\(\int \frac {\text {arcsinh}(a+b x)^3}{(1+a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\) [278]

Optimal result

Integrand size = 30, antiderivative size = 115 \[ \int \frac {\text {arcsinh}(a+b x)^3}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {\text {arcsinh}(a+b x)^3}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^3}{b \sqrt {1+(a+b x)^2}}-\frac {3 \text {arcsinh}(a+b x)^2 \log \left (1+e^{2 \text {arcsinh}(a+b x)}\right )}{b}-\frac {3 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(a+b x)}\right )}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5860, 5787, 5797, 3799, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arcsinh}(a+b x)^3}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {3 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(a+b x)}\right )}{2 b}+\frac {(a+b x) \text {arcsinh}(a+b x)^3}{b \sqrt {(a+b x)^2+1}}+\frac {\text {arcsinh}(a+b x)^3}{b}-\frac {3 \text {arcsinh}(a+b x)^2 \log \left (e^{2 \text {arcsinh}(a+b x)}+1\right )}{b} \]

[In]

Int[ArcSinh[a + b*x]^3/(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

Rule 2221

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]
- Dist[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 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 3799

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

Rule 5787

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcS
inh[c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5797

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]

Rule 5860

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> D
ist[1/d, Subst[Int[(C/d^2 + (C/d^2)*x^2)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A,
B, C, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 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]

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.11 \[ \int \frac {\text {arcsinh}(a+b x)^3}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {2 \text {arcsinh}(a+b x)^2 \left (\frac {\left (a+b x-\sqrt {1+a^2+2 a b x+b^2 x^2}\right ) \text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}}-3 \log \left (1+e^{-2 \text {arcsinh}(a+b x)}\right )\right )+6 \text {arcsinh}(a+b x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(a+b x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(a+b x)}\right )}{2 b} \]

[In]

Integrate[ArcSinh[a + b*x]^3/(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.77

method	result	size
default	\(-\frac {\left (b^{2} x^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x +2 a b x -a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a^{2}+1\right ) \operatorname {arcsinh}\left (b x +a \right )^{3}}{b \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}+\frac {2 \operatorname {arcsinh}\left (b x +a \right )^{3}}{b}-\frac {3 \operatorname {arcsinh}\left (b x +a \right )^{2} \ln \left (1+\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{b}-\frac {3 \,\operatorname {arcsinh}\left (b x +a \right ) \operatorname {polylog}\left (2, -\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{b}+\frac {3 \operatorname {polylog}\left (3, -\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{2 b}\)	\(203\)

[In]

int(arcsinh(b*x+a)^3/(b^2*x^2+2*a*b*x+a^2+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

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

[In]

integrate(arcsinh(b*x+a)^3/(b^2*x^2+2*a*b*x+a^2+1)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*arcsinh(b*x + a)^3/(b^4*x^4 + 4*a*b^3*x^3 + 2*(3*a^2 + 1)*b^2*x^2 +
a^4 + 4*(a^3 + a)*b*x + 2*a^2 + 1), x)

Sympy [F]

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

[In]

integrate(asinh(b*x+a)**3/(b**2*x**2+2*a*b*x+a**2+1)**(3/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate(arcsinh(b*x+a)^3/(b^2*x^2+2*a*b*x+a^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(b*x + a)^3/(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2), x)

Giac [F]

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

[In]

integrate(arcsinh(b*x+a)^3/(b^2*x^2+2*a*b*x+a^2+1)^(3/2),x, algorithm="giac")

[Out]

integrate(arcsinh(b*x + a)^3/(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^3}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^3}{{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}} \,d x \]

[In]

int(asinh(a + b*x)^3/(a^2 + b^2*x^2 + 2*a*b*x + 1)^(3/2),x)

[Out]

int(asinh(a + b*x)^3/(a^2 + b^2*x^2 + 2*a*b*x + 1)^(3/2), x)