\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^2} \, dx\) [17]

Optimal result

Integrand size = 18, antiderivative size = 263 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^2} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^2} \, dx=\frac {-\frac {(a+b \text {arcsinh}(c x))^2}{d+e x}+\frac {2 b c \left ((a+b \text {arcsinh}(c x)) \left (\log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )-\log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )}{\sqrt {c^2 d^2+e^2}}}{e} \] Input:

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

Output:

(-((a + b*ArcSinh[c*x])^2/(d + e*x)) + (2*b*c*((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 + Sqr 
t[c^2*d^2 + e^2]))]))/Sqrt[c^2*d^2 + e^2])/e
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6243, 6258, 3042, 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.

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

\(\Big \downarrow \) 6243

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

\(\Big \downarrow \) 6258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

\(\displaystyle \frac {4 b c \int -\frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{-2 c e^{\text {arcsinh}(c x)} d-e e^{2 \text {arcsinh}(c x)}+e}d\text {arcsinh}(c x)}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 b c \int \frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{-2 c e^{\text {arcsinh}(c x)} d-e e^{2 \text {arcsinh}(c x)}+e}d\text {arcsinh}(c x)}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}\)

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {4 b c \left (\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}-\frac {b \int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d+\sqrt {c^2 d^2+e^2}}+1\right )de^{\text {arcsinh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}-\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}-\frac {b \int e^{-\text {arcsinh}(c x)} \log \left (\frac {e^{\text {arcsinh}(c x)} e}{c d-\sqrt {c^2 d^2+e^2}}+1\right )de^{\text {arcsinh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {4 b c \left (\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}-\frac {e \left (\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}\)

Input:

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

Output:

-((a + b*ArcSinh[c*x])^2/(e*(d + e*x))) - (4*b*c*(-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*Pol 
yLog[2, -((e*E^ArcSinh[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 + Sq 
rt[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])))/e
 

Defintions of rubi rules used

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 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 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])
 

Maple [A] (verified)

Time = 5.75 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.00

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} c^{2}}{\left (c e x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{e \left (c e x +c d \right )}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}+\frac {2 \operatorname {dilog}\left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 \operatorname {dilog}\left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{\left (c e x +c d \right ) e}-\frac {\ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{x c +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(525\)
default	\(\frac {-\frac {a^{2} c^{2}}{\left (c e x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{e \left (c e x +c d \right )}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}+\frac {2 \operatorname {dilog}\left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 \operatorname {dilog}\left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{\left (c e x +c d \right ) e}-\frac {\ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{x c +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(525\)
parts	\(-\frac {a^{2}}{\left (e x +d \right ) e}+\frac {b^{2} \left (-\frac {c^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{e \left (c e x +c d \right )}+\frac {2 c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 c^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}+\frac {2 c^{2} \operatorname {dilog}\left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 c^{2} \operatorname {dilog}\left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}\right )}{c}-\frac {2 a b c \,\operatorname {arcsinh}\left (x c \right )}{\left (c e x +c d \right ) e}-\frac {2 a b c \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (x c +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (x c +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{x c +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\)	\(528\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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