\(\int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^3} \, dx\) [108]

Optimal result

Integrand size = 23, antiderivative size = 246 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^3} \, dx=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)}{b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 b^3 d}+\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{8 b^3 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 b^3 d} \] Output:

-1/2*e^2*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^2-e^2*(d*x 
+c)/b^2/d/(a+b*arcsinh(d*x+c))-3/2*e^2*(d*x+c)^3/b^2/d/(a+b*arcsinh(d*x+c) 
)-1/8*e^2*cosh(a/b)*Chi((a+b*arcsinh(d*x+c))/b)/b^3/d+9/8*e^2*cosh(3*a/b)* 
Chi(3*(a+b*arcsinh(d*x+c))/b)/b^3/d+1/8*e^2*sinh(a/b)*Shi((a+b*arcsinh(d*x 
+c))/b)/b^3/d-9/8*e^2*sinh(3*a/b)*Shi(3*(a+b*arcsinh(d*x+c))/b)/b^3/d
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.88 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^3} \, dx=\frac {e^2 \left (-\frac {4 b^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{(a+b \text {arcsinh}(c+d x))^2}+\frac {4 b \left (-2 (c+d x)-3 (c+d x)^3\right )}{a+b \text {arcsinh}(c+d x)}+8 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )-8 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+9 \left (-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )\right )}{8 b^3 d} \] Input:

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

Output:

(e^2*((-4*b^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^ 
2 + (4*b*(-2*(c + d*x) - 3*(c + d*x)^3))/(a + b*ArcSinh[c + d*x]) + 8*Cosh 
[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]] - 8*Sinh[a/b]*SinhIntegral[a/b 
+ ArcSinh[c + d*x]] + 9*(-(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]]) 
 + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c + d*x])] + Sinh[a/b]*Sinh 
Integral[a/b + ArcSinh[c + d*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + Arc 
Sinh[c + d*x])])))/(8*b^3*d)
 

Rubi [A] (verified)

Time = 2.11 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6274, 27, 6194, 6233, 6189, 3042, 3784, 26, 3042, 26, 3779, 3782, 6195, 5971, 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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6194

\(\displaystyle \frac {e^2 \left (\frac {\int \frac {c+d x}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{b}+\frac {3 \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 6233

\(\displaystyle \frac {e^2 \left (\frac {\frac {\int \frac {1}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 6189

\(\displaystyle \frac {e^2 \left (\frac {\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^2 \left (\frac {-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}+\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {e^2 \left (\frac {-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \sinh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {e^2 \left (\frac {\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-\sinh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^2 \left (\frac {-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-\sinh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {e^2 \left (\frac {-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {e^2 \left (\frac {-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}+\frac {-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 3782

\(\displaystyle \frac {e^2 \left (\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}+\frac {\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b^2}-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {e^2 \left (\frac {3 \left (\frac {3 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}+\frac {\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b^2}-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {e^2 \left (\frac {3 \left (\frac {3 \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 (a+b \text {arcsinh}(c+d x))}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 (a+b \text {arcsinh}(c+d x))}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}+\frac {\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b^2}-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \left (\frac {\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{b^2}-\frac {c+d x}{b (a+b \text {arcsinh}(c+d x))}}{b}+\frac {3 \left (\frac {3 \left (-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {1}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {1}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^3}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6189

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

rule 6194

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

rule 6195

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

rule 6233

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

rule 6274

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(506\) vs. \(2(232)=464\).

Time = 1.20 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.06

method	result	size
derivativedivides	\(\frac {-\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (3 b \,\operatorname {arcsinh}\left (d x +c \right )+3 a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}-\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}+\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b \,\operatorname {arcsinh}\left (d x +c \right )+a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d}\)	\(507\)
default	\(\frac {-\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (3 b \,\operatorname {arcsinh}\left (d x +c \right )+3 a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}-\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}+\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b \,\operatorname {arcsinh}\left (d x +c \right )+a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d}\)	\(507\)

Input:

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

Output:

1/d*(-1/16*(-4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)^3-(1+(d*x+c)^2)^(1/ 
2)+3*d*x+3*c)*e^2*(3*b*arcsinh(d*x+c)+3*a-b)/b^2/(b^2*arcsinh(d*x+c)^2+2*a 
*b*arcsinh(d*x+c)+a^2)-9/16*e^2/b^3*exp(3*a/b)*Ei(1,3*arcsinh(d*x+c)+3*a/b 
)+1/16*(-(1+(d*x+c)^2)^(1/2)+d*x+c)*e^2*(b*arcsinh(d*x+c)+a-b)/b^2/(b^2*ar 
csinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2)+1/16*e^2/b^3*exp(a/b)*Ei(1,arcsin 
h(d*x+c)+a/b)+1/16/b*e^2*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^ 
2+1/16/b^2*e^2*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))+1/16/b^3*e 
^2*exp(-a/b)*Ei(1,-arcsinh(d*x+c)-a/b)-1/16/b*e^2*(4*(d*x+c)^3+3*d*x+3*c+4 
*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2 
-3/16/b^2*e^2*(4*(d*x+c)^3+3*d*x+3*c+4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d 
*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))-9/16/b^3*e^2*exp(-3*a/b)*Ei(1,-3*arcs 
inh(d*x+c)-3*a/b))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*((3*a*d^9*e^2 + b*d^9*e^2)*x^9 + 9*(3*a*c*d^8*e^2 + b*c*d^8*e^2)*x^8 
+ 3*(3*(12*c^2*d^7*e^2 + d^7*e^2)*a + (12*c^2*d^7*e^2 + d^7*e^2)*b)*x^7 + 
21*(3*(4*c^3*d^6*e^2 + c*d^6*e^2)*a + (4*c^3*d^6*e^2 + c*d^6*e^2)*b)*x^6 + 
 3*(3*(42*c^4*d^5*e^2 + 21*c^2*d^5*e^2 + d^5*e^2)*a + (42*c^4*d^5*e^2 + 21 
*c^2*d^5*e^2 + d^5*e^2)*b)*x^5 + 3*(3*(42*c^5*d^4*e^2 + 35*c^3*d^4*e^2 + 5 
*c*d^4*e^2)*a + (42*c^5*d^4*e^2 + 35*c^3*d^4*e^2 + 5*c*d^4*e^2)*b)*x^4 + ( 
3*(84*c^6*d^3*e^2 + 105*c^4*d^3*e^2 + 30*c^2*d^3*e^2 + d^3*e^2)*a + (84*c^ 
6*d^3*e^2 + 105*c^4*d^3*e^2 + 30*c^2*d^3*e^2 + d^3*e^2)*b)*x^3 + 3*(3*(12* 
c^7*d^2*e^2 + 21*c^5*d^2*e^2 + 10*c^3*d^2*e^2 + c*d^2*e^2)*a + (12*c^7*d^2 
*e^2 + 21*c^5*d^2*e^2 + 10*c^3*d^2*e^2 + c*d^2*e^2)*b)*x^2 + ((3*a*d^6*e^2 
 + b*d^6*e^2)*x^6 + 6*(3*a*c*d^5*e^2 + b*c*d^5*e^2)*x^5 + ((45*c^2*d^4*e^2 
 + 4*d^4*e^2)*a + (15*c^2*d^4*e^2 + d^4*e^2)*b)*x^4 + 4*((15*c^3*d^3*e^2 + 
 4*c*d^3*e^2)*a + (5*c^3*d^3*e^2 + c*d^3*e^2)*b)*x^3 + ((45*c^4*d^2*e^2 + 
24*c^2*d^2*e^2 + d^2*e^2)*a + 3*(5*c^4*d^2*e^2 + 2*c^2*d^2*e^2)*b)*x^2 + ( 
3*c^6*e^2 + 4*c^4*e^2 + c^2*e^2)*a + (c^6*e^2 + c^4*e^2)*b + 2*((9*c^5*d*e 
^2 + 8*c^3*d*e^2 + c*d*e^2)*a + (3*c^5*d*e^2 + 2*c^3*d*e^2)*b)*x)*(d^2*x^2 
 + 2*c*d*x + c^2 + 1)^(3/2) + (3*(3*a*d^7*e^2 + b*d^7*e^2)*x^7 + 21*(3*a*c 
*d^6*e^2 + b*c*d^6*e^2)*x^6 + ((189*c^2*d^5*e^2 + 17*d^5*e^2)*a + (63*c^2* 
d^5*e^2 + 5*d^5*e^2)*b)*x^5 + 5*((63*c^3*d^4*e^2 + 17*c*d^4*e^2)*a + (21*c 
^3*d^4*e^2 + 5*c*d^4*e^2)*b)*x^4 + (5*(63*c^4*d^3*e^2 + 34*c^2*d^3*e^2 ...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^3} \, dx=e^{2} \left (\left (\int \frac {x^{2}}{\mathit {asinh} \left (d x +c \right )^{3} b^{3}+3 \mathit {asinh} \left (d x +c \right )^{2} a \,b^{2}+3 \mathit {asinh} \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) d^{2}+2 \left (\int \frac {x}{\mathit {asinh} \left (d x +c \right )^{3} b^{3}+3 \mathit {asinh} \left (d x +c \right )^{2} a \,b^{2}+3 \mathit {asinh} \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c d +\left (\int \frac {1}{\mathit {asinh} \left (d x +c \right )^{3} b^{3}+3 \mathit {asinh} \left (d x +c \right )^{2} a \,b^{2}+3 \mathit {asinh} \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c^{2}\right ) \] Input:

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

Output:

e**2*(int(x**2/(asinh(c + d*x)**3*b**3 + 3*asinh(c + d*x)**2*a*b**2 + 3*as 
inh(c + d*x)*a**2*b + a**3),x)*d**2 + 2*int(x/(asinh(c + d*x)**3*b**3 + 3* 
asinh(c + d*x)**2*a*b**2 + 3*asinh(c + d*x)*a**2*b + a**3),x)*c*d + int(1/ 
(asinh(c + d*x)**3*b**3 + 3*asinh(c + d*x)**2*a*b**2 + 3*asinh(c + d*x)*a* 
*2*b + a**3),x)*c**2)