\(\int \frac {a+b \text {arcsinh}(c x)}{x (d+c^2 d x^2)} \, dx\) [33]

Optimal result

Integrand size = 24, antiderivative size = 61 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(61)=122\).

Time = 0.06 (sec) , antiderivative size = 207, normalized size of antiderivative = 3.39 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )} \, dx=-\frac {a \text {arcsinh}(c x)}{d}-\frac {b \text {arcsinh}(c x) \log \left (1-\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}-\frac {b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}+\frac {a \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {a \log \left (1+c^2 x^2\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \] Input:

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

Output:

-((a*ArcSinh[c*x])/d) - (b*ArcSinh[c*x]*Log[1 - (Sqrt[-c^2]*E^ArcSinh[c*x] 
)/c])/d - (b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/d + (a*L 
og[1 - E^(2*ArcSinh[c*x])])/d + (b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x]) 
])/d - (a*Log[1 + c^2*x^2])/(2*d) - (b*PolyLog[2, -((Sqrt[-c^2]*E^ArcSinh[ 
c*x])/c)])/d - (b*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/d + (b*PolyLo 
g[2, E^(2*ArcSinh[c*x])])/(2*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6214, 5984, 3042, 26, 4670, 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.

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

((2*I)*(I*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])] + (I/4)*b*PolyL 
og[2, -E^(2*ArcSinh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcSinh[c*x])]))/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 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 4670

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

rule 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

rule 6214

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

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.64

method	result	size
parts	\(\frac {a \left (\ln \left (x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b \left (-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\)	\(161\)
derivativedivides	\(\frac {a \left (\ln \left (x c \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b \left (-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\)	\(163\)
default	\(\frac {a \left (\ln \left (x c \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b \left (-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\)	\(163\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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