\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx\) [263]

Optimal result

Integrand size = 28, antiderivative size = 209 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {c \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {1+c^2 x^2}}+\frac {2 b c \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5805, 5775, 3797, 2221, 2317, 2438, 5783} \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {c \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {c^2 x^2+1}}+\frac {c \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}+\frac {2 b c \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b^2 c \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

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

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5783

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

Rule 5805

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

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

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=-\frac {a^2 \sqrt {d+c^2 d x^2}}{x}+\frac {a b \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2+2 c x \log (c x)\right )}{x \sqrt {1+c^2 x^2}}+a^2 c \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b^2 c \sqrt {d+c^2 d x^2} \left (\text {arcsinh}(c x) \left (\left (3-\frac {3 \sqrt {1+c^2 x^2}}{c x}\right ) \text {arcsinh}(c x)+\text {arcsinh}(c x)^2+6 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )-3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 \sqrt {1+c^2 x^2}} \]

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^2,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(578\) vs. \(2(207)=414\).

Time = 0.29 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.77

method	result	size
default	\(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+a^{2} c^{2} x \sqrt {c^{2} d \,x^{2}+d}+\frac {a^{2} c^{2} d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{3} c}{3 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )^{2}}{x \left (c^{2} x^{2}+1\right )}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{x \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}}\right )\)	\(579\)
parts	\(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+a^{2} c^{2} x \sqrt {c^{2} d \,x^{2}+d}+\frac {a^{2} c^{2} d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{3} c}{3 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )^{2}}{x \left (c^{2} x^{2}+1\right )}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{x \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}}\right )\)	\(579\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/x^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**2/x**2, x)

Maxima [F]

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

[In]

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

[Out]

(c*sqrt(d)*arcsinh(c*x) - sqrt(c^2*d*x^2 + d)/x)*a^2 + integrate(sqrt(c^2*d*x^2 + d)*b^2*log(c*x + sqrt(c^2*x^
2 + 1))^2/x^2 + 2*sqrt(c^2*d*x^2 + d)*a*b*log(c*x + sqrt(c^2*x^2 + 1))/x^2, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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