\(\int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx\) [26]

Optimal result

Integrand size = 16, antiderivative size = 180 \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {e x \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {d \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5829, 5773, 5819, 3384, 3379, 3382, 5778} \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {e x \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))} \]

[In]

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

[Out]

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

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 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]

Rule 5829

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

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

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83 \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\frac {b c d \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {b c e x \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+c d \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-c d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{b^2 c^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.51

method	result	size
derivativedivides	\(\frac {\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) e}{4 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{4 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c}\)	\(272\)
default	\(\frac {\frac {\left (-\sqrt {c^{2} x^{2}+1}+c x \right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) e}{4 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{4 c b \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c}\)	\(272\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-(c^3*e*x^4 + c^3*d*x^3 + c*e*x^2 + c*d*x + (c^2*e*x^3 + c^2*d*x^2 + e*x + 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((2*c^5*e*x^5 + c^5*d*x^4 + 4*c^3*e*x^3 + 2*c^3*d*x^2 + 2*c*e*x + (2*c^3*e*x^3 + c^3*d*
x^2 - c*d)*(c^2*x^2 + 1) + c*d + (4*c^4*e*x^4 + 2*c^4*d*x^3 + 4*c^2*e*x^2 + c^2*d*x + e)*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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((d + e*x)/(a + b*asinh(c*x))^2,x)

[Out]

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