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

Optimal result

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

[Out]

2*arcsinh(a*x)^3*arctan(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+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*polylog(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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5789, 4265, 2611, 6744, 2320, 6724} \[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \operatorname {PolyLog}\left (4,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \operatorname {PolyLog}\left (4,i e^{\text {arcsinh}(a x)}\right )}{a c} \]

[In]

Int[ArcSinh[a*x]^3/(c + a^2*c*x^2),x]

[Out]

(2*ArcSinh[a*x]^3*ArcTan[E^ArcSinh[a*x]])/(a*c) - ((3*I)*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]])/(a*c)
+ ((3*I)*ArcSinh[a*x]^2*PolyLog[2, I*E^ArcSinh[a*x]])/(a*c) + ((6*I)*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a
*x]])/(a*c) - ((6*I)*ArcSinh[a*x]*PolyLog[3, I*E^ArcSinh[a*x]])/(a*c) - ((6*I)*PolyLog[4, (-I)*E^ArcSinh[a*x]]
)/(a*c) + ((6*I)*PolyLog[4, I*E^ArcSinh[a*x]])/(a*c)

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 4265

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] + (-Dist[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] + Dist[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 5789

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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 6744

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\text {arcsinh}(a x)\right )}{a c} \\ & = \frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {(3 i) \text {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c}+\frac {(3 i) \text {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c} \\ & = \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 {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c}-\frac {(6 i) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c} \\ & = \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) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c}+\frac {(6 i) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c} \\ & = \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) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {(6 i) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )}{a c} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(-(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^A
rcSinh[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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(asinh(a*x)^3/(c + a^2*c*x^2),x)

[Out]

int(asinh(a*x)^3/(c + a^2*c*x^2), x)