\(\int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {d+c^2 d x^2}} \, dx\) [296]

Optimal result

Integrand size = 28, antiderivative size = 223 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {d+c^2 d x^2}} \, dx=-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}+\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}+\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5816, 4267, 2611, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {d+c^2 d x^2}} \, dx=-\frac {2 \sqrt {c^2 x^2+1} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}+\frac {2 b \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}+\frac {2 b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 d x^2+d}}-\frac {2 b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

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

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 4267

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

Rule 5816

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

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

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {d+c^2 d x^2}} \, dx=\frac {a^2 \log (c x)}{\sqrt {d}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{\sqrt {d}}+\frac {2 a b \sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x) \left (\log \left (1-e^{-\text {arcsinh}(c x)}\right )-\log \left (1+e^{-\text {arcsinh}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x)^2 \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x)^2 \log \left (1+e^{-\text {arcsinh}(c x)}\right )+2 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-2 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(545\) vs. \(2(250)=500\).

Time = 0.28 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.45

method	result	size
default	\(-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{\sqrt {d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}\right )\)	\(546\)
parts	\(-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{\sqrt {d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {c^{2} x^{2}+1}\, d}\right )\)	\(546\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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