\(\int \frac {a+b \text {arcsinh}(c x)}{x (d+c^2 d x^2)^2} \, dx\) [42]

Optimal result

Integrand size = 24, antiderivative size = 110 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c x}{2 d^2 \sqrt {1+c^2 x^2}}+\frac {a+b \text {arcsinh}(c x)}{2 d^2 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^2}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(234\) vs. \(2(110)=220\).

Time = 0.31 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.13 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx=-\frac {\frac {a^2}{b}-\frac {a}{1+c^2 x^2}+\frac {b c x}{\sqrt {1+c^2 x^2}}+2 a \text {arcsinh}(c x)-\frac {b \text {arcsinh}(c x)}{1+c^2 x^2}+2 b \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+2 b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-2 a \log \left (1-e^{2 \text {arcsinh}(c x)}\right )-2 b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+a \log \left (1+c^2 x^2\right )+2 b \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+2 b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \] Input:

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

Output:

-1/2*(a^2/b - a/(1 + c^2*x^2) + (b*c*x)/Sqrt[1 + c^2*x^2] + 2*a*ArcSinh[c* 
x] - (b*ArcSinh[c*x])/(1 + c^2*x^2) + 2*b*ArcSinh[c*x]*Log[1 + (c*E^ArcSin 
h[c*x])/Sqrt[-c^2]] + 2*b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x]) 
/c] - 2*a*Log[1 - E^(2*ArcSinh[c*x])] - 2*b*ArcSinh[c*x]*Log[1 - E^(2*ArcS 
inh[c*x])] + a*Log[1 + c^2*x^2] + 2*b*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[- 
c^2]] + 2*b*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - b*PolyLog[2, E^(2* 
ArcSinh[c*x])])/d^2
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.89 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6226, 27, 208, 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.

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/2*(b*c*x)/(d^2*Sqrt[1 + c^2*x^2]) + (a + b*ArcSinh[c*x])/(2*d^2*(1 + c^ 
2*x^2)) + ((2*I)*(I*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])] + (I/ 
4)*b*PolyLog[2, -E^(2*ArcSinh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcSinh[c*x 
])]))/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 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

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]
 

rule 6226

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
 + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 
))   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ 
b*c*(n/(2*f*(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]) /; FreeQ[{ 
a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.97

method	result	size
derivativedivides	\(\frac {a \left (\ln \left (x c \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {1}{2 c^{2} x^{2}+2}\right )}{d^{2}}+\frac {b \left (\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(217\)
default	\(\frac {a \left (\ln \left (x c \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {1}{2 c^{2} x^{2}+2}\right )}{d^{2}}+\frac {b \left (\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(217\)
parts	\(\frac {a \ln \left (x \right )}{d^{2}}+\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {b \left (\frac {-\sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )+1}{2 c^{2} x^{2}+2}-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(222\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*arcsinh(c*x) + a)/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{5} + 2 c^{2} x^{3} + x}\, dx}{d^{2}} \] Input:

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

Output:

(Integral(a/(c**4*x**5 + 2*c**2*x**3 + x), x) + Integral(b*asinh(c*x)/(c** 
4*x**5 + 2*c**2*x**3 + x), x))/d**2
 

Maxima [F]

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

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

Output:

1/2*a*(1/(c^2*d^2*x^2 + d^2) - log(c^2*x^2 + 1)/d^2 + 2*log(x)/d^2) + b*in 
tegrate(log(c*x + sqrt(c^2*x^2 + 1))/(c^4*d^2*x^5 + 2*c^2*d^2*x^3 + d^2*x) 
, x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{4} x^{5}+2 c^{2} x^{3}+x}d x \right ) b \,c^{2} x^{2}+2 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{4} x^{5}+2 c^{2} x^{3}+x}d x \right ) b -\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,c^{2} x^{2}-\mathrm {log}\left (c^{2} x^{2}+1\right ) a +2 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) a -a \,c^{2} x^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )} \] Input:

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

Output:

(2*int(asinh(c*x)/(c**4*x**5 + 2*c**2*x**3 + x),x)*b*c**2*x**2 + 2*int(asi 
nh(c*x)/(c**4*x**5 + 2*c**2*x**3 + x),x)*b - log(c**2*x**2 + 1)*a*c**2*x** 
2 - log(c**2*x**2 + 1)*a + 2*log(x)*a*c**2*x**2 + 2*log(x)*a - a*c**2*x**2 
)/(2*d**2*(c**2*x**2 + 1))