\(\int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx\) [116]

Optimal result

Integrand size = 22, antiderivative size = 362 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx=\frac {b^2}{16 d^3 (1+c x)^2}+\frac {11 b^2}{16 d^3 (1+c x)}-\frac {11 b^2 \text {arctanh}(c x)}{16 d^3}+\frac {b (a+b \text {arctanh}(c x))}{4 d^3 (1+c x)^2}+\frac {5 b (a+b \text {arctanh}(c x))}{4 d^3 (1+c x)}-\frac {5 (a+b \text {arctanh}(c x))^2}{8 d^3}+\frac {(a+b \text {arctanh}(c x))^2}{2 d^3 (1+c x)^2}+\frac {(a+b \text {arctanh}(c x))^2}{d^3 (1+c x)}+\frac {2 (a+b \text {arctanh}(c x))^2 \text {arctanh}\left (1-\frac {2}{1-c x}\right )}{d^3}+\frac {(a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{d^3}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{d^3}+\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )}{d^3}-\frac {b (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-c x}\right )}{2 d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 d^3} \] Output:

1/16*b^2/d^3/(c*x+1)^2+11/16*b^2/d^3/(c*x+1)-11/16*b^2*arctanh(c*x)/d^3+1/ 
4*b*(a+b*arctanh(c*x))/d^3/(c*x+1)^2+5/4*b*(a+b*arctanh(c*x))/d^3/(c*x+1)- 
5/8*(a+b*arctanh(c*x))^2/d^3+1/2*(a+b*arctanh(c*x))^2/d^3/(c*x+1)^2+(a+b*a 
rctanh(c*x))^2/d^3/(c*x+1)-2*(a+b*arctanh(c*x))^2*arctanh(-1+2/(-c*x+1))/d 
^3+(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/d^3-b*(a+b*arctanh(c*x))*polylog(2,1 
-2/(-c*x+1))/d^3+b*(a+b*arctanh(c*x))*polylog(2,-1+2/(-c*x+1))/d^3-b*(a+b* 
arctanh(c*x))*polylog(2,1-2/(c*x+1))/d^3+1/2*b^2*polylog(3,1-2/(-c*x+1))/d 
^3-1/2*b^2*polylog(3,-1+2/(-c*x+1))/d^3-1/2*b^2*polylog(3,1-2/(c*x+1))/d^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.22 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx=\frac {\frac {96 a^2}{(1+c x)^2}+\frac {192 a^2}{1+c x}+192 a^2 \log (c x)-192 a^2 \log (1+c x)+12 a b \left (12 \cosh (2 \text {arctanh}(c x))+\cosh (4 \text {arctanh}(c x))-16 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-12 \sinh (2 \text {arctanh}(c x))+4 \text {arctanh}(c x) \left (6 \cosh (2 \text {arctanh}(c x))+\cosh (4 \text {arctanh}(c x))+8 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-6 \sinh (2 \text {arctanh}(c x))-\sinh (4 \text {arctanh}(c x))\right )-\sinh (4 \text {arctanh}(c x))\right )+b^2 \left (8 i \pi ^3-128 \text {arctanh}(c x)^3+72 \cosh (2 \text {arctanh}(c x))+144 \text {arctanh}(c x) \cosh (2 \text {arctanh}(c x))+144 \text {arctanh}(c x)^2 \cosh (2 \text {arctanh}(c x))+3 \cosh (4 \text {arctanh}(c x))+12 \text {arctanh}(c x) \cosh (4 \text {arctanh}(c x))+24 \text {arctanh}(c x)^2 \cosh (4 \text {arctanh}(c x))+192 \text {arctanh}(c x)^2 \log \left (1-e^{2 \text {arctanh}(c x)}\right )+192 \text {arctanh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(c x)}\right )-96 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(c x)}\right )-72 \sinh (2 \text {arctanh}(c x))-144 \text {arctanh}(c x) \sinh (2 \text {arctanh}(c x))-144 \text {arctanh}(c x)^2 \sinh (2 \text {arctanh}(c x))-3 \sinh (4 \text {arctanh}(c x))-12 \text {arctanh}(c x) \sinh (4 \text {arctanh}(c x))-24 \text {arctanh}(c x)^2 \sinh (4 \text {arctanh}(c x))\right )}{192 d^3} \] Input:

Integrate[(a + b*ArcTanh[c*x])^2/(x*(d + c*d*x)^3),x]
 

Output:

((96*a^2)/(1 + c*x)^2 + (192*a^2)/(1 + c*x) + 192*a^2*Log[c*x] - 192*a^2*L 
og[1 + c*x] + 12*a*b*(12*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] - 16* 
PolyLog[2, E^(-2*ArcTanh[c*x])] - 12*Sinh[2*ArcTanh[c*x]] + 4*ArcTanh[c*x] 
*(6*Cosh[2*ArcTanh[c*x]] + Cosh[4*ArcTanh[c*x]] + 8*Log[1 - E^(-2*ArcTanh[ 
c*x])] - 6*Sinh[2*ArcTanh[c*x]] - Sinh[4*ArcTanh[c*x]]) - Sinh[4*ArcTanh[c 
*x]]) + b^2*((8*I)*Pi^3 - 128*ArcTanh[c*x]^3 + 72*Cosh[2*ArcTanh[c*x]] + 1 
44*ArcTanh[c*x]*Cosh[2*ArcTanh[c*x]] + 144*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c 
*x]] + 3*Cosh[4*ArcTanh[c*x]] + 12*ArcTanh[c*x]*Cosh[4*ArcTanh[c*x]] + 24* 
ArcTanh[c*x]^2*Cosh[4*ArcTanh[c*x]] + 192*ArcTanh[c*x]^2*Log[1 - E^(2*ArcT 
anh[c*x])] + 192*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTanh[c*x])] - 96*PolyLog[ 
3, E^(2*ArcTanh[c*x])] - 72*Sinh[2*ArcTanh[c*x]] - 144*ArcTanh[c*x]*Sinh[2 
*ArcTanh[c*x]] - 144*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]] - 3*Sinh[4*ArcTan 
h[c*x]] - 12*ArcTanh[c*x]*Sinh[4*ArcTanh[c*x]] - 24*ArcTanh[c*x]^2*Sinh[4* 
ArcTanh[c*x]]))/(192*d^3)
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6502, 2009}

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.

\(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x (c d x+d)^3} \, dx\)

\(\Big \downarrow \) 6502

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))}{d^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^3}+\frac {5 b (a+b \text {arctanh}(c x))}{4 d^3 (c x+1)}+\frac {b (a+b \text {arctanh}(c x))}{4 d^3 (c x+1)^2}+\frac {(a+b \text {arctanh}(c x))^2}{d^3 (c x+1)}+\frac {(a+b \text {arctanh}(c x))^2}{2 d^3 (c x+1)^2}-\frac {5 (a+b \text {arctanh}(c x))^2}{8 d^3}+\frac {2 \text {arctanh}\left (1-\frac {2}{1-c x}\right ) (a+b \text {arctanh}(c x))^2}{d^3}+\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{d^3}-\frac {11 b^2 \text {arctanh}(c x)}{16 d^3}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,\frac {2}{1-c x}-1\right )}{2 d^3}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right )}{2 d^3}+\frac {11 b^2}{16 d^3 (c x+1)}+\frac {b^2}{16 d^3 (c x+1)^2}\)

Input:

Int[(a + b*ArcTanh[c*x])^2/(x*(d + c*d*x)^3),x]
 

Output:

b^2/(16*d^3*(1 + c*x)^2) + (11*b^2)/(16*d^3*(1 + c*x)) - (11*b^2*ArcTanh[c 
*x])/(16*d^3) + (b*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)^2) + (5*b*(a + b 
*ArcTanh[c*x]))/(4*d^3*(1 + c*x)) - (5*(a + b*ArcTanh[c*x])^2)/(8*d^3) + ( 
a + b*ArcTanh[c*x])^2/(2*d^3*(1 + c*x)^2) + (a + b*ArcTanh[c*x])^2/(d^3*(1 
 + c*x)) + (2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^3 + ((a + 
 b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^3 - (b*(a + b*ArcTanh[c*x])*PolyLog 
[2, 1 - 2/(1 - c*x)])/d^3 + (b*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - 
 c*x)])/d^3 - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^3 + ( 
b^2*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^3) - (b^2*PolyLog[3, -1 + 2/(1 - c*x 
)])/(2*d^3) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6502

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( 
f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.71 (sec) , antiderivative size = 1321, normalized size of antiderivative = 3.65

method	result	size
parts	\(\text {Expression too large to display}\)	\(1321\)
derivativedivides	\(\text {Expression too large to display}\)	\(1323\)
default	\(\text {Expression too large to display}\)	\(1323\)

Input:

int((a+b*arctanh(c*x))^2/x/(c*d*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

a^2/d^3*(1/2/(c*x+1)^2+1/(c*x+1)-ln(c*x+1)+ln(x))+b^2/d^3*(arctanh(c*x)^2* 
ln(c*x)+1/2/(c*x+1)^2*arctanh(c*x)^2+1/(c*x+1)*arctanh(c*x)^2-arctanh(c*x) 
^2*ln(c*x+1)+2*arctanh(c*x)^2*ln((c*x+1)/(-c^2*x^2+1)^(1/2))-2/3*arctanh(c 
*x)^3+1/8*(4*I*Pi*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2- 
1)))^3-4*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/( 
1-(c*x+1)^2/(c^2*x^2-1)))^2+4*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1))^3-4*I*Pi* 
csgn(I*(c*x+1)^2/(c^2*x^2-1))*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c 
^2*x^2-1)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))-4*I*Pi*csgn(I*(-(c*x+1)^2/(c 
^2*x^2-1)-1))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1))) 
^2+4*I*Pi*csgn(I*(c*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2*csgn(I 
/(1-(c*x+1)^2/(c^2*x^2-1)))-4*I*Pi*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))*csgn( 
I*(-(c*x+1)^2/(c^2*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^2+4*I*Pi*csgn(I*(c 
*x+1)^2/(c^2*x^2-1)/(1-(c*x+1)^2/(c^2*x^2-1)))^3+4*I*Pi*csgn(I*(c*x+1)/(-c 
^2*x^2+1)^(1/2))^2*csgn(I*(c*x+1)^2/(c^2*x^2-1))+4*I*Pi*csgn(I/(1-(c*x+1)^ 
2/(c^2*x^2-1)))*csgn(I*(-(c*x+1)^2/(c^2*x^2-1)-1))*csgn(I*(-(c*x+1)^2/(c^2 
*x^2-1)-1)/(1-(c*x+1)^2/(c^2*x^2-1)))+8*I*Pi*csgn(I*(c*x+1)/(-c^2*x^2+1)^( 
1/2))*csgn(I*(c*x+1)^2/(c^2*x^2-1))^2-5+8*ln(2))*arctanh(c*x)^2-arctanh(c* 
x)^2*ln((c*x+1)^2/(-c^2*x^2+1)-1)+arctanh(c*x)^2*ln(1-(c*x+1)/(-c^2*x^2+1) 
^(1/2))+2*arctanh(c*x)*polylog(2,(c*x+1)/(-c^2*x^2+1)^(1/2))-2*polylog(3,( 
c*x+1)/(-c^2*x^2+1)^(1/2))+arctanh(c*x)^2*ln(1+(c*x+1)/(-c^2*x^2+1)^(1/...
 

Fricas [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{3} x} \,d x } \] Input:

integrate((a+b*arctanh(c*x))^2/x/(c*d*x+d)^3,x, algorithm="fricas")
 

Output:

integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(c^3*d^3*x^4 + 3* 
c^2*d^3*x^3 + 3*c*d^3*x^2 + d^3*x), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx=\frac {\int \frac {a^{2}}{c^{3} x^{4} + 3 c^{2} x^{3} + 3 c x^{2} + x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{4} + 3 c^{2} x^{3} + 3 c x^{2} + x}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{4} + 3 c^{2} x^{3} + 3 c x^{2} + x}\, dx}{d^{3}} \] Input:

integrate((a+b*atanh(c*x))**2/x/(c*d*x+d)**3,x)
 

Output:

(Integral(a**2/(c**3*x**4 + 3*c**2*x**3 + 3*c*x**2 + x), x) + Integral(b** 
2*atanh(c*x)**2/(c**3*x**4 + 3*c**2*x**3 + 3*c*x**2 + x), x) + Integral(2* 
a*b*atanh(c*x)/(c**3*x**4 + 3*c**2*x**3 + 3*c*x**2 + x), x))/d**3
 

Maxima [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{3} x} \,d x } \] Input:

integrate((a+b*arctanh(c*x))^2/x/(c*d*x+d)^3,x, algorithm="maxima")
 

Output:

1/2*a^2*((2*c*x + 3)/(c^2*d^3*x^2 + 2*c*d^3*x + d^3) - 2*log(c*x + 1)/d^3 
+ 2*log(x)/d^3) + 1/8*(2*b^2*c*x + 3*b^2 - 2*(b^2*c^2*x^2 + 2*b^2*c*x + b^ 
2)*log(c*x + 1))*log(-c*x + 1)^2/(c^2*d^3*x^2 + 2*c*d^3*x + d^3) + integra 
te(1/4*((b^2*c*x - b^2)*log(c*x + 1)^2 + 4*(a*b*c*x - a*b)*log(c*x + 1) - 
(2*b^2*c^3*x^3 + 5*b^2*c^2*x^2 - 4*a*b + (4*a*b*c + 3*b^2*c)*x - 2*(b^2*c^ 
4*x^4 + 3*b^2*c^3*x^3 + 3*b^2*c^2*x^2 + b^2)*log(c*x + 1))*log(-c*x + 1))/ 
(c^4*d^3*x^5 + 2*c^3*d^3*x^4 - 2*c*d^3*x^2 - d^3*x), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{3} x} \,d x } \] Input:

integrate((a+b*arctanh(c*x))^2/x/(c*d*x+d)^3,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x) + a)^2/((c*d*x + d)^3*x), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x\,{\left (d+c\,d\,x\right )}^3} \,d x \] Input:

int((a + b*atanh(c*x))^2/(x*(d + c*d*x)^3),x)
 

Output:

int((a + b*atanh(c*x))^2/(x*(d + c*d*x)^3), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x (d+c d x)^3} \, dx =\text {Too large to display} \] Input:

int((a+b*atanh(c*x))^2/x/(c*d*x+d)^3,x)
 

Output:

( - 16*atanh(c*x)**3*b**2*c**2*x**2 - 32*atanh(c*x)**3*b**2*c*x - 16*atanh 
(c*x)**3*b**2 - 48*atanh(c*x)**2*a*b*c**2*x**2 - 96*atanh(c*x)**2*a*b*c*x 
- 48*atanh(c*x)**2*a*b - 36*atanh(c*x)**2*b**2*c**2*x**2 - 24*atanh(c*x)** 
2*b**2*c*x + 60*atanh(c*x)**2*b**2 - 48*atanh(c*x)*a*b*c**2*x**2 + 144*ata 
nh(c*x)*a*b - 36*atanh(c*x)*b**2*c**2*x**2 + 60*atanh(c*x)*b**2 - 384*int( 
atanh(c*x)/(c**4*x**5 + 2*c**3*x**4 - 2*c*x**2 - x),x)*a*b*c**2*x**2 - 768 
*int(atanh(c*x)/(c**4*x**5 + 2*c**3*x**4 - 2*c*x**2 - x),x)*a*b*c*x - 384* 
int(atanh(c*x)/(c**4*x**5 + 2*c**3*x**4 - 2*c*x**2 - x),x)*a*b - 192*int(a 
tanh(c*x)**2/(c**4*x**5 + 2*c**3*x**4 - 2*c*x**2 - x),x)*b**2*c**2*x**2 - 
384*int(atanh(c*x)**2/(c**4*x**5 + 2*c**3*x**4 - 2*c*x**2 - x),x)*b**2*c*x 
 - 192*int(atanh(c*x)**2/(c**4*x**5 + 2*c**3*x**4 - 2*c*x**2 - x),x)*b**2 
+ 12*log(c*x - 1)*a*b*c**2*x**2 + 24*log(c*x - 1)*a*b*c*x + 12*log(c*x - 1 
)*a*b + 3*log(c*x - 1)*b**2*c**2*x**2 + 6*log(c*x - 1)*b**2*c*x + 3*log(c* 
x - 1)*b**2 - 192*log(c*x + 1)*a**2*c**2*x**2 - 384*log(c*x + 1)*a**2*c*x 
- 192*log(c*x + 1)*a**2 - 12*log(c*x + 1)*a*b*c**2*x**2 - 24*log(c*x + 1)* 
a*b*c*x - 12*log(c*x + 1)*a*b - 3*log(c*x + 1)*b**2*c**2*x**2 - 6*log(c*x 
+ 1)*b**2*c*x - 3*log(c*x + 1)*b**2 + 192*log(x)*a**2*c**2*x**2 + 384*log( 
x)*a**2*c*x + 192*log(x)*a**2 - 96*a**2*c**2*x**2 + 192*a**2 - 36*a*b*c**2 
*x**2 + 60*a*b - 21*b**2*c**2*x**2 + 27*b**2)/(192*d**3*(c**2*x**2 + 2*c*x 
 + 1))