\(\int (a+b \text {csch}^{-1}(c+d x))^2 \, dx\) [10]

Optimal result

Integrand size = 12, antiderivative size = 85 \[ \int \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}+\frac {4 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d}+\frac {2 b^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d}-\frac {2 b^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d} \] Output:

(d*x+c)*(a+b*arccsch(d*x+c))^2/d+4*b*(a+b*arccsch(d*x+c))*arctanh(1/(d*x+c 
)+(1+1/(d*x+c)^2)^(1/2))/d+2*b^2*polylog(2,-1/(d*x+c)-(1+1/(d*x+c)^2)^(1/2 
))/d-2*b^2*polylog(2,1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(176\) vs. \(2(85)=170\).

Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.07 \[ \int \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {a^2 c+a^2 d x+2 a b (c+d x) \text {csch}^{-1}(c+d x)+b^2 c \text {csch}^{-1}(c+d x)^2+b^2 d x \text {csch}^{-1}(c+d x)^2-2 b^2 \text {csch}^{-1}(c+d x) \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 b^2 \text {csch}^{-1}(c+d x) \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )+2 a b \log \left (\cosh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )-2 a b \log \left (\sinh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )-2 b^2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )+2 b^2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )}{d} \] Input:

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

Output:

(a^2*c + a^2*d*x + 2*a*b*(c + d*x)*ArcCsch[c + d*x] + b^2*c*ArcCsch[c + d* 
x]^2 + b^2*d*x*ArcCsch[c + d*x]^2 - 2*b^2*ArcCsch[c + d*x]*Log[1 - E^(-Arc 
Csch[c + d*x])] + 2*b^2*ArcCsch[c + d*x]*Log[1 + E^(-ArcCsch[c + d*x])] + 
2*a*b*Log[Cosh[ArcCsch[c + d*x]/2]] - 2*a*b*Log[Sinh[ArcCsch[c + d*x]/2]] 
- 2*b^2*PolyLog[2, -E^(-ArcCsch[c + d*x])] + 2*b^2*PolyLog[2, E^(-ArcCsch[ 
c + d*x])])/d
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6870, 6834, 5975, 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.

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

\(\Big \downarrow \) 6870

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

\(\Big \downarrow \) 6834

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

\(\Big \downarrow \) 5975

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((-((c + d*x)*(a + b*ArcCsch[c + d*x])^2) + (2*I)*b*((2*I)*(a + b*ArcCsch 
[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]] - I*b*PolyLog[2, E^ArcCsch[c + d*x] 
] + I*b*PolyLog[2, -c - d*x]))/d)
 

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 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 5975

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)) 
, x] + Simp[d*(m/(b*n))   Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /; 
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
 

rule 6834

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-c^(-1)   S 
ubst[Int[(a + b*x)^n*Csch[x]*Coth[x], x], x, ArcCsch[c*x]], x] /; FreeQ[{a, 
 b, c, n}, x] && IGtQ[n, 0]
 

rule 6870

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[1/d 
   Subst[Int[(a + b*ArcCsch[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d 
}, x] && IGtQ[p, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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