\(\int \frac {\text {arcsinh}(a+b x)^2}{(1+a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\) [279]

Optimal result

Integrand size = 30, antiderivative size = 86 \[ \int \frac {\text {arcsinh}(a+b x)^2}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {\text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \text {arcsinh}(a+b x) \log \left (1+e^{2 \text {arcsinh}(a+b x)}\right )}{b}-\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a+b x)}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {5860, 5787, 5797, 3799, 2221, 2317, 2438} \[ \int \frac {\text {arcsinh}(a+b x)^2}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a+b x)}\right )}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^2}{b \sqrt {(a+b x)^2+1}}+\frac {\text {arcsinh}(a+b x)^2}{b}-\frac {2 \text {arcsinh}(a+b x) \log \left (e^{2 \text {arcsinh}(a+b x)}+1\right )}{b} \]

[In]

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

[Out]

ArcSinh[a + b*x]^2/b + ((a + b*x)*ArcSinh[a + b*x]^2)/(b*Sqrt[1 + (a + b*x)^2]) - (2*ArcSinh[a + b*x]*Log[1 +
E^(2*ArcSinh[a + b*x])])/b - PolyLog[2, -E^(2*ArcSinh[a + b*x])]/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 2317

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

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)^2}{\left (1+x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \text {arcsinh}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \text {Subst}\left (\int \frac {x \text {arcsinh}(x)}{1+x^2} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \text {arcsinh}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \text {Subst}(\int x \tanh (x) \, dx,x,\text {arcsinh}(a+b x))}{b} \\ & = \frac {\text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {4 \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \text {arcsinh}(a+b x) \log \left (1+e^{2 \text {arcsinh}(a+b x)}\right )}{b}+\frac {2 \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \text {arcsinh}(a+b x) \log \left (1+e^{2 \text {arcsinh}(a+b x)}\right )}{b}+\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(a+b x)}\right )}{b} \\ & = \frac {\text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^2}{b \sqrt {1+(a+b x)^2}}-\frac {2 \text {arcsinh}(a+b x) \log \left (1+e^{2 \text {arcsinh}(a+b x)}\right )}{b}-\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(a+b x)}\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.14 \[ \int \frac {\text {arcsinh}(a+b x)^2}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {\text {arcsinh}(a+b x) \left (\frac {\left (a+b x-\sqrt {1+a^2+2 a b x+b^2 x^2}\right ) \text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}}-2 \log \left (1+e^{-2 \text {arcsinh}(a+b x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(a+b x)}\right )}{b} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.95

method	result	size
default	\(-\frac {\left (b^{2} x^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x +2 a b x -a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a^{2}+1\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}}{b \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}+\frac {2 \operatorname {arcsinh}\left (b x +a \right )^{2}}{b}-\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (1+\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{b}-\frac {\operatorname {polylog}\left (2, -\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{b}\)	\(168\)

[In]

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

Fricas [F]

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

[In]

integrate(arcsinh(b*x+a)^2/(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)^2/(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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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