\(\int \frac {a+b \text {arcsinh}(c x)}{x^3 (d+c^2 d x^2)^2} \, dx\) [44]

Optimal result

Integrand size = 24, antiderivative size = 137 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c}{2 d^2 x \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{2 d^2 x^2}-\frac {c^2 (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2} \] Output:

-1/2*b*c/d^2/x/(c^2*x^2+1)^(1/2)-1/2*(a+b*arcsinh(c*x))/d^2/x^2-1/2*c^2*(a 
+b*arcsinh(c*x))/d^2/(c^2*x^2+1)+4*c^2*(a+b*arcsinh(c*x))*arctanh((c*x+(c^ 
2*x^2+1)^(1/2))^2)/d^2+b*c^2*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/d^2-b*c 
^2*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. \(326\) vs. \(2(137)=274\).

Time = 0.31 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.38 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\frac {2 a^2 c^2}{b}-\frac {2 a}{x^2}+\frac {b c}{x \sqrt {1+c^2 x^2}}+\frac {2 b c^3 x}{\sqrt {1+c^2 x^2}}-\frac {2 b c \sqrt {1+c^2 x^2}}{x}+\frac {a}{x^2+c^2 x^4}+4 a c^2 \text {arcsinh}(c x)-\frac {2 b \text {arcsinh}(c x)}{x^2}+\frac {b \text {arcsinh}(c x)}{x^2+c^2 x^4}+4 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+4 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-4 a c^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )-4 b c^2 \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+2 a c^2 \log \left (1+c^2 x^2\right )+4 b c^2 \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+4 b c^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-2 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^2} \] Input:

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

Output:

((2*a^2*c^2)/b - (2*a)/x^2 + (b*c)/(x*Sqrt[1 + c^2*x^2]) + (2*b*c^3*x)/Sqr 
t[1 + c^2*x^2] - (2*b*c*Sqrt[1 + c^2*x^2])/x + a/(x^2 + c^2*x^4) + 4*a*c^2 
*ArcSinh[c*x] - (2*b*ArcSinh[c*x])/x^2 + (b*ArcSinh[c*x])/(x^2 + c^2*x^4) 
+ 4*b*c^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 4*b*c^2*Ar 
cSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 4*a*c^2*Log[1 - E^(2*A 
rcSinh[c*x])] - 4*b*c^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 2*a*c^2 
*Log[1 + c^2*x^2] + 4*b*c^2*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 4* 
b*c^2*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 2*b*c^2*PolyLog[2, E^(2* 
ArcSinh[c*x])])/(2*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.97 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.39, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6224, 27, 245, 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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 245

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(b*c*(-(1/(x*Sqrt[1 + c^2*x^2])) - (2*c^2*x)/Sqrt[1 + c^2*x^2]))/(2*d^2) - 
 (a + b*ArcSinh[c*x])/(2*d^2*x^2*(1 + c^2*x^2)) - (2*c^2*(-1/2*(b*c*x)/Sqr 
t[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*ArcSi 
nh[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 245

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + 
 b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) 
   Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si 
mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -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]
 

rule 6224

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/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 
1)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim 
p[b*c*(n/(f*(m + 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
 

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 = 242, normalized size of antiderivative = 1.77

method	result	size
derivativedivides	\(c^{2} \left (\frac {a \left (-\frac {1}{2 \left (c^{2} x^{2}+1\right )}+\ln \left (c^{2} x^{2}+1\right )-\frac {1}{2 c^{2} x^{2}}-2 \ln \left (x c \right )\right )}{d^{2}}+\frac {b \left (-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}+2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\)	\(242\)
default	\(c^{2} \left (\frac {a \left (-\frac {1}{2 \left (c^{2} x^{2}+1\right )}+\ln \left (c^{2} x^{2}+1\right )-\frac {1}{2 c^{2} x^{2}}-2 \ln \left (x c \right )\right )}{d^{2}}+\frac {b \left (-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}+2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\)	\(242\)
parts	\(\frac {a \left (-\frac {1}{2 x^{2}}-2 c^{2} \ln \left (x \right )+\frac {c^{4} \left (-\frac {1}{c^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 \ln \left (c^{2} x^{2}+1\right )}{c^{2}}\right )}{2}\right )}{d^{2}}+\frac {b \,c^{2} \left (-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}+2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(253\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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