\(\int (d+e x)^2 (a+b \text {sech}^{-1}(c x)) \, dx\) [75]

Optimal result

Integrand size = 16, antiderivative size = 201 \[ \int (d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b d e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2}-\frac {b e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^2}+\frac {(d+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{6 c^3}-\frac {b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{3 e} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73 \[ \int (d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {-b c e \sqrt {\frac {1-c x}{1+c x}} (1+c x) (6 d+e x)+2 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \text {sech}^{-1}(c x)+i b \left (6 c^2 d^2+e^2\right ) \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{6 c^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.74, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6842, 541, 25, 2340, 25, 27, 538, 223, 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.

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

\(\Big \downarrow \) 6842

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

\(\Big \downarrow \) 541

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {-2 d^3 \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}+\frac {e \arcsin (c x) \left (6 c^2 d^2+e^2\right )}{c}-6 d e^2 \sqrt {1-c^2 x^2}}{2 c^2}-\frac {e^3 x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{3 e}+\frac {(d+e x)^3 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e}\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

rule 6842

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.99

method	result	size
parts	\(\frac {a \left (e x +d \right )^{3}}{3 e}+\frac {b \left (\frac {c \,e^{2} \operatorname {arcsech}\left (c x \right ) x^{3}}{3}+c e \,\operatorname {arcsech}\left (c x \right ) x^{2} d +\operatorname {arcsech}\left (c x \right ) c x \,d^{2}+\frac {c \,\operatorname {arcsech}\left (c x \right ) d^{3}}{3 e}+\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (-2 c^{3} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \arcsin \left (c x \right )-6 c d \,e^{2} \sqrt {-c^{2} x^{2}+1}-e^{3} c x \sqrt {-c^{2} x^{2}+1}+e^{3} \arcsin \left (c x \right )\right )}{6 c e \sqrt {-c^{2} x^{2}+1}}\right )}{c}\)	\(199\)
derivativedivides	\(\frac {\frac {a \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arcsech}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsech}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsech}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsech}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-2 c^{3} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \arcsin \left (c x \right )-6 c d \,e^{2} \sqrt {-c^{2} x^{2}+1}-e^{3} c x \sqrt {-c^{2} x^{2}+1}+e^{3} \arcsin \left (c x \right )\right )}{6 e \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c}\)	\(215\)
default	\(\frac {\frac {a \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arcsech}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsech}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsech}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsech}\left (c x \right ) c^{3} x^{3}}{3}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-2 c^{3} d^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \arcsin \left (c x \right )-6 c d \,e^{2} \sqrt {-c^{2} x^{2}+1}-e^{3} c x \sqrt {-c^{2} x^{2}+1}+e^{3} \arcsin \left (c x \right )\right )}{6 e \sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}}{c}\)	\(215\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (107) = 214\).

Time = 0.16 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.39 \[ \int (d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {2 \, a c^{3} e^{2} x^{3} + 6 \, a c^{3} d e x^{2} + 6 \, a c^{3} d^{2} x - 2 \, {\left (6 \, b c^{2} d^{2} + b e^{2}\right )} \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 2 \, {\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 2 \, {\left (b c^{3} e^{2} x^{3} + 3 \, b c^{3} d e x^{2} + 3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} - 3 \, b c^{3} d e - b c^{3} e^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} e^{2} x^{2} + 6 \, b c^{2} d e x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{6 \, c^{3}} \] Input:

integrate((e*x+d)^2*(a+b*arcsech(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*(6*b*c^2*d^2 + 
b*e^2)*arctan((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(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)) - 
1)/x) + 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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.76 \[ \int (d+e x)^2 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + {\left (x^{2} \operatorname {arsech}\left (c x\right ) - \frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c}\right )} b d e + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d^{2}}{c} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*int(asech(c*x),x)*b*d**2 + 3*int(asech(c*x)*x**2,x)*b*e**2 + 6*int(asec 
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