\(\int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx\) [25]

Optimal result

Integrand size = 21, antiderivative size = 110 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {b^2 e (c+d x)^2}{4 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{2 d}-\frac {e (a+b \text {arccosh}(c+d x))^2}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d} \] Output:

1/4*b^2*e*(d*x+c)^2/d-1/2*b*e*(d*x+c-1)^(1/2)*(d*x+c)*(d*x+c+1)^(1/2)*(a+b 
*arccosh(d*x+c))/d-1/4*e*(a+b*arccosh(d*x+c))^2/d+1/2*e*(d*x+c)^2*(a+b*arc 
cosh(d*x+c))^2/d
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.52 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e \left ((c+d x) \left (2 a^2 (c+d x)+b^2 (c+d x)-2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right )-2 b (c+d x) \left (-2 a (c+d x)+b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+b^2 \left (-1+2 c^2+4 c d x+2 d^2 x^2\right ) \text {arccosh}(c+d x)^2-2 a b \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{4 d} \] Input:

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

Output:

(e*((c + d*x)*(2*a^2*(c + d*x) + b^2*(c + d*x) - 2*a*b*Sqrt[-1 + c + d*x]* 
Sqrt[1 + c + d*x]) - 2*b*(c + d*x)*(-2*a*(c + d*x) + b*Sqrt[-1 + c + d*x]* 
Sqrt[1 + c + d*x])*ArcCosh[c + d*x] + b^2*(-1 + 2*c^2 + 4*c*d*x + 2*d^2*x^ 
2)*ArcCosh[c + d*x]^2 - 2*a*b*Log[c + d*x + Sqrt[-1 + c + d*x]*Sqrt[1 + c 
+ d*x]]))/(4*d)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6411, 27, 6298, 6354, 15, 6308}

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.

\(\displaystyle \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int e (c+d x) (a+b \text {arccosh}(c+d x))^2d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \int (c+d x) (a+b \text {arccosh}(c+d x))^2d(c+d x)}{d}\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6354

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6308

\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \left (\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))+\frac {(a+b \text {arccosh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )\right )}{d}\)

Input:

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

Output:

(e*(((c + d*x)^2*(a + b*ArcCosh[c + d*x])^2)/2 - b*(-1/4*(b*(c + d*x)^2) + 
 (Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])) 
/2 + (a + b*ArcCosh[c + d*x])^2/(4*b))))/d
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 6298

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

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
 

rule 6354

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 
1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( 
m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1)))   Int[(f*x)^(m 
 - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f 
*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/( 
-1 + c*x)^p]   Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*( 
a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, 
p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N 
eQ[m + 2*p + 1, 0]
 

rule 6411

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{2} \left (d x +c \right )^{2}}{2}+e \,b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(157\)
default	\(\frac {\frac {e \,a^{2} \left (d x +c \right )^{2}}{2}+e \,b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(157\)
parts	\(e \,a^{2} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{2} \left (\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right ) \operatorname {arccosh}\left (d x +c \right )}{4}+\frac {\cosh \left (2 \,\operatorname {arccosh}\left (d x +c \right )\right )}{8}\right )}{d}+\frac {2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(161\)
orering	\(\frac {\left (7 d^{4} x^{4}+28 c \,d^{3} x^{3}+41 c^{2} d^{2} x^{2}+26 c^{3} d x +6 c^{4}-6 d^{2} x^{2}-12 c d x -4 c^{2}\right ) \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}{8 d \left (d x +c \right )^{3}}-\frac {\left (3 d^{4} x^{4}+12 c \,d^{3} x^{3}+17 c^{2} d^{2} x^{2}+10 c^{3} d x +2 c^{4}-4 d^{2} x^{2}-8 c d x -2 c^{2}\right ) \left (d e \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}+\frac {2 \left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b d}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}\right )}{8 \left (d x +c \right )^{2} d^{2}}+\frac {x \left (d x +2 c \right ) \left (d x +c -1\right ) \left (d x +c +1\right ) \left (\frac {4 d^{2} e \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b}{\sqrt {d x +c -1}\, \sqrt {d x +c +1}}+\frac {2 \left (d e x +c e \right ) b^{2} d^{2}}{\left (d x +c -1\right ) \left (d x +c +1\right )}-\frac {\left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b \,d^{2}}{\left (d x +c -1\right )^{\frac {3}{2}} \sqrt {d x +c +1}}-\frac {\left (d e x +c e \right ) \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right ) b \,d^{2}}{\sqrt {d x +c -1}\, \left (d x +c +1\right )^{\frac {3}{2}}}\right )}{8 d^{2} \left (d x +c \right )}\)	\(396\)

Input:

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

Output:

1/d*(1/2*e*a^2*(d*x+c)^2+e*b^2*(1/4*cosh(2*arccosh(d*x+c))*arccosh(d*x+c)^ 
2-1/4*sinh(2*arccosh(d*x+c))*arccosh(d*x+c)+1/8*cosh(2*arccosh(d*x+c)))+2* 
e*a*b*(1/2*(d*x+c)^2*arccosh(d*x+c)-1/4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(( 
d*x+c)*((d*x+c)^2-1)^(1/2)+ln(d*x+c+((d*x+c)^2-1)^(1/2)))/((d*x+c)^2-1)^(1 
/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (98) = 196\).

Time = 0.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.12 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c d e x + {\left (2 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} c d e x + {\left (2 \, b^{2} c^{2} - b^{2}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 2 \, {\left (2 \, a b d^{2} e x^{2} + 4 \, a b c d e x + {\left (2 \, a b c^{2} - a b\right )} e - {\left (b^{2} d e x + b^{2} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (a b d e x + a b c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

e*(Integral(a**2*c, x) + Integral(a**2*d*x, x) + Integral(b**2*c*acosh(c + 
 d*x)**2, x) + Integral(2*a*b*c*acosh(c + d*x), x) + Integral(b**2*d*x*aco 
sh(c + d*x)**2, x) + Integral(2*a*b*d*x*acosh(c + d*x), x))
 

Maxima [F]

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

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

Output:

1/2*a^2*d*e*x^2 + 1/2*(2*x^2*arccosh(d*x + c) - d*(3*c^2*log(2*d^2*x + 2*c 
*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + 
 c^2 - 1)*x/d^2 - (c^2 - 1)*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x 
 + c^2 - 1)*d)/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d^3))*a*b*d*e + 
 a^2*c*e*x + 2*((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a*b*c* 
e/d + 1/2*(b^2*d*e*x^2 + 2*b^2*c*e*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x 
 + c - 1) + c)^2 - integrate((b^2*d^4*e*x^4 + 4*b^2*c*d^3*e*x^3 + (5*c^2*d 
^2*e - d^2*e)*b^2*x^2 + 2*(c^3*d*e - c*d*e)*b^2*x + (b^2*d^3*e*x^3 + 3*b^2 
*c*d^2*e*x^2 + 2*b^2*c^2*d*e*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d 
*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/(d^3*x^3 + 3*c*d^2*x^2 + c^3 
 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3* 
c^2*d - d)*x - c), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e \left (4 \mathit {acosh} \left (d x +c \right ) a b \,c^{2}+4 \mathit {acosh} \left (d x +c \right ) a b c d x +2 \mathit {acosh} \left (d x +c \right ) a b \,d^{2} x^{2}+3 \sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b c -\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}\, a b d x -4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, a b c +2 \left (\int \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} c d +2 \left (\int \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) b^{2} d^{2}-2 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a b \,c^{2}-\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}-1}+c +d x \right ) a b +2 a^{2} c d x +a^{2} d^{2} x^{2}\right )}{2 d} \] Input:

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

Output:

(e*(4*acosh(c + d*x)*a*b*c**2 + 4*acosh(c + d*x)*a*b*c*d*x + 2*acosh(c + d 
*x)*a*b*d**2*x**2 + 3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a*b*c - sqrt(c* 
*2 + 2*c*d*x + d**2*x**2 - 1)*a*b*d*x - 4*sqrt(c + d*x + 1)*sqrt(c + d*x - 
 1)*a*b*c + 2*int(acosh(c + d*x)**2,x)*b**2*c*d + 2*int(acosh(c + d*x)**2* 
x,x)*b**2*d**2 - 2*log(sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a*b 
*c**2 - log(sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a*b + 2*a**2*c 
*d*x + a**2*d**2*x**2))/(2*d)