\(\int \frac {x (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx\) [31]

Optimal result

Integrand size = 22, antiderivative size = 73 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^2 d}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 c^2 d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(167\) vs. \(2(73)=146\).

Time = 0.05 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.29 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{d+c^2 d x^2} \, dx=-\frac {b \text {arcsinh}(c x)^2}{2 c^2 d}+\frac {b \text {arcsinh}(c x) \log \left (1-\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{c^2 d}+\frac {b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{c^2 d}+\frac {a \log \left (1+c^2 x^2\right )}{2 c^2 d}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{c^2 d}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{c^2 d} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {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.

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

((-I)*(((-1/2*I)*(a + b*ArcSinh[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^2*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 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]
 

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}}{c^{2}}\)	\(84\)
default	\(\frac {\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}}{c^{2}}\)	\(84\)
parts	\(\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d \,c^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d \,c^{2}}\)	\(86\)

Input:

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

Output:

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

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/8*b*((log(c^2*x^2 + 1)^2 - 4*log(c^2*x^2 + 1)*log(c*x + sqrt(c^2*x^2 + 
1)))/(c^2*d) + 8*integrate(1/2*log(c^2*x^2 + 1)/(c^4*d*x^3 + c^2*d*x + (c^ 
3*d*x^2 + c*d)*sqrt(c^2*x^2 + 1)), x)) + 1/2*a*log(c^2*d*x^2 + d)/(c^2*d)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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