3.2.1 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x (d+e x^2)} \, dx\) [101]

3.2.1.1 Optimal result

Integrand size = 21, antiderivative size = 425 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )} \, dx=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b d}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d} \]

output

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

3.2.1.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 1141, normalized size of antiderivative = 2.68 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )} \, dx =\text {Too large to display} \]

input

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

output

-1/8*(b*Pi^2 - (4*I)*b*Pi*ArcCsch[c*x] - 12*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 - S 
qrt[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]] - 8*b*ArcCsch[c*x]*Log[1 - E^(-2*Ar 
cCsch[c*x])] + (2*I)*b*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^Arc 
Csch[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^ArcCsc 
h[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[Sq 
rt[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*A...
 

3.2.1.3 Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.

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

\(\Big \downarrow \) 6858

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

\(\Big \downarrow \) 6238

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.2.1.3.1 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]
 

3.2.1.4 Maple [F]

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

input

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

output

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

3.2.1.5 Fricas [F]

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

input

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

output

integral((b*arccsch(c*x) + a)/(e*x^3 + d*x), x)
 

3.2.1.6 Sympy [F]

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

input

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

output

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

3.2.1.7 Maxima [F]

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

input

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

output

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

3.2.1.8 Giac [F]

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

input

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

output

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

3.2.1.9 Mupad [F(-1)]

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

input

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

output

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