3.7.25 \(\int \frac {d+e x^2}{(a+b \sinh ^{-1}(c x))^2} \, dx\) [625]

Optimal. Leaf size=247 \[ -\frac {d \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {d \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {e \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^3}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3} \]

[Out]

d*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c-1/4*e*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b^2/c^3+3/4*e*cosh(3*a/b
)*Shi(3*(a+b*arcsinh(c*x))/b)/b^2/c^3-d*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c+1/4*e*Chi((a+b*arcsinh(c*x))
/b)*sinh(a/b)/b^2/c^3-3/4*e*Chi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/b^2/c^3-d*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsi
nh(c*x))-e*x^2*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))

________________________________________________________________________________________

Rubi [A]
time = 0.29, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5793, 5773, 5819, 3384, 3379, 3382, 5778} \begin {gather*} \frac {e \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac {e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (e*x^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) - (
d*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b^2*c) + (e*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])
/(4*b^2*c^3) - (3*e*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(4*b^2*c^3) + (d*Cosh[a/b]*SinhInt
egral[(a + b*ArcSinh[c*x])/b])/(b^2*c) - (e*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^3) + (3*e
*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^3)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5773

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1
)/(b*c*(n + 1))), x] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5778

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSi
nh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Si
nh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}
, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5793

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

Rule 5819

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

Rubi steps

\begin {align*} \int \frac {d+e x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac {d}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac {e x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d \int \frac {1}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+e \int \frac {x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ &=-\frac {d \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {(c d) \int \frac {x}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b}+\frac {e \text {Subst}\left (\int \left (-\frac {\sinh (x)}{4 (a+b x)}+\frac {3 \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {d \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {d \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac {e \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {(3 e) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac {d \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\left (d \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac {\left (e \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {\left (3 e \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {\left (d \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}+\frac {\left (e \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {\left (3 e \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac {d \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {d \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {e \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^3}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac {e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^3}+\frac {3 e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.59, size = 190, normalized size = 0.77 \begin {gather*} -\frac {\frac {4 b c^2 d \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)}+\frac {4 b c^2 e x^2 \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)}+\left (4 c^2 d-e\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 e \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-4 c^2 d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-3 e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )}{4 b^2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((4*b*c^2*d*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (4*b*c^2*e*x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*
x]) + (4*c^2*d - e)*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 3*e*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh
[(3*a)/b] - 4*c^2*d*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]]
- 3*e*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(b^2*c^3)

________________________________________________________________________________________

Maple [A]
time = 7.01, size = 438, normalized size = 1.77

method	result	size
derivativedivides	\(\frac {\frac {\left (-4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c^{3} x^{3}-\sqrt {c^{2} x^{2}+1}+3 c x \right ) e}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \arcsinh \left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e \left (4 c^{3} x^{3}+3 c x +4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{8 b \,c^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \arcsinh \left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2} c^{2}}+\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d}{2 b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {d \,{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) e}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {e \,{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{8 c^{2} b^{2}}-\frac {d \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {e \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right )}{8 c^{2} b^{2}}}{c}\)	\(438\)
default	\(\frac {\frac {\left (-4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c^{3} x^{3}-\sqrt {c^{2} x^{2}+1}+3 c x \right ) e}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \arcsinh \left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b^{2}}-\frac {e \left (4 c^{3} x^{3}+3 c x +4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+\sqrt {c^{2} x^{2}+1}\right )}{8 b \,c^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \arcsinh \left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2} c^{2}}+\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d}{2 b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {d \,{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) e}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {e \,{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{8 c^{2} b^{2}}-\frac {d \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {e \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{8 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {e \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right )}{8 c^{2} b^{2}}}{c}\)	\(438\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(1/8*(-4*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c^3*x^3-(c^2*x^2+1)^(1/2)+3*c*x)*e/c^2/b/(a+b*arcsinh(c*x))+3/8*e/c^2
/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)-1/8/b*e/c^2*(4*c^3*x^3+3*c*x+4*c^2*x^2*(c^2*x^2+1)^(1/2)+(c^2*x^2+1
)^(1/2))/(a+b*arcsinh(c*x))-3/8/b^2*e/c^2*exp(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b)+1/2*(-(c^2*x^2+1)^(1/2)+c*x)
*d/b/(a+b*arcsinh(c*x))+1/2*d/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-1/8*(-(c^2*x^2+1)^(1/2)+c*x)*e/c^2/b/(a+b*ar
csinh(c*x))-1/8/c^2*e/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)-1/2/b*d*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))-1
/2/b^2*d*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)+1/8/c^2/b*e*(c*x+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(c*x))+1/8/c^2/b^2*
e*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^5*e + (c^3*d + c*e)*x^3 + c*d*x + (c^2*x^4*e + (c^2*d + e)*x^2 + d)*sqrt(c^2*x^2 + 1))/(a*b*c^3*x^2 +
sqrt(c^2*x^2 + 1)*a*b*c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x
^2 + 1))) + integrate((3*c^5*x^6*e + (c^5*d + 6*c^3*e)*x^4 + (2*c^3*d + 3*c*e)*x^2 + (3*c^3*x^4*e + (c^3*d + c
*e)*x^2 - c*d)*(c^2*x^2 + 1) + c*d + (6*c^4*x^5*e + (2*c^4*d + 7*c^2*e)*x^3 + (c^2*d + 2*e)*x)*sqrt(c^2*x^2 +
1))/(a*b*c^5*x^4 + (c^2*x^2 + 1)*a*b*c^3*x^2 + 2*a*b*c^3*x^2 + a*b*c + (b^2*c^5*x^4 + (c^2*x^2 + 1)*b^2*c^3*x^
2 + 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 + b^2*c^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a
*b*c^4*x^3 + a*b*c^2*x)*sqrt(c^2*x^2 + 1)), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d + e x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________