\(\int \frac {(a+b \text {csch}^{-1}(c+d x))^2}{(e+f x)^2} \, dx\) [12]

Optimal result

Integrand size = 20, antiderivative size = 448 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\frac {d \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {2 b^2 d \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b^2 d \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 12.51 (sec) , antiderivative size = 1874, normalized size of antiderivative = 4.18 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-(a^2/(f*(e + f*x))) - (2*a*b*(c + d*x)^2*(f + (d*e - c*f)/(c + d*x))^2*(A 
rcCsch[c + d*x]/(f + (d*e)/(c + d*x) - (c*f)/(c + d*x)) - (2*ArcTan[(d*e - 
 c*f - f*Tanh[ArcCsch[c + d*x]/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2) 
*f^2]])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]))/(d*(-(d*e) + c*f)*( 
e + f*x)^2) - (b^2*(c + d*x)^2*(f + (d*e - c*f)/(c + d*x))^2*(ArcCsch[c + 
d*x]^2/((-(d*e) + c*f)*(f + (d*e - c*f)/(c + d*x))) + (2*(((-I)*Pi*ArcTanh 
[(-(d*e) + c*f + f*Tanh[ArcCsch[c + d*x]/2])/Sqrt[f^2 + (d*e - c*f)^2]])/S 
qrt[f^2 + (d*e - c*f)^2] - ((2*I)*ArcCos[(I*f)/(-(d*e) + c*f)]*ArcTan[((d* 
e - (I + c)*f)*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c 
*d*e*f - (1 + c^2)*f^2]] + (Pi - (2*I)*ArcCsch[c + d*x])*ArcTanh[(((-I)*d* 
e + f + I*c*f)*Tan[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c 
*d*e*f - (1 + c^2)*f^2]] + (ArcCos[(I*f)/(-(d*e) + c*f)] + 2*ArcTan[((d*e 
- (I + c)*f)*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d 
*e*f - (1 + c^2)*f^2]] - (2*I)*ArcTanh[(((-I)*d*e + f + I*c*f)*Tan[(Pi + ( 
2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]])* 
Log[-(((-1)^(3/4)*Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2])/(Sqrt[2]*E 
^(ArcCsch[c + d*x]/2)*Sqrt[I*(-(d*e) + c*f)]*Sqrt[f + (d*e - c*f)/(c + d*x 
)]))] + (ArcCos[(I*f)/(-(d*e) + c*f)] - 2*ArcTan[((d*e - (I + c)*f)*Cot[(P 
i + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^ 
2]] + (2*I)*ArcTanh[(((-I)*d*e + f + I*c*f)*Tan[(Pi + (2*I)*ArcCsch[c +...
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6876, 5992, 3042, 4679, 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.

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

\(\Big \downarrow \) 6876

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

\(\Big \downarrow \) 5992

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(d*((a + b*ArcCsch[c + d*x])^2/(f*(d*e - c*f + f*(c + d*x))) - (2*b*((a + 
 b*ArcCsch[c + d*x])^2/(2*b*(d*e - c*f)) - (f*(a + b*ArcCsch[c + d*x])*Log 
[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + 
 c^2)*f^2])])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (f 
*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sq 
rt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c 
*d*e*f + (1 + c^2)*f^2]) - (b*f*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c* 
f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)*Sqrt[d 
^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (b*f*PolyLog[2, -((E^ArcCsch[c + d* 
x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - 
 c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])))/f))
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 5992

Int[Coth[(c_.) + (d_.)*(x_)]*Csch[(c_.) + (d_.)*(x_)]*(Csch[(c_.) + (d_.)*( 
x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(e 
 + f*x)^m)*((a + b*Csch[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[f*(m/(b 
*d*(n + 1)))   Int[(e + f*x)^(m - 1)*(a + b*Csch[c + d*x])^(n + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 6876

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Csch[x]*C 
oth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]], x] /; FreeQ[{a, 
 b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-b^2*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2/(f^2*x + e*f) - a^2/(f^2 
*x + e*f) - integrate(-((b^2*d^2*f*x^2 + 2*b^2*c*d*f*x + (c^2*f + f)*b^2)* 
log(d*x + c)^2 - 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + f)*a*b)*log(d 
*x + c) + 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + f)*a*b - (b^2*d^2*f* 
x^2 + 2*b^2*c*d*f*x + (c^2*f + f)*b^2)*log(d*x + c) + (b^2*c*d*e + (c^2*f 
+ f)*a*b + (a*b*d^2*f + b^2*d^2*f)*x^2 + (2*a*b*c*d*f + (d^2*e + c*d*f)*b^ 
2)*x - (b^2*d^2*f*x^2 + 2*b^2*c*d*f*x + (c^2*f + f)*b^2)*log(d*x + c))*sqr 
t(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1) 
 + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((b^2*d^2*f*x^2 + 2*b^2*c*d*f*x + (c^ 
2*f + f)*b^2)*log(d*x + c)^2 - 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + 
 f)*a*b)*log(d*x + c)))/(d^2*f^3*x^4 + c^2*e^2*f + 2*(d^2*e*f^2 + c*d*f^3) 
*x^3 + e^2*f + (d^2*e^2*f + 4*c*d*e*f^2 + c^2*f^3 + f^3)*x^2 + 2*(c*d*e^2* 
f + c^2*e*f^2 + e*f^2)*x + (d^2*f^3*x^4 + c^2*e^2*f + 2*(d^2*e*f^2 + c*d*f 
^3)*x^3 + e^2*f + (d^2*e^2*f + 4*c*d*e*f^2 + c^2*f^3 + f^3)*x^2 + 2*(c*d*e 
^2*f + c^2*e*f^2 + e*f^2)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(acsch(c + d*x)/(e**2 + 2*e*f*x + f**2*x**2),x)*a*b*e**2 + 2*int(acs 
ch(c + d*x)/(e**2 + 2*e*f*x + f**2*x**2),x)*a*b*e*f*x + int(acsch(c + d*x) 
**2/(e**2 + 2*e*f*x + f**2*x**2),x)*b**2*e**2 + int(acsch(c + d*x)**2/(e** 
2 + 2*e*f*x + f**2*x**2),x)*b**2*e*f*x + a**2*x)/(e*(e + f*x))