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

Optimal result

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

[Out]

(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*pol
ylog(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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6451, 6415, 5560, 4267, 2317, 2438} \[ \int \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {4 b \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}+\frac {2 b^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d}-\frac {2 b^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d} \]

[In]

Int[(a + b*ArcCsch[c + d*x])^2,x]

[Out]

((c + d*x)*(a + b*ArcCsch[c + d*x])^2)/d + (4*b*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d + (2*b
^2*PolyLog[2, -E^ArcCsch[c + d*x]])/d - (2*b^2*PolyLog[2, E^ArcCsch[c + d*x]])/d

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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 5560

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)), x] + Dist[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 6415

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[-c^(-1), Subst[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 6451

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b \text {csch}^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d} \\ & = -\frac {\text {Subst}\left (\int (a+b x)^2 \coth (x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d} \\ & = \frac {(c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d} \\ & = \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 {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d} \\ & = \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 {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d} \\ & = \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} \\ \end{align*}

Mathematica [B] (verified)

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

Time = 0.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.07 \[ \int \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {a^2 c+a^2 d x+2 a b (c+d x) \text {csch}^{-1}(c+d x)+b^2 c \text {csch}^{-1}(c+d x)^2+b^2 d x \text {csch}^{-1}(c+d x)^2-2 b^2 \text {csch}^{-1}(c+d x) \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 b^2 \text {csch}^{-1}(c+d x) \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )+2 a b \log \left (\cosh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )-2 a b \log \left (\sinh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )-2 b^2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )+2 b^2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )}{d} \]

[In]

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

[Out]

(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^(-ArcCsch[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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

(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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*asinh(1/(c + d*x)))^2,x)

[Out]

int((a + b*asinh(1/(c + d*x)))^2, x)