\(\int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx\) [52]

Optimal result

Integrand size = 30, antiderivative size = 444 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=-\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {b c^2 f \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {b c^2 f \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5845, 5843, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=-\frac {g \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right ) (f+g x)}+\frac {c^2 f \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac {c^2 f \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac {b c^2 f \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac {b c^2 f \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac {b c \sqrt {c^2 x^2+1} \log (f+g x)}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )} \]

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5843

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

Rule 5845

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

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx=\frac {c \sqrt {1+c^2 x^2} \left (-\frac {g \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c f+c g x}+b \log (f+g x)+\frac {c f \left ((a+b \text {arcsinh}(c x)) \left (\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )-b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1769\) vs. \(2(442)=884\).

Time = 0.85 (sec) , antiderivative size = 1770, normalized size of antiderivative = 3.99

method	result	size
default	\(\text {Expression too large to display}\)	\(1770\)
parts	\(\text {Expression too large to display}\)	\(1770\)

[In]

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

[Out]

-a/d/(c^2*f^2+g^2)/(x+f/g)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)-a/g*c^2*f/(c^2*f^2+
g^2)/(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*(
(x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/
d/(c^2*x^2+1)/(c^2*f^2+g^2)/(g*x+f)*x^3*c^4*f-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*f^2+g^2)/(g*x+f)*x*c
^2*f-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^2+1)/(c^2*f^2+g^2)/(g*x+f)*x^2*c^2*g+b*(d*(c^2*x^2+1))^(1/2
)*arcsinh(c*x)/d/(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)/(g*x+f)*x*c*g+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^2
+1)/(c^2*f^2+g^2)/(g*x+f)*x*c^2*f+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)/(g*x+
f)*c*f-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/d/(c^2*x^2+1)/(c^2*f^2+g^2)/(g*x+f)*g+b*(d*(c^2*x^2+1))^(1/2)*(c^2
*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2+g^4)*c^3*ln((c*x+(c^2*x^2+1)^
(1/2))^2*g+2*c*f*(c*x+(c^2*x^2+1)^(1/2))-g)*f^2-2*b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c
^4*f^2*g^2*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2+g^4)*c^3*ln(c*x+(c^2*x^2+1)^(1/2))*f^2+b*(d*(c^2*x^2+1))^(1/2
)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2+g^4)*c^2*ln((-(c*x+(c^2
*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))*arcsinh(c*x)*(c^2*f^2+g^2)^(1/2)*f-b*(d*
(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2+g^4)*c
^2*ln(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))*arcsinh(c*x)*(c^2*f^2+g^2
)^(1/2)*f+b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2
*f^2*g^2+g^4)*c^2*dilog((-(c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*
f^2+g^2)^(1/2)*f-b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2*x^2+c^4*f^4+c^2*g^4*x^
2+2*c^2*f^2*g^2+g^4)*c^2*dilog(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))*
(c^2*f^2+g^2)^(1/2)*f+b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2*x^2+c^4*f^4+c^2*g
^4*x^2+2*c^2*f^2*g^2+g^4)*c*ln((c*x+(c^2*x^2+1)^(1/2))^2*g+2*c*f*(c*x+(c^2*x^2+1)^(1/2))-g)*g^2-2*b*(d*(c^2*x^
2+1))^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^6*f^4*x^2+2*c^4*f^2*g^2*x^2+c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2+g^4)*c*ln(c*x
+(c^2*x^2+1)^(1/2))*g^2

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^2*d*g^2*x^4 + 2*c^2*d*f*g*x^3 + 2*d*f*g*x + d*f^2 + (c^2*
d*f^2 + d*g^2)*x^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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