinh[c*x])/(c*d 
 + Sqrt[c^2*d^2 + e^2])])/e - (3*(-(ArcSinh[c*x]^2*PolyLog[2, -((e*E^ArcSi 
nh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))]) + 2*(ArcSinh[c*x]*PolyLog[3, -((e* 
E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))] - PolyLog[4, -((e*E^ArcSinh[ 
c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])))/e - (3*(-(ArcSinh[c*x]^2*PolyLog[2, 
 -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))]) + 2*(ArcSinh[c*x]*Pol 
yLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))] - PolyLog[4, -( 
(e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])))/e
 

3.1.3.3.1 Defintions of rubi rules used

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 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]
 

3.1.3.4 Maple [F]

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

input

int(arcsinh(c*x)^3/(e*x+d),x)
 

output

int(arcsinh(c*x)^3/(e*x+d),x)
 

3.1.3.5 Fricas [F]

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

input

integrate(arcsinh(c*x)^3/(e*x+d),x, algorithm="fricas")
 

output

integral(arcsinh(c*x)^3/(e*x + d), x)
 

3.1.3.6 Sympy [F]

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

input

integrate(asinh(c*x)**3/(e*x+d),x)
 

output

Integral(asinh(c*x)**3/(d + e*x), x)
 

3.1.3.7 Maxima [F]

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

input

integrate(arcsinh(c*x)^3/(e*x+d),x, algorithm="maxima")
 

output

integrate(arcsinh(c*x)^3/(e*x + d), x)
 

3.1.3.8 Giac [F]

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

input

integrate(arcsinh(c*x)^3/(e*x+d),x, algorithm="giac")
 

output

integrate(arcsinh(c*x)^3/(e*x + d), x)
 

3.1.3.9 Mupad [F(-1)]

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

input

int(asinh(c*x)^3/(d + e*x),x)
 

output

int(asinh(c*x)^3/(d + e*x), x)