\(\int \frac {x^3 (a+b \text {arcsinh}(c x))}{(d+c^2 d x^2)^2} \, dx\) [38]

Optimal result

Integrand size = 24, antiderivative size = 145 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=-\frac {b x}{2 c^3 d^2 \sqrt {1+c^2 x^2}}+\frac {b \text {arcsinh}(c x)}{2 c^4 d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^4 d^2}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^4 d^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 c^4 d^2} \] Output:

-1/2*b*x/c^3/d^2/(c^2*x^2+1)^(1/2)+1/2*b*arcsinh(c*x)/c^4/d^2-1/2*x^2*(a+b 
*arcsinh(c*x))/c^2/d^2/(c^2*x^2+1)-1/2*(a+b*arcsinh(c*x))^2/b/c^4/d^2+(a+b 
*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/c^4/d^2+1/2*b*polylog(2,-(c 
*x+(c^2*x^2+1)^(1/2))^2)/c^4/d^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.66 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {a-b c x \sqrt {1+c^2 x^2}+b \text {arcsinh}(c x)-b \text {arcsinh}(c x)^2-b c^2 x^2 \text {arcsinh}(c x)^2+2 b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+2 b c^2 x^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+2 b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+2 b c^2 x^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+a \log \left (1+c^2 x^2\right )+a c^2 x^2 \log \left (1+c^2 x^2\right )+2 b \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+2 b \left (1+c^2 x^2\right ) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c^4 d^2 \left (1+c^2 x^2\right )} \] Input:

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

Output:

(a - b*c*x*Sqrt[1 + c^2*x^2] + b*ArcSinh[c*x] - b*ArcSinh[c*x]^2 - b*c^2*x 
^2*ArcSinh[c*x]^2 + 2*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2*b*c^2*x 
^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2*b*ArcSinh[c*x]*Log[1 + I*E^A 
rcSinh[c*x]] + 2*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + a*Log[ 
1 + c^2*x^2] + a*c^2*x^2*Log[1 + c^2*x^2] + 2*b*(1 + c^2*x^2)*PolyLog[2, ( 
-I)*E^ArcSinh[c*x]] + 2*b*(1 + c^2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c 
^4*d^2*(1 + c^2*x^2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6225, 27, 252, 222, 6212, 3042, 26, 4201, 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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c^4 d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {\text {arcsinh}(c x)}{c^3}-\frac {x}{c^2 \sqrt {c^2 x^2+1}}\right )}{2 c d^2}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c^4 d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {\text {arcsinh}(c x)}{c^3}-\frac {x}{c^2 \sqrt {c^2 x^2+1}}\right )}{2 c d^2}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c^4 d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {\text {arcsinh}(c x)}{c^3}-\frac {x}{c^2 \sqrt {c^2 x^2+1}}\right )}{2 c d^2}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c^4 d^2}-\frac {x^2 (a+b \text {arcsinh}(c x))}{2 c^2 d^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {\text {arcsinh}(c x)}{c^3}-\frac {x}{c^2 \sqrt {c^2 x^2+1}}\right )}{2 c d^2}\)

Input:

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

Output:

-1/2*(x^2*(a + b*ArcSinh[c*x]))/(c^2*d^2*(1 + c^2*x^2)) + (b*(-(x/(c^2*Sqr 
t[1 + c^2*x^2])) + ArcSinh[c*x]/c^3))/(2*c*d^2) - (I*(((-1/2*I)*(a + b*Arc 
Sinh[c*x])^2)/b + (2*I)*(((a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])] 
)/2 + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/4)))/(c^4*d^2)
 

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, 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 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 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6212

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[1/e   Subst[Int[(a + b*x)^n*Tanh[x], x], x, ArcSinh[c*x] 
], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
 

rule 6225

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^( 
m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; Fre 
eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 1]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{2 c^{2} x^{2}+2}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2}}}{c^{4}}\)	\(140\)
default	\(\frac {\frac {a \left (\frac {1}{2 c^{2} x^{2}+2}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2}}}{c^{4}}\)	\(140\)
parts	\(\frac {a \left (\frac {1}{2 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2 c^{4}}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{2} c^{4}}\)	\(145\)

Input:

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

Output:

1/c^4*(a/d^2*(1/2/(c^2*x^2+1)+1/2*ln(c^2*x^2+1))+b/d^2*(-1/2*arcsinh(x*c)^ 
2+1/2*(-(c^2*x^2+1)^(1/2)*x*c+c^2*x^2+arcsinh(x*c)+1)/(c^2*x^2+1)+arcsinh( 
x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)+1/2*polylog(2,-(x*c+(c^2*x^2+1)^(1/2) 
)^2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a x^{3}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{3} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a*x**3/(c**4*x**4 + 2*c**2*x**2 + 1), x) + Integral(b*x**3*asinh 
(c*x)/(c**4*x**4 + 2*c**2*x**2 + 1), x))/d**2
 

Maxima [F]

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

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

Output:

-1/8*b*(((c^2*x^2 + 1)*log(c^2*x^2 + 1)^2 - 4*((c^2*x^2 + 1)*log(c^2*x^2 + 
 1) + 1)*log(c*x + sqrt(c^2*x^2 + 1)) - 2)/(c^6*d^2*x^2 + c^4*d^2) + 8*int 
egrate(1/2*((c^2*x^2 + 1)*log(c^2*x^2 + 1) + 1)/(c^8*d^2*x^5 + 2*c^6*d^2*x 
^3 + c^4*d^2*x + (c^7*d^2*x^4 + 2*c^5*d^2*x^2 + c^3*d^2)*sqrt(c^2*x^2 + 1) 
), x)) + 1/2*a*(1/(c^6*d^2*x^2 + c^4*d^2) + log(c^2*x^2 + 1)/(c^4*d^2))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{c^{4} x^{4}+2 c^{2} x^{2}+1}d x \right ) b \,c^{6} x^{2}+2 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{c^{4} x^{4}+2 c^{2} x^{2}+1}d x \right ) b \,c^{4}+\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,c^{2} x^{2}+\mathrm {log}\left (c^{2} x^{2}+1\right ) a -a \,c^{2} x^{2}}{2 c^{4} d^{2} \left (c^{2} x^{2}+1\right )} \] Input:

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

Output:

(2*int((asinh(c*x)*x**3)/(c**4*x**4 + 2*c**2*x**2 + 1),x)*b*c**6*x**2 + 2* 
int((asinh(c*x)*x**3)/(c**4*x**4 + 2*c**2*x**2 + 1),x)*b*c**4 + log(c**2*x 
**2 + 1)*a*c**2*x**2 + log(c**2*x**2 + 1)*a - a*c**2*x**2)/(2*c**4*d**2*(c 
**2*x**2 + 1))