3.4.31 \(\int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx\) [331]

3.4.31.1 Optimal result

Integrand size = 19, antiderivative size = 174 \[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \operatorname {PolyLog}\left (4,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \operatorname {PolyLog}\left (4,i e^{\text {arcsinh}(a x)}\right )}{a c} \]

output

2*arcsinh(a*x)^3*arctan(a*x+(a^2*x^2+1)^(1/2))/a/c-3*I*arcsinh(a*x)^2*poly 
log(2,-I*(a*x+(a^2*x^2+1)^(1/2)))/a/c+3*I*arcsinh(a*x)^2*polylog(2,I*(a*x+ 
(a^2*x^2+1)^(1/2)))/a/c+6*I*arcsinh(a*x)*polylog(3,-I*(a*x+(a^2*x^2+1)^(1/ 
2)))/a/c-6*I*arcsinh(a*x)*polylog(3,I*(a*x+(a^2*x^2+1)^(1/2)))/a/c-6*I*pol 
ylog(4,-I*(a*x+(a^2*x^2+1)^(1/2)))/a/c+6*I*polylog(4,I*(a*x+(a^2*x^2+1)^(1 
/2)))/a/c
 

3.4.31.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.28 \[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\frac {-\text {arcsinh}(a x)^3 \log \left (1+\frac {a e^{\text {arcsinh}(a x)}}{\sqrt {-a^2}}\right )+\text {arcsinh}(a x)^3 \log \left (1+\frac {\sqrt {-a^2} e^{\text {arcsinh}(a x)}}{a}\right )+3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {arcsinh}(a x)}}{\sqrt {-a^2}}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} e^{\text {arcsinh}(a x)}}{a}\right )-6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {arcsinh}(a x)}}{\sqrt {-a^2}}\right )+6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,\frac {\sqrt {-a^2} e^{\text {arcsinh}(a x)}}{a}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {arcsinh}(a x)}}{\sqrt {-a^2}}\right )-6 \operatorname {PolyLog}\left (4,\frac {\sqrt {-a^2} e^{\text {arcsinh}(a x)}}{a}\right )}{\sqrt {-a^2} c} \]

input

Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2),x]
 

output

(-(ArcSinh[a*x]^3*Log[1 + (a*E^ArcSinh[a*x])/Sqrt[-a^2]]) + ArcSinh[a*x]^3 
*Log[1 + (Sqrt[-a^2]*E^ArcSinh[a*x])/a] + 3*ArcSinh[a*x]^2*PolyLog[2, (a*E 
^ArcSinh[a*x])/Sqrt[-a^2]] - 3*ArcSinh[a*x]^2*PolyLog[2, (Sqrt[-a^2]*E^Arc 
Sinh[a*x])/a] - 6*ArcSinh[a*x]*PolyLog[3, (a*E^ArcSinh[a*x])/Sqrt[-a^2]] + 
 6*ArcSinh[a*x]*PolyLog[3, (Sqrt[-a^2]*E^ArcSinh[a*x])/a] + 6*PolyLog[4, ( 
a*E^ArcSinh[a*x])/Sqrt[-a^2]] - 6*PolyLog[4, (Sqrt[-a^2]*E^ArcSinh[a*x])/a 
])/(Sqrt[-a^2]*c)
 

3.4.31.3 Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6204, 3042, 4668, 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.

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \text {arcsinh}(a x)^3 \csc \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )d\text {arcsinh}(a x)}{a c}\)

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

\(\displaystyle \frac {3 i \left (2 \int \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )\right )-3 i \left (2 \int \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )\right )+2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )\right )-3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )\right )+2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )-\int e^{-\text {arcsinh}(a x)} \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )\right )-3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )-\int e^{-\text {arcsinh}(a x)} \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )de^{\text {arcsinh}(a x)}\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )\right )+2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )+3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (4,-i e^{\text {arcsinh}(a x)}\right )\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )\right )-3 i \left (2 \left (\text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )-\operatorname {PolyLog}\left (4,i e^{\text {arcsinh}(a x)}\right )\right )-\text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )\right )}{a c}\)

input

Int[ArcSinh[a*x]^3/(c + a^2*c*x^2),x]
 

output

(2*ArcSinh[a*x]^3*ArcTan[E^ArcSinh[a*x]] + (3*I)*(-(ArcSinh[a*x]^2*PolyLog 
[2, (-I)*E^ArcSinh[a*x]]) + 2*(ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a*x] 
] - PolyLog[4, (-I)*E^ArcSinh[a*x]])) - (3*I)*(-(ArcSinh[a*x]^2*PolyLog[2, 
 I*E^ArcSinh[a*x]]) + 2*(ArcSinh[a*x]*PolyLog[3, I*E^ArcSinh[a*x]] - PolyL 
og[4, I*E^ArcSinh[a*x]])))/(a*c)
 

3.4.31.3.1 Defintions of rubi rules used

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 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 6204

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 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]
 

3.4.31.4 Maple [F]

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

input

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

output

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

3.4.31.5 Fricas [F]

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

input

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

output

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

3.4.31.6 Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

input

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

output

Integral(asinh(a*x)**3/(a**2*x**2 + 1), x)/c
 

3.4.31.7 Maxima [F]

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

input

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

output

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

3.4.31.8 Giac [F]

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

input

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

output

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

3.4.31.9 Mupad [F(-1)]

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

input

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

output

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