3.1.18 \(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx\) [18]

3.1.18.1 Optimal result

Integrand size = 18, antiderivative size = 349 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx=-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}+\frac {b c^3 d (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b c^3 d (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2+e^2\right )}+\frac {b^2 c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}}-\frac {b^2 c^3 d \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \left (c^2 d^2+e^2\right )^{3/2}} \]

output

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

3.1.18.2 Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx=\frac {-\frac {2 b c e \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^2}+\frac {2 b^2 c^2 \log (d+e x)}{c^2 d^2+e^2}+\frac {2 b c^3 d \left ((a+b \text {arcsinh}(c x)) \left (\log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )-\log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )}{\left (c^2 d^2+e^2\right )^{3/2}}}{2 e} \]

input

Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x)^3,x]
 

output

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

3.1.18.3 Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.95, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {6243, 6258, 3042, 3805, 3042, 3147, 16, 3803, 25, 2694, 27, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 6243

\(\displaystyle \frac {b c \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^2 \sqrt {c^2 x^2+1}}dx}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 6258

\(\displaystyle \frac {b c^2 \int \frac {a+b \text {arcsinh}(c x)}{(c d+c e x)^2}d\text {arcsinh}(c x)}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \int \frac {a+b \text {arcsinh}(c x)}{(c d-i e \sin (i \text {arcsinh}(c x)))^2}d\text {arcsinh}(c x)}{e}\)

\(\Big \downarrow \) 3805

\(\displaystyle \frac {b c^2 \left (\frac {c d \int \frac {a+b \text {arcsinh}(c x)}{c d+c e x}d\text {arcsinh}(c x)}{c^2 d^2+e^2}+\frac {b e \int \frac {\sqrt {c^2 x^2+1}}{c d+c e x}d\text {arcsinh}(c x)}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \left (\frac {c d \int \frac {a+b \text {arcsinh}(c x)}{c d-i e \sin (i \text {arcsinh}(c x))}d\text {arcsinh}(c x)}{c^2 d^2+e^2}+\frac {b e \int \frac {\cos (i \text {arcsinh}(c x))}{c d-i e \sin (i \text {arcsinh}(c x))}d\text {arcsinh}(c x)}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}\right )}{e}\)

\(\Big \downarrow \) 3147

\(\displaystyle -\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \left (\frac {c d \int \frac {a+b \text {arcsinh}(c x)}{c d-i e \sin (i \text {arcsinh}(c x))}d\text {arcsinh}(c x)}{c^2 d^2+e^2}+\frac {b \int \frac {1}{c d+c e x}d(c e x)}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}\right )}{e}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \left (\frac {c d \int \frac {a+b \text {arcsinh}(c x)}{c d-i e \sin (i \text {arcsinh}(c x))}d\text {arcsinh}(c x)}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}+\frac {b \log (c d+c e x)}{c^2 d^2+e^2}\right )}{e}\)

\(\Big \downarrow \) 3803

\(\displaystyle \frac {b c^2 \left (\frac {2 c d \int -\frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{-2 c e^{\text {arcsinh}(c x)} d-e e^{2 \text {arcsinh}(c x)}+e}d\text {arcsinh}(c x)}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}+\frac {b \log (c d+c e x)}{c^2 d^2+e^2}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b c^2 \left (-\frac {2 c d \int \frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{-2 c e^{\text {arcsinh}(c x)} d-e e^{2 \text {arcsinh}(c x)}+e}d\text {arcsinh}(c x)}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}+\frac {b \log (c d+c e x)}{c^2 d^2+e^2}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {b c^2 \left (-\frac {2 c d \left (\frac {e \int -\frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{2 \left (c d+e e^{\text {arcsinh}(c x)}-\sqrt {c^2 d^2+e^2}\right )}d\text {arcsinh}(c x)}{\sqrt {c^2 d^2+e^2}}-\frac {e \int -\frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{2 \left (c d+e e^{\text {arcsinh}(c x)}+\sqrt {c^2 d^2+e^2}\right )}d\text {arcsinh}(c x)}{\sqrt {c^2 d^2+e^2}}\right )}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}+\frac {b \log (c d+c e x)}{c^2 d^2+e^2}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c^2 \left (-\frac {2 c d \left (\frac {e \int \frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{c d+e e^{\text {arcsinh}(c x)}+\sqrt {c^2 d^2+e^2}}d\text {arcsinh}(c x)}{2 \sqrt {c^2 d^2+e^2}}-\frac {e \int \frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{c d+e e^{\text {arcsinh}(c x)}-\sqrt {c^2 d^2+e^2}}d\text {arcsinh}(c x)}{2 \sqrt {c^2 d^2+e^2}}\right )}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}+\frac {b \log (c d+c e x)}{c^2 d^2+e^2}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {b c^2 \left (-\frac {2 c d \left (\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}-\frac {b \int \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d+\sqrt {c^2 d^2+e^2}}+1\right )d\text {arcsinh}(c x)}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}-\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}-\frac {b \int \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d-\sqrt {c^2 d^2+e^2}}+1\right )d\text {arcsinh}(c x)}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}\right )}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}+\frac {b \log (c d+c e x)}{c^2 d^2+e^2}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {b c^2 \left (-\frac {2 c d \left (\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}-\frac {b \int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d+\sqrt {c^2 d^2+e^2}}+1\right )de^{\text {arcsinh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}-\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}-\frac {b \int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d-\sqrt {c^2 d^2+e^2}}+1\right )de^{\text {arcsinh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}\right )}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}+\frac {b \log (c d+c e x)}{c^2 d^2+e^2}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {b c^2 \left (-\frac {2 c d \left (\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}-\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}\right )}{c^2 d^2+e^2}-\frac {e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{\left (c^2 d^2+e^2\right ) (c d+c e x)}+\frac {b \log (c d+c e x)}{c^2 d^2+e^2}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{2 e (d+e x)^2}\)

input

Int[(a + b*ArcSinh[c*x])^2/(d + e*x)^3,x]
 

output

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

3.1.18.3.1 Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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] - Si 
mp[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 2694

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]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[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] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* 
(f_.)*(x_)]), x_Symbol] :> Simp[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 3805

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] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[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 6243

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] 
 - Simp[b*c*(n/(e*(m + 1)))   Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^( 
n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n 
, 0] && NeQ[m, -1]
 

rule 6258

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/S 
qrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[I 
nt[(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] && (Gt 
Q[m, 0] || IGtQ[n, 0])
 

3.1.18.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(814\) vs. \(2(365)=730\).

Time = 0.65 (sec) , antiderivative size = 815, normalized size of antiderivative = 2.34

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} c^{3}}{2 \left (e c x +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\operatorname {arcsinh}\left (c x \right ) \left (2 d c e \sqrt {c^{2} x^{2}+1}-4 d \,c^{2} e x +e^{2} \operatorname {arcsinh}\left (c x \right )+2 \sqrt {c^{2} x^{2}+1}\, e^{2} c x -2 c^{2} d^{2}-2 e^{2} c^{2} x^{2}+c^{2} d^{2} \operatorname {arcsinh}\left (c x \right )\right )}{2 e \left (e c x +c d \right )^{2} \left (c^{2} d^{2}+e^{2}\right )}-\frac {2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{e \left (c^{2} d^{2}+e^{2}\right )}+\frac {\ln \left (2 d \left (c x +\sqrt {c^{2} x^{2}+1}\right ) c +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-e \right )}{e \left (c^{2} d^{2}+e^{2}\right )}+\frac {d c \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}}}-\frac {d c \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}}}+\frac {d c \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}}}-\frac {d c \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}}}\right )-\frac {a b \,c^{3} \operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right )^{2} e}-\frac {a b \,c^{3} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{e \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {a b \,c^{4} d \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\)	\(815\)
default	\(\frac {-\frac {a^{2} c^{3}}{2 \left (e c x +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\operatorname {arcsinh}\left (c x \right ) \left (2 d c e \sqrt {c^{2} x^{2}+1}-4 d \,c^{2} e x +e^{2} \operatorname {arcsinh}\left (c x \right )+2 \sqrt {c^{2} x^{2}+1}\, e^{2} c x -2 c^{2} d^{2}-2 e^{2} c^{2} x^{2}+c^{2} d^{2} \operatorname {arcsinh}\left (c x \right )\right )}{2 e \left (e c x +c d \right )^{2} \left (c^{2} d^{2}+e^{2}\right )}-\frac {2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{e \left (c^{2} d^{2}+e^{2}\right )}+\frac {\ln \left (2 d \left (c x +\sqrt {c^{2} x^{2}+1}\right ) c +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-e \right )}{e \left (c^{2} d^{2}+e^{2}\right )}+\frac {d c \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}}}-\frac {d c \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}}}+\frac {d c \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}}}-\frac {d c \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}}}\right )-\frac {a b \,c^{3} \operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right )^{2} e}-\frac {a b \,c^{3} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{e \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {a b \,c^{4} d \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\)	\(815\)
parts	\(-\frac {a^{2}}{2 \left (e x +d \right )^{2} e}+\frac {b^{2} \left (-\frac {c^{3} \operatorname {arcsinh}\left (c x \right ) \left (2 d c e \sqrt {c^{2} x^{2}+1}-4 d \,c^{2} e x +e^{2} \operatorname {arcsinh}\left (c x \right )+2 \sqrt {c^{2} x^{2}+1}\, e^{2} c x -2 c^{2} d^{2}-2 e^{2} c^{2} x^{2}+c^{2} d^{2} \operatorname {arcsinh}\left (c x \right )\right )}{2 e \left (c^{2} d^{2}+e^{2}\right ) \left (e c x +c d \right )^{2}}+\frac {c^{3} \ln \left (2 d \left (c x +\sqrt {c^{2} x^{2}+1}\right ) c +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-e \right )}{\left (c^{2} d^{2}+e^{2}\right ) e}-\frac {2 c^{3} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\left (c^{2} d^{2}+e^{2}\right ) e}+\frac {c^{4} d \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{4} d \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{4} d \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{\frac {3}{2}} e}\right )}{c}-\frac {a b \,c^{2} \operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right )^{2} e}-\frac {a b \,c^{2} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{e \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {a b \,c^{3} d \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\)	\(822\)

input

int((a+b*arcsinh(c*x))^2/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

output

1/c*(-1/2*a^2*c^3/(c*e*x+c*d)^2/e+b^2*c^3*(-1/2*arcsinh(c*x)*(2*d*c*e*(c^2 
*x^2+1)^(1/2)-4*d*c^2*e*x+e^2*arcsinh(c*x)+2*(c^2*x^2+1)^(1/2)*e^2*c*x-2*c 
^2*d^2-2*e^2*c^2*x^2+c^2*d^2*arcsinh(c*x))/e/(c*e*x+c*d)^2/(c^2*d^2+e^2)-2 
/e/(c^2*d^2+e^2)*ln(c*x+(c^2*x^2+1)^(1/2))+1/e/(c^2*d^2+e^2)*ln(2*d*(c*x+( 
c^2*x^2+1)^(1/2))*c+e*(c*x+(c^2*x^2+1)^(1/2))^2-e)+1/e/(c^2*d^2+e^2)^(3/2) 
*d*c*arcsinh(c*x)*ln((-c*d-e*(c*x+(c^2*x^2+1)^(1/2))+(c^2*d^2+e^2)^(1/2))/ 
(-c*d+(c^2*d^2+e^2)^(1/2)))-1/e/(c^2*d^2+e^2)^(3/2)*d*c*arcsinh(c*x)*ln((c 
*d+e*(c*x+(c^2*x^2+1)^(1/2))+(c^2*d^2+e^2)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2) 
))+1/e/(c^2*d^2+e^2)^(3/2)*d*c*dilog((-c*d-e*(c*x+(c^2*x^2+1)^(1/2))+(c^2* 
d^2+e^2)^(1/2))/(-c*d+(c^2*d^2+e^2)^(1/2)))-1/e/(c^2*d^2+e^2)^(3/2)*d*c*di 
log((c*d+e*(c*x+(c^2*x^2+1)^(1/2))+(c^2*d^2+e^2)^(1/2))/(c*d+(c^2*d^2+e^2) 
^(1/2))))-a*b*c^3/(c*e*x+c*d)^2/e*arcsinh(c*x)-a*b*c^3/e/(c^2*d^2+e^2)/(c* 
x+d*c/e)*((c*x+d*c/e)^2-2*d*c/e*(c*x+d*c/e)+(c^2*d^2+e^2)/e^2)^(1/2)-a*b*c 
^4/e^2*d/(c^2*d^2+e^2)/((c^2*d^2+e^2)/e^2)^(1/2)*ln((2*(c^2*d^2+e^2)/e^2-2 
*d*c/e*(c*x+d*c/e)+2*((c^2*d^2+e^2)/e^2)^(1/2)*((c*x+d*c/e)^2-2*d*c/e*(c*x 
+d*c/e)+(c^2*d^2+e^2)/e^2)^(1/2))/(c*x+d*c/e)))
 

3.1.18.5 Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]

input

integrate((a+b*arcsinh(c*x))^2/(e*x+d)^3,x, algorithm="fricas")
 

output

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(e^3*x^3 + 3*d*e^ 
2*x^2 + 3*d^2*e*x + d^3), x)
 

3.1.18.6 Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]

input

integrate((a+b*asinh(c*x))**2/(e*x+d)**3,x)
 

output

Integral((a + b*asinh(c*x))**2/(d + e*x)**3, x)
 

3.1.18.7 Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]

input

integrate((a+b*arcsinh(c*x))^2/(e*x+d)^3,x, algorithm="maxima")
 

output

-(c*(sqrt(c^2*x^2 + 1)/(c^2*d^2*e*x + c^2*d^3 + e^3*x + d*e^2) - c^2*d*arc 
sinh(c*d*x/(e*abs(x + d/e)) - 1/(c*abs(x + d/e)))/((c^2*d^2/e^2 + 1)^(3/2) 
*e^4)) + arcsinh(c*x)/(e^3*x^2 + 2*d*e^2*x + d^2*e))*a*b - 1/2*b^2*(log(c* 
x + sqrt(c^2*x^2 + 1))^2/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 2*integrate((c^3* 
x^2 + sqrt(c^2*x^2 + 1)*c^2*x + c)*log(c*x + sqrt(c^2*x^2 + 1))/(c^3*e^3*x 
^5 + 2*c^3*d*e^2*x^4 + 2*c*d*e^2*x^2 + c*d^2*e*x + (c^3*d^2*e + c*e^3)*x^3 
 + (c^2*e^3*x^4 + 2*c^2*d*e^2*x^3 + 2*d*e^2*x + d^2*e + (c^2*d^2*e + e^3)* 
x^2)*sqrt(c^2*x^2 + 1)), x)) - 1/2*a^2/(e^3*x^2 + 2*d*e^2*x + d^2*e)
 

3.1.18.8 Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]

input

integrate((a+b*arcsinh(c*x))^2/(e*x+d)^3,x, algorithm="giac")
 

output

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

3.1.18.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]

input

int((a + b*asinh(c*x))^2/(d + e*x)^3,x)
 

output

int((a + b*asinh(c*x))^2/(d + e*x)^3, x)