3.278 \(\int \frac{\sinh ^{-1}(a+b x)^3}{(1+a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{3 \sinh ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac{3 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(a+b x)}\right )}{2 b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{(a+b x)^2+1}}+\frac{\sinh ^{-1}(a+b x)^3}{b}-\frac{3 \sinh ^{-1}(a+b x)^2 \log \left (e^{2 \sinh ^{-1}(a+b x)}+1\right )}{b} \]

[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)

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Rubi [A]  time = 0.206168, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {5867, 5687, 5714, 3718, 2190, 2531, 2282, 6589} \[ -\frac{3 \sinh ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac{3 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(a+b x)}\right )}{2 b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{(a+b x)^2+1}}+\frac{\sinh ^{-1}(a+b x)^3}{b}-\frac{3 \sinh ^{-1}(a+b x)^2 \log \left (e^{2 \sinh ^{-1}(a+b x)}+1\right )}{b} \]

Antiderivative was successfully verified.

[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 5867

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*x^2)/d^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 5687

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*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[
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 5714

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 3718

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2531

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 2282

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 6589

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*} \int \frac{\sinh ^{-1}(a+b x)^3}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)^3}{\left (1+x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{1+(a+b x)^2}}-\frac{3 \operatorname{Subst}\left (\int \frac{x \sinh ^{-1}(x)^2}{1+x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{1+(a+b x)^2}}-\frac{3 \operatorname{Subst}\left (\int x^2 \tanh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\sinh ^{-1}(a+b x)^3}{b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{1+(a+b x)^2}}-\frac{6 \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\sinh ^{-1}(a+b x)^3}{b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{1+(a+b x)^2}}-\frac{3 \sinh ^{-1}(a+b x)^2 \log \left (1+e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac{6 \operatorname{Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\sinh ^{-1}(a+b x)^3}{b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{1+(a+b x)^2}}-\frac{3 \sinh ^{-1}(a+b x)^2 \log \left (1+e^{2 \sinh ^{-1}(a+b x)}\right )}{b}-\frac{3 \sinh ^{-1}(a+b x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac{3 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac{\sinh ^{-1}(a+b x)^3}{b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{1+(a+b x)^2}}-\frac{3 \sinh ^{-1}(a+b x)^2 \log \left (1+e^{2 \sinh ^{-1}(a+b x)}\right )}{b}-\frac{3 \sinh ^{-1}(a+b x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a+b x)}\right )}{2 b}\\ &=\frac{\sinh ^{-1}(a+b x)^3}{b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{1+(a+b x)^2}}-\frac{3 \sinh ^{-1}(a+b x)^2 \log \left (1+e^{2 \sinh ^{-1}(a+b x)}\right )}{b}-\frac{3 \sinh ^{-1}(a+b x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a+b x)}\right )}{b}+\frac{3 \text{Li}_3\left (-e^{2 \sinh ^{-1}(a+b x)}\right )}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.569996, size = 128, normalized size = 1.11 \[ \frac{6 \sinh ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(a+b x)}\right )+3 \text{PolyLog}\left (3,-e^{-2 \sinh ^{-1}(a+b x)}\right )+2 \sinh ^{-1}(a+b x)^2 \left (\frac{\left (-\sqrt{a^2+2 a b x+b^2 x^2+1}+a+b x\right ) \sinh ^{-1}(a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2+1}}-3 \log \left (e^{-2 \sinh ^{-1}(a+b x)}+1\right )\right )}{2 b} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [A]  time = 0.1, size = 203, normalized size = 1.8 \begin{align*} -{\frac{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{3}}{b \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) } \left ({b}^{2}{x}^{2}-\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}xb+2\,xab-\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}a+{a}^{2}+1 \right ) }+2\,{\frac{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{3}}{b}}-3\,{\frac{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}\ln \left ( 1+ \left ( bx+a+\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) ^{2} \right ) }{b}}-3\,{\frac{{\it Arcsinh} \left ( bx+a \right ){\it polylog} \left ( 2,- \left ( bx+a+\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) ^{2} \right ) }{b}}+{\frac{3}{2\,b}{\it polylog} \left ( 3,- \left ( bx+a+\sqrt{1+ \left ( bx+a \right ) ^{2}} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*x^2-(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x*b+2*x*a*b-(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a+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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname{arsinh}\left (b x + a\right )^{3}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \,{\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + a^{4} + 4 \,{\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)