\(\int \frac {a+b \text {arcsinh}(c x)}{x (d+c^2 d x^2)^3} \, dx\) [51]

Optimal result

Integrand size = 24, antiderivative size = 159 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^3} \, dx=-\frac {b c x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {a+b \text {arcsinh}(c x)}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {a+b \text {arcsinh}(c x)}{2 d^3 \left (1+c^2 x^2\right )}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}+\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3} \] Output:

-1/12*b*c*x/d^3/(c^2*x^2+1)^(3/2)-2/3*b*c*x/d^3/(c^2*x^2+1)^(1/2)+1/4*(a+b 
*arcsinh(c*x))/d^3/(c^2*x^2+1)^2+1/2*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)-2* 
(a+b*arcsinh(c*x))*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d^3-1/2*b*polylog(2, 
-(c*x+(c^2*x^2+1)^(1/2))^2)/d^3+1/2*b*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2) 
/d^3
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.82 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^3} \, dx=\frac {-\frac {2 a^2}{b}+\frac {a}{\left (1+c^2 x^2\right )^2}-\frac {b c x}{3 \left (1+c^2 x^2\right )^{3/2}}+\frac {2 a}{1+c^2 x^2}-\frac {8 b c x}{3 \sqrt {1+c^2 x^2}}-4 a \text {arcsinh}(c x)+\frac {b \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^2}+\frac {2 b \text {arcsinh}(c x)}{1+c^2 x^2}-4 b \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-4 b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+4 a \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+4 b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )-2 a \log \left (1+c^2 x^2\right )-4 b \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-4 b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+2 b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{4 d^3} \] Input:

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

Output:

((-2*a^2)/b + a/(1 + c^2*x^2)^2 - (b*c*x)/(3*(1 + c^2*x^2)^(3/2)) + (2*a)/ 
(1 + c^2*x^2) - (8*b*c*x)/(3*Sqrt[1 + c^2*x^2]) - 4*a*ArcSinh[c*x] + (b*Ar 
cSinh[c*x])/(1 + c^2*x^2)^2 + (2*b*ArcSinh[c*x])/(1 + c^2*x^2) - 4*b*ArcSi 
nh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 4*b*ArcSinh[c*x]*Log[1 + 
(Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 4*a*Log[1 - E^(2*ArcSinh[c*x])] + 4*b*Arc 
Sinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] - 2*a*Log[1 + c^2*x^2] - 4*b*PolyLog 
[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 4*b*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[ 
c*x])/c] + 2*b*PolyLog[2, E^(2*ArcSinh[c*x])])/(4*d^3)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.01 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6226, 27, 209, 208, 6226, 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.

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

\(\Big \downarrow \) 6226

\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{d^2 x \left (c^2 x^2+1\right )^2}dx}{d}-\frac {b c \int \frac {1}{\left (c^2 x^2+1\right )^{5/2}}dx}{4 d^3}+\frac {a+b \text {arcsinh}(c x)}{4 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx}{d^3}-\frac {b c \int \frac {1}{\left (c^2 x^2+1\right )^{5/2}}dx}{4 d^3}+\frac {a+b \text {arcsinh}(c x)}{4 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/4*(b*c*(x/(3*(1 + c^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 + c^2*x^2])))/d^3 + 
 (a + b*ArcSinh[c*x])/(4*d^3*(1 + c^2*x^2)^2) + (-1/2*(b*c*x)/Sqrt[1 + c^2 
*x^2] + (a + b*ArcSinh[c*x])/(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^3
 

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 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 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 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 = 1.16 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.76

method	result	size
derivativedivides	\(\frac {a \left (\ln \left (x c \right )+\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {1}{2 c^{2} x^{2}+2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{3}}+\frac {b \left (\frac {-8 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+8 c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, x c +16 c^{2} x^{2}+9 \,\operatorname {arcsinh}\left (x c \right )+8}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{3}}\)	\(280\)
default	\(\frac {a \left (\ln \left (x c \right )+\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {1}{2 c^{2} x^{2}+2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{3}}+\frac {b \left (\frac {-8 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+8 c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, x c +16 c^{2} x^{2}+9 \,\operatorname {arcsinh}\left (x c \right )+8}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{3}}\)	\(280\)
parts	\(\frac {a \left (\ln \left (x \right )-\frac {c^{2} \left (\frac {\ln \left (c^{2} x^{2}+1\right )}{c^{2}}-\frac {1}{2 c^{2} \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{c^{2} \left (c^{2} x^{2}+1\right )}\right )}{2}\right )}{d^{3}}+\frac {b \left (\frac {-8 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+8 c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-9 \sqrt {c^{2} x^{2}+1}\, x c +16 c^{2} x^{2}+9 \,\operatorname {arcsinh}\left (x c \right )+8}{12 c^{4} x^{4}+24 c^{2} x^{2}+12}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{3}}\)	\(292\)

Input:

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

Output:

a/d^3*(ln(x*c)+1/4/(c^2*x^2+1)^2+1/2/(c^2*x^2+1)-1/2*ln(c^2*x^2+1))+b/d^3* 
(1/12*(-8*(c^2*x^2+1)^(1/2)*c^3*x^3+8*c^4*x^4+6*arcsinh(x*c)*c^2*x^2-9*(c^ 
2*x^2+1)^(1/2)*x*c+16*c^2*x^2+9*arcsinh(x*c)+8)/(c^4*x^4+2*c^2*x^2+1)+arcs 
inh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))+polylog(2,x*c+(c^2*x^2+1)^(1/2))+arcs 
inh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))+polylog(2,-x*c-(c^2*x^2+1)^(1/2))-arc 
sinh(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]

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

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

Output:

integral((b*arcsinh(c*x) + a)/(c^6*d^3*x^7 + 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 
 + d^3*x), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a}{c^{6} x^{7} + 3 c^{4} x^{5} + 3 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{7} + 3 c^{4} x^{5} + 3 c^{2} x^{3} + x}\, dx}{d^{3}} \] Input:

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

Output:

(Integral(a/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x), x) + Integral(b*a 
sinh(c*x)/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x), x))/d**3
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) b \,c^{4} x^{4}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) b \,c^{2} x^{2}+4 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{7}+3 c^{4} x^{5}+3 c^{2} x^{3}+x}d x \right ) b -2 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,c^{4} x^{4}-4 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a +4 \,\mathrm {log}\left (x \right ) a \,c^{4} x^{4}+8 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a -a \,c^{4} x^{4}+2 a}{4 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

(4*int(asinh(c*x)/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x),x)*b*c**4*x* 
*4 + 8*int(asinh(c*x)/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x),x)*b*c** 
2*x**2 + 4*int(asinh(c*x)/(c**6*x**7 + 3*c**4*x**5 + 3*c**2*x**3 + x),x)*b 
 - 2*log(c**2*x**2 + 1)*a*c**4*x**4 - 4*log(c**2*x**2 + 1)*a*c**2*x**2 - 2 
*log(c**2*x**2 + 1)*a + 4*log(x)*a*c**4*x**4 + 8*log(x)*a*c**2*x**2 + 4*lo 
g(x)*a - a*c**4*x**4 + 2*a)/(4*d**3*(c**4*x**4 + 2*c**2*x**2 + 1))