\(\int (d+e x)^2 (a+b \text {csch}^{-1}(c x)) \, dx\) [45]

Optimal result

Integrand size = 16, antiderivative size = 122 \[ \int (d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b d e \sqrt {1+\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \text {csch}^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2-e^2\right ) \text {arctanh}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{6 c^3} \] Output:

b*d*e*(1+1/c^2/x^2)^(1/2)*x/c+1/6*b*e^2*(1+1/c^2/x^2)^(1/2)*x^2/c-1/3*b*d^ 
3*arccsch(c*x)/e+1/3*(e*x+d)^3*(a+b*arccsch(c*x))/e+1/6*b*(6*c^2*d^2-e^2)* 
arctanh((1+1/c^2/x^2)^(1/2))/c^3
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {c^2 x \left (b e \sqrt {1+\frac {1}{c^2 x^2}} (6 d+e x)+2 a c \left (3 d^2+3 d e x+e^2 x^2\right )\right )+2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {csch}^{-1}(c x)+b \left (6 c^2 d^2-e^2\right ) \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{6 c^3} \] Input:

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

Output:

(c^2*x*(b*e*Sqrt[1 + 1/(c^2*x^2)]*(6*d + e*x) + 2*a*c*(3*d^2 + 3*d*e*x + e 
^2*x^2)) + 2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcCsch[c*x] + b*(6*c^2*d 
^2 - e^2)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(6*c^3)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6844, 1892, 1803, 540, 25, 2338, 25, 538, 222, 243, 73, 221}

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.

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

\(\Big \downarrow \) 6844

\(\displaystyle \frac {b \int \frac {(d+e x)^3}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{3 c e}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\)

\(\Big \downarrow \) 1892

\(\displaystyle \frac {b \int \frac {\left (\frac {d}{x}+e\right )^3 x}{\sqrt {1+\frac {1}{c^2 x^2}}}dx}{3 c e}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}\)

\(\Big \downarrow \) 1803

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \int \frac {\left (\frac {d}{x}+e\right )^3 x^3}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}}{3 c e}\)

\(\Big \downarrow \) 540

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (-\frac {1}{2} \int -\frac {\left (\frac {2 d^3}{x^2}+6 e^2 d+\frac {e \left (6 d^2-\frac {e^2}{c^2}\right )}{x}\right ) x^2}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (\frac {1}{2} \int \frac {\left (\frac {2 d^3}{x^2}+6 e^2 d+\frac {e \left (6 d^2-\frac {e^2}{c^2}\right )}{x}\right ) x^2}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (\frac {1}{2} \left (-\int -\frac {\left (\frac {2 d^3}{x}+e \left (6 d^2-\frac {e^2}{c^2}\right )\right ) x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-6 d e^2 x \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (\frac {1}{2} \left (\int \frac {\left (\frac {2 d^3}{x}+e \left (6 d^2-\frac {e^2}{c^2}\right )\right ) x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-6 d e^2 x \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (\frac {1}{2} \left (2 d^3 \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}+e \left (6 d^2-\frac {e^2}{c^2}\right ) \int \frac {x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-6 d e^2 x \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (\frac {1}{2} \left (e \left (6 d^2-\frac {e^2}{c^2}\right ) \int \frac {x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}+2 c d^3 \text {arcsinh}\left (\frac {1}{c x}\right )-6 d e^2 x \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (\frac {1}{2} \left (\frac {1}{2} e \left (6 d^2-\frac {e^2}{c^2}\right ) \int \frac {x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x^2}+2 c d^3 \text {arcsinh}\left (\frac {1}{c x}\right )-6 d e^2 x \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (\frac {1}{2} \left (c^2 e \left (6 d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{c^2 \sqrt {1+\frac {1}{c^2 x^2}}-c^2}d\sqrt {1+\frac {1}{c^2 x^2}}+2 c d^3 \text {arcsinh}\left (\frac {1}{c x}\right )-6 d e^2 x \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \left (\frac {1}{2} \left (2 c d^3 \text {arcsinh}\left (\frac {1}{c x}\right )-e \text {arctanh}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right ) \left (6 d^2-\frac {e^2}{c^2}\right )-6 d e^2 x \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{2} e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{3 c e}\)

Input:

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

Output:

((d + e*x)^3*(a + b*ArcCsch[c*x]))/(3*e) - (b*(-1/2*(e^3*Sqrt[1 + 1/(c^2*x 
^2)]*x^2) + (-6*d*e^2*Sqrt[1 + 1/(c^2*x^2)]*x + 2*c*d^3*ArcSinh[1/(c*x)] - 
 e*(6*d^2 - e^2/c^2)*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/2))/(3*c*e)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1892

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^ 
(p_.), x_Symbol] :> Int[x^(m + mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; F 
reeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n 
2] ||  !IntegerQ[p])
 

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

rule 6844

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo 
l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsch[c*x])/(e*(m + 1))), x] + Simp[ 
b/(c*e*(m + 1))   Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] 
 /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.52

method	result	size
parts	\(\frac {a \left (e x +d \right )^{3}}{3 e}+\frac {b \left (\frac {c \,e^{2} \operatorname {arccsch}\left (c x \right ) x^{3}}{3}+c e \,\operatorname {arccsch}\left (c x \right ) x^{2} d +\operatorname {arccsch}\left (c x \right ) c x \,d^{2}+\frac {c \,\operatorname {arccsch}\left (c x \right ) d^{3}}{3 e}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 c^{3} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \,\operatorname {arcsinh}\left (c x \right )+6 c d \,e^{2} \sqrt {c^{2} x^{2}+1}+e^{3} c x \sqrt {c^{2} x^{2}+1}-e^{3} \operatorname {arcsinh}\left (c x \right )\right )}{6 c^{3} e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x}\right )}{c}\)	\(186\)
derivativedivides	\(\frac {\frac {a \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arccsch}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arccsch}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arccsch}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 c^{3} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \,\operatorname {arcsinh}\left (c x \right )+6 c d \,e^{2} \sqrt {c^{2} x^{2}+1}+e^{3} c x \sqrt {c^{2} x^{2}+1}-e^{3} \operatorname {arcsinh}\left (c x \right )\right )}{6 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c}\)	\(204\)
default	\(\frac {\frac {a \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arccsch}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arccsch}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arccsch}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 c^{3} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \,\operatorname {arcsinh}\left (c x \right )+6 c d \,e^{2} \sqrt {c^{2} x^{2}+1}+e^{3} c x \sqrt {c^{2} x^{2}+1}-e^{3} \operatorname {arcsinh}\left (c x \right )\right )}{6 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c}\)	\(204\)

Input:

int((e*x+d)^2*(a+b*arccsch(c*x)),x,method=_RETURNVERBOSE)
 

Output:

1/3*a*(e*x+d)^3/e+b/c*(1/3*c*e^2*arccsch(c*x)*x^3+c*e*arccsch(c*x)*x^2*d+a 
rccsch(c*x)*c*x*d^2+1/3*c/e*arccsch(c*x)*d^3+1/6/c^3/e*(c^2*x^2+1)^(1/2)*( 
-2*c^3*d^3*arctanh(1/(c^2*x^2+1)^(1/2))+6*c^2*d^2*e*arcsinh(c*x)+6*c*d*e^2 
*(c^2*x^2+1)^(1/2)+e^3*c*x*(c^2*x^2+1)^(1/2)-e^3*arcsinh(c*x))/((c^2*x^2+1 
)/c^2/x^2)^(1/2)/x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (108) = 216\).

Time = 0.14 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.69 \[ \int (d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {2 \, a c^{3} e^{2} x^{3} + 6 \, a c^{3} d e x^{2} + 6 \, a c^{3} d^{2} x + 2 \, {\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (6 \, b c^{2} d^{2} - b e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 2 \, {\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{3} e^{2} x^{3} + 3 \, b c^{3} d e x^{2} + 3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} - 3 \, b c^{3} d e - b c^{3} e^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (b c^{2} e^{2} x^{2} + 6 \, b c^{2} d e x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{6 \, c^{3}} \] Input:

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

Output:

1/6*(2*a*c^3*e^2*x^3 + 6*a*c^3*d*e*x^2 + 6*a*c^3*d^2*x + 2*(3*b*c^3*d^2 + 
3*b*c^3*d*e + b*c^3*e^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) 
- (6*b*c^2*d^2 - b*e^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x) - 2*( 
3*b*c^3*d^2 + 3*b*c^3*d*e + b*c^3*e^2)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2 
)) - c*x - 1) + 2*(b*c^3*e^2*x^3 + 3*b*c^3*d*e*x^2 + 3*b*c^3*d^2*x - 3*b*c 
^3*d^2 - 3*b*c^3*d*e - b*c^3*e^2)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 
 1)/(c*x)) + (b*c^2*e^2*x^2 + 6*b*c^2*d*e*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) 
)/c^3
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.57 \[ \int (d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d e + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d^{2}}{2 \, c} \] Input:

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

Output:

1/3*a*e^2*x^3 + a*d*e*x^2 + (x^2*arccsch(c*x) + x*sqrt(1/(c^2*x^2) + 1)/c) 
*b*d*e + 1/12*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x^2) + 1)/(c^2*(1/(c^2* 
x^2) + 1) - c^2) - log(sqrt(1/(c^2*x^2) + 1) + 1)/c^2 + log(sqrt(1/(c^2*x^ 
2) + 1) - 1)/c^2)/c)*b*e^2 + a*d^2*x + 1/2*(2*c*x*arccsch(c*x) + log(sqrt( 
1/(c^2*x^2) + 1) + 1) - log(sqrt(1/(c^2*x^2) + 1) - 1))*b*d^2/c
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\left (\int \mathit {acsch} \left (c x \right )d x \right ) b \,d^{2}+\left (\int \mathit {acsch} \left (c x \right ) x^{2}d x \right ) b \,e^{2}+2 \left (\int \mathit {acsch} \left (c x \right ) x d x \right ) b d e +a \,d^{2} x +a d e \,x^{2}+\frac {a \,e^{2} x^{3}}{3} \] Input:

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

Output:

(3*int(acsch(c*x),x)*b*d**2 + 3*int(acsch(c*x)*x**2,x)*b*e**2 + 6*int(acsc 
h(c*x)*x,x)*b*d*e + 3*a*d**2*x + 3*a*d*e*x**2 + a*e**2*x**3)/3