Integrand size = 30, antiderivative size = 781 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {b c^2 f \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c^2 f \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1+c^2 x^2}}-\frac {b c^2 f \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c^2 f \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}} \]
-a*(c^2*d*x^2+d)^(1/2)/g/(g*x+f)-b*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/g/(g*x +f)+a*c^3*f^2*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)/(c^2*x^2+ 1)^(1/2)+1/2*b*c^3*f^2*arcsinh(c*x)^2*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2 )/(c^2*x^2+1)^(1/2)-1/2*(-c^2*f*x+g)^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^ (1/2)/b/c/(c^2*f^2+g^2)/(g*x+f)^2/(c^2*x^2+1)^(1/2)+b*c*ln(g*x+f)*(c^2*d*x ^2+d)^(1/2)/g^2/(c^2*x^2+1)^(1/2)+a*c^2*f*arctanh((-c^2*f*x+g)/(c^2*f^2+g^ 2)^(1/2)/(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c ^2*x^2+1)^(1/2)-b*c^2*f*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*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+ 1)^(1/2)+b*c^2*f*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*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(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*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)+b*c^2*f*poly log(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^ (1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)+1/2*(a+b*arcsinh(c*x))^2*( c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)/b/c/(g*x+f)^2
Result contains complex when optimal does not.
Time = 7.67 (sec) , antiderivative size = 1384, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx =\text {Too large to display} \]
((-2*a*g*Sqrt[d + c^2*d*x^2])/(f + g*x) - (2*a*c^2*Sqrt[d]*f*Log[f + g*x]) /Sqrt[c^2*f^2 + g^2] + 2*a*c*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^ 2]] + (2*a*c^2*Sqrt[d]*f*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^2] *Sqrt[d + c^2*d*x^2]])/Sqrt[c^2*f^2 + g^2] + b*c*Sqrt[d + c^2*d*x^2]*((-2* g*ArcSinh[c*x])/(c*f + c*g*x) + ArcSinh[c*x]^2/Sqrt[1 + c^2*x^2] + ((2*I)* c*f*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/Sqrt[c^2*f^2 + g^2]])/(Sqrt [c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) + (2*Log[1 + (g*x)/f])/Sqrt[1 + c^2*x^2 ] + (2*c*f*(2*ArcCos[((-I)*c*f)/g]*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*Ar cSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*ArcTan h[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSin h[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-( c^2*f^2) - g^2])/(E^(ArcSinh[c*x]/2)*Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] + (A rcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[ c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*A rcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c*x ]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] - (ArcCos[( (-I)*c*f)/g] + (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4] )/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*((-I)*c*f + g + Sqrt[-(c^2*...
Time = 2.74 (sec) , antiderivative size = 572, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6260, 6254, 27, 6249, 27, 6271, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx\) |
| \(\Big \downarrow \) 6260 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{(f+g x)^2}dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6254 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}-\frac {\int -\frac {2 \left (g-c^2 f x\right ) (a+b \text {arcsinh}(c x))^2}{(f+g x)^3}dx}{2 b c}\right )}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {\int \frac {\left (g-c^2 f x\right ) (a+b \text {arcsinh}(c x))^2}{(f+g x)^3}dx}{b c}+\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6249 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {-2 b c \int -\frac {\left (g-c^2 f x\right )^2 (a+b \text {arcsinh}(c x))}{2 \left (c^2 f^2+g^2\right ) (f+g x)^2 \sqrt {c^2 x^2+1}}dx-\frac {\left (g-c^2 f x\right )^2 (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 f^2+g^2\right ) (f+g x)^2}}{b c}+\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {\frac {b c \int \frac {\left (g-c^2 f x\right )^2 (a+b \text {arcsinh}(c x))}{(f+g x)^2 \sqrt {c^2 x^2+1}}dx}{c^2 f^2+g^2}-\frac {\left (g-c^2 f x\right )^2 (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 f^2+g^2\right ) (f+g x)^2}}{b c}+\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6271 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {\frac {b c \int \left (\frac {b \text {arcsinh}(c x) \left (c^2 f x-g\right )^2}{(f+g x)^2 \sqrt {c^2 x^2+1}}+\frac {a \left (c^2 f x-g\right )^2}{(f+g x)^2 \sqrt {c^2 x^2+1}}\right )dx}{c^2 f^2+g^2}-\frac {\left (g-c^2 f x\right )^2 (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 f^2+g^2\right ) (f+g x)^2}}{b c}+\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {\frac {b c \left (\frac {a c^3 f^2 \text {arcsinh}(c x)}{g^2}+\frac {a c^2 f \sqrt {c^2 f^2+g^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}\right )}{g^2}-\frac {a \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right )}{g (f+g x)}+\frac {b c^3 f^2 \text {arcsinh}(c x)^2}{2 g^2}-\frac {b c^2 f \sqrt {c^2 f^2+g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2}+\frac {b c^2 f \sqrt {c^2 f^2+g^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2}-\frac {b \sqrt {c^2 x^2+1} \text {arcsinh}(c x) \left (c^2 f^2+g^2\right )}{g (f+g x)}-\frac {b c^2 f \text {arcsinh}(c x) \sqrt {c^2 f^2+g^2} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^2}+\frac {b c^2 f \text {arcsinh}(c x) \sqrt {c^2 f^2+g^2} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g^2}+\frac {b c \left (c^2 f^2+g^2\right ) \log (f+g x)}{g^2}\right )}{c^2 f^2+g^2}-\frac {\left (g-c^2 f x\right )^2 (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 f^2+g^2\right ) (f+g x)^2}}{b c}+\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {c^2 x^2+1}}\) |
(Sqrt[d + c^2*d*x^2]*(((1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(2*b*c*(f + g *x)^2) + (-1/2*((g - c^2*f*x)^2*(a + b*ArcSinh[c*x])^2)/((c^2*f^2 + g^2)*( f + g*x)^2) + (b*c*(-((a*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2])/(g*(f + g*x))) + (a*c^3*f^2*ArcSinh[c*x])/g^2 - (b*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*Arc Sinh[c*x])/(g*(f + g*x)) + (b*c^3*f^2*ArcSinh[c*x]^2)/(2*g^2) + (a*c^2*f*S qrt[c^2*f^2 + g^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2 *x^2])])/g^2 - (b*c^2*f*Sqrt[c^2*f^2 + g^2]*ArcSinh[c*x]*Log[1 + (E^ArcSin h[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/g^2 + (b*c^2*f*Sqrt[c^2*f^2 + g^2] *ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/g^2 + (b*c*(c^2*f^2 + g^2)*Log[f + g*x])/g^2 - (b*c^2*f*Sqrt[c^2*f^2 + g^2]*P olyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/g^2 + (b*c^2 *f*Sqrt[c^2*f^2 + g^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/g^2))/(c^2*f^2 + g^2))/(b*c)))/Sqrt[1 + c^2*x^2]
3.1.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_. ) + (g_.)*(x_))^(p_.), x_Symbol] :> With[{u = IntHide[(f + g*x)^p*(d + e*x) ^m, x]}, Simp[(a + b*ArcSinh[c*x])^n u, x] - Simp[b*c*n Int[SimplifyInt egrand[u*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x], x]] /; F reeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.) + (g_.)*(x_))^(m_)*Sqr t[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f + g*x)^m*(d + e*x^2)*((a + b*A rcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] - Simp[1/(b*c*Sqrt[d]*(n + 1)) Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcS inh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2* d] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[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] /; Fre eQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ [p - 1/2] && !GtQ[d, 0]
Int[(ArcSinh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^( p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSinh[c*x ])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && I GtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]
Leaf count of result is larger than twice the leaf count of optimal. \(1700\) vs. \(2(745)=1490\).
Time = 0.82 (sec) , antiderivative size = 1701, normalized size of antiderivative = 2.18
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1701\) |
| parts | \(\text {Expression too large to display}\) | \(1701\) |
a/g^2*(-1/d/(c^2*f^2+g^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)^(3/2)-c^2*f*g/(c^2*f^2+g^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)-c^2*d*f/g*ln((-c^2*d*f/g+c^2*d*(x +f/g))/(c^2*d)^(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))/(c^2*d)^(1/2)-d*(c^2*f^2+g^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) ))+2*c^2/(c^2*f^2+g^2)*g^2*(1/4*(2*c^2*d*(x+f/g)-2*c^2*d*f/g)/c^2/d*((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)+1/8*(4*c^2*d^2*( c^2*f^2+g^2)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d*ln((-c^2*d*f/g+c^2*d*(x+f/g))/(c ^2*d)^(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 ))/(c^2*d)^(1/2)))+1/2*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c *x)^2*c/g^2+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)/g^2/(g*x+f)*x ^3*c^4*f-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/g^2/(g*x+f)*x*c^2*f-b*(d*(c^ 2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)/g/(g*x+f)*x^2*c^2+b*(d*(c^2*x^2+1 ))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)^(1/2)/g/(g*x+f)*x*c+b*(d*(c^2*x^2+1))^(1 /2)*arcsinh(c*x)/(c^2*x^2+1)/g^2/(g*x+f)*x*c^2*f+b*(d*(c^2*x^2+1))^(1/2)*a rcsinh(c*x)/(c^2*x^2+1)^(1/2)/g^2/(g*x+f)*c*f-b*(d*(c^2*x^2+1))^(1/2)*arcs inh(c*x)/(c^2*x^2+1)/g/(g*x+f)-b*(d*(c^2*x^2+1))^(1/2)*c^2/(c^2*x^2+1)^(1/ 2)/g^2/(c^2*f^2+g^2)^(1/2)*arcsinh(c*x)*ln((-(c*x+(c^2*x^2+1)^(1/2))*g-...
\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=\int { \frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (g x + f\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=\int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=\int { \frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (g x + f\right )}^{2}} \,d x } \]
-(c^2*d*f*arcsinh(c*f*x/(g*abs(x + f/g)) - 1/(c*abs(x + f/g)))/(sqrt(c^2*d *f^2/g^2 + d)*g^3) - c*sqrt(d)*arcsinh(c*x)/g^2 + sqrt(c^2*d*x^2 + d)/(g^2 *x + f*g))*a + b*integrate(sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1) )/(g^2*x^2 + 2*f*g*x + f^2), x)
Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d}}{{\left (f+g\,x\right )}^2} \,d x \]