\(\int \frac {(d+e x^2) (a+b \text {csch}^{-1}(c x))}{x} \, dx\) [87]

Optimal result

Integrand size = 19, antiderivative size = 115 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} b d \text {csch}^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \text {csch}^{-1}(c x)\right )-b d \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b d \text {csch}^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \] Output:

1/2*b*e*(1+1/c^2/x^2)^(1/2)*x/c+1/2*b*d*arccsch(c*x)^2+1/2*e*x^2*(a+b*arcc 
sch(c*x))-b*d*arccsch(c*x)*ln(1-(1/c/x+(1+1/c^2/x^2)^(1/2))^2)+b*d*arccsch 
(c*x)*ln(1/x)-d*(a+b*arccsch(c*x))*ln(1/x)-1/2*b*d*polylog(2,(1/c/x+(1+1/c 
^2/x^2)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx=\frac {1}{2} a e x^2+\frac {b e x \sqrt {\frac {1+c^2 x^2}{c^2 x^2}}}{2 c}+\frac {1}{2} b e x^2 \text {csch}^{-1}(c x)+\frac {1}{2} b d \text {csch}^{-1}(c x)^2-b d \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+a d \log (x)-\frac {1}{2} b d \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \] Input:

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

Output:

(a*e*x^2)/2 + (b*e*x*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(2*c) + (b*e*x^2*ArcCs 
ch[c*x])/2 + (b*d*ArcCsch[c*x]^2)/2 - b*d*ArcCsch[c*x]*Log[1 - E^(2*ArcCsc 
h[c*x])] + a*d*Log[x] - (b*d*PolyLog[2, E^(2*ArcCsch[c*x])])/2
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6858, 6237, 27, 7293, 2009}

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.

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

\(\Big \downarrow \) 6858

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

\(\Big \downarrow \) 6237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6237

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x 
_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim 
p[(a + b*ArcSinh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 
+ c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[e, c^2*d] 
 && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
 

rule 6858

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSinh[x/c])^n/x 
^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0 
] && IntegersQ[m, p]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int \frac {\left (x^{2} e +d \right ) \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{x}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*acsch(c*x))*(d + e*x**2)/x, x)
 

Maxima [F]

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

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

Output:

2*b*c^2*d*integrate(1/2*x*log(x)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sq 
rt(c^2*x^2 + 1) + 1), x) - 1/2*b*e*x^2*log(c) - 1/2*b*e*x^2*log(x) + 1/2*a 
*e*x^2 - b*d*log(c)*log(x) - 1/2*b*d*log(x)^2 - 1/4*(2*log(c^2*x^2 + 1)*lo 
g(x) + dilog(-c^2*x^2))*b*d + a*d*log(x) + 1/2*(b*e*x^2 + 2*b*d*log(x))*lo 
g(sqrt(c^2*x^2 + 1) + 1) + 1/4*b*e*(2*sqrt(c^2*x^2 + 1) - log(c^2*x^2 + 1) 
)/c^2 + 1/4*b*e*log(c^2*x^2 + 1)/c^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(acsch(c*x)/x,x)*b*d + 2*int(acsch(c*x)*x,x)*b*e + 2*log(x)*a*d + a* 
e*x**2)/2