\(\int \frac {x (a+b \text {csch}^{-1}(c x))}{d+e x^2} \, dx\) [100]

Optimal result

Integrand size = 19, antiderivative size = 449 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{d+e x^2} \, dx=\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )}{2 e} \] Output:

1/2*(a+b*arccsch(c*x))*ln(1-c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1 
/2)-(-c^2*d+e)^(1/2)))/e+1/2*(a+b*arccsch(c*x))*ln(1+c*(-d)^(1/2)*(1/c/x+( 
1+1/c^2/x^2)^(1/2))/(e^(1/2)-(-c^2*d+e)^(1/2)))/e+1/2*(a+b*arccsch(c*x))*l 
n(1-c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2)+(-c^2*d+e)^(1/2)))/e 
+1/2*(a+b*arccsch(c*x))*ln(1+c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^( 
1/2)+(-c^2*d+e)^(1/2)))/e-(a+b*arccsch(c*x))*ln(1-(1/c/x+(1+1/c^2/x^2)^(1/ 
2))^2)/e+1/2*b*polylog(2,-c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2 
)-(-c^2*d+e)^(1/2)))/e+1/2*b*polylog(2,c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^( 
1/2))/(e^(1/2)-(-c^2*d+e)^(1/2)))/e+1/2*b*polylog(2,-c*(-d)^(1/2)*(1/c/x+( 
1+1/c^2/x^2)^(1/2))/(e^(1/2)+(-c^2*d+e)^(1/2)))/e+1/2*b*polylog(2,c*(-d)^( 
1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2)+(-c^2*d+e)^(1/2)))/e-1/2*b*polyl 
og(2,(1/c/x+(1+1/c^2/x^2)^(1/2))^2)/e
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 1103, normalized size of antiderivative = 2.46 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{d+e x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(b*Pi^2 - (4*I)*b*Pi*ArcCsch[c*x] - 8*b*ArcCsch[c*x]^2 + 16*b*ArcSin[Sqrt[ 
1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + 
(2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 16*b*ArcSin[Sqrt[1 - Sqrt[e] 
/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCs 
ch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*b*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[ 
c*x])] + (2*I)*b*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c 
*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) 
+ e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sq 
rt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x] 
)/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^A 
rcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[- 
(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*ArcSin[Sqrt[1 - Sqrt[ 
e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcC 
sch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + 
e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + 
Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*ArcSin[Sqrt[1 - 
 Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E 
^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2* 
d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 + (I*(Sqrt[ 
e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*ArcSin[...
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6858, 6238, 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.

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

\(\Big \downarrow \) 6858

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

\(\Big \downarrow \) 6238

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 6238

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, 
 (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[e, c^ 
2*d] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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