\(\int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx\) [363]

Optimal result

Integrand size = 19, antiderivative size = 143 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {4 \sqrt {1-a^2 x^2}}{a^5 c}+\frac {11 x \sqrt {1-a^2 x^2}}{8 a^4 c}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}+\frac {2 \sqrt {1-a^2 x^2}}{a^5 c (1-a x)}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^5 c}-\frac {27 \arcsin (a x)}{8 a^5 c} \] Output:

4*(-a^2*x^2+1)^(1/2)/a^5/c+11/8*x*(-a^2*x^2+1)^(1/2)/a^4/c+1/4*x^3*(-a^2*x 
^2+1)^(1/2)/a^2/c+2*(-a^2*x^2+1)^(1/2)/a^5/c/(-a*x+1)-2/3*(-a^2*x^2+1)^(3/ 
2)/a^5/c-27/8*arcsin(a*x)/a^5/c
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.57 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {-\frac {\sqrt {1+a x} \left (-128+47 a x+17 a^2 x^2+10 a^3 x^3+6 a^4 x^4\right )}{\sqrt {1-a x}}+162 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{24 a^5 c} \] Input:

Integrate[(E^ArcTanh[a*x]*x^4)/(c - a*c*x),x]
 

Output:

(-((Sqrt[1 + a*x]*(-128 + 47*a*x + 17*a^2*x^2 + 10*a^3*x^3 + 6*a^4*x^4))/S 
qrt[1 - a*x]) + 162*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])/(24*a^5*c)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {6678, 27, 563, 25, 2346, 25, 2346, 25, 2346, 25, 27, 455, 223}

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.

\(\displaystyle \int \frac {x^4 e^{\text {arctanh}(a x)}}{c-a c x} \, dx\)

\(\Big \downarrow \) 6678

\(\displaystyle c \int \frac {x^4 \sqrt {1-a^2 x^2}}{c^2 (1-a x)^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {x^4 \sqrt {1-a^2 x^2}}{(1-a x)^2}dx}{c}\)

\(\Big \downarrow \) 563

\(\displaystyle \frac {\frac {\int -\frac {a^4 x^4+2 a^3 x^3+2 a^2 x^2+2 a x+2}{\sqrt {1-a^2 x^2}}dx}{a^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\int \frac {a^4 x^4+2 a^3 x^3+2 a^2 x^2+2 a x+2}{\sqrt {1-a^2 x^2}}dx}{a^4}}{c}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {-\frac {\int -\frac {8 x^3 a^5+11 x^2 a^4+8 x a^3+8 a^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\int \frac {8 x^3 a^5+11 x^2 a^4+8 x a^3+8 a^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {-\frac {\int -\frac {33 x^2 a^6+40 x a^5+24 a^4}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\int \frac {33 x^2 a^6+40 x a^5+24 a^4}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {-\frac {\int -\frac {a^6 (80 a x+81)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\frac {\int \frac {a^6 (80 a x+81)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\frac {1}{2} a^4 \int \frac {80 a x+81}{\sqrt {1-a^2 x^2}}dx-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\frac {1}{2} a^4 \left (81 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {80 \sqrt {1-a^2 x^2}}{a}\right )-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {2 \sqrt {1-a^2 x^2}}{a^5 (1-a x)}-\frac {\frac {\frac {\frac {1}{2} a^4 \left (\frac {81 \arcsin (a x)}{a}-\frac {80 \sqrt {1-a^2 x^2}}{a}\right )-\frac {33}{2} a^4 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {8}{3} a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}}{a^4}}{c}\)

Input:

Int[(E^ArcTanh[a*x]*x^4)/(c - a*c*x),x]
 

Output:

((2*Sqrt[1 - a^2*x^2])/(a^5*(1 - a*x)) - (-1/4*(a^2*x^3*Sqrt[1 - a^2*x^2]) 
 + ((-8*a^3*x^2*Sqrt[1 - a^2*x^2])/3 + ((-33*a^4*x*Sqrt[1 - a^2*x^2])/2 + 
(a^4*((-80*Sqrt[1 - a^2*x^2])/a + (81*ArcSin[a*x])/a))/2)/(3*a^2))/(4*a^2) 
)/a^4)/c
 

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 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 455

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

rule 563

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* 
b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1)   Int[(1/Sqrt[a 
+ b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n 
 - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 
, 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

rule 6678

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* 
(x_))^(m_.), x_Symbol] :> Simp[c^n   Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - 
 a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 
0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 
, 0]) && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.91

method	result	size
risch	\(-\frac {\left (6 a^{3} x^{3}+16 a^{2} x^{2}+33 a x +80\right ) \left (a^{2} x^{2}-1\right )}{24 a^{5} \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\frac {27 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{4} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}}{c}\)	\(130\)
default	\(-\frac {-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {11 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {11 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {-\frac {2 x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {4 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}}{a}}{c}\)	\(212\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/24*(6*a^3*x^3+16*a^2*x^2+33*a*x+80)*(a^2*x^2-1)/a^5/(-a^2*x^2+1)^(1/2)/ 
c-(27/8/a^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/a^6/(x- 
1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))/c
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {128 \, a x + 162 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (6 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 47 \, a x - 128\right )} \sqrt {-a^{2} x^{2} + 1} - 128}{24 \, {\left (a^{6} c x - a^{5} c\right )}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c),x, algorithm="fricas")
 

Output:

1/24*(128*a*x + 162*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (6* 
a^4*x^4 + 10*a^3*x^3 + 17*a^2*x^2 + 47*a*x - 128)*sqrt(-a^2*x^2 + 1) - 128 
)/(a^6*c*x - a^5*c)
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=- \frac {\int \frac {x^{4}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a*c*x+c),x)
 

Output:

-(Integral(x**4/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + In 
tegral(a*x**5/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{4 \, a^{2} c} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6} c x - a^{5} c} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a^{3} c} + \frac {11 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{4} c} - \frac {27 \, \arcsin \left (a x\right )}{8 \, a^{5} c} + \frac {10 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} c} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c),x, algorithm="maxima")
 

Output:

1/4*sqrt(-a^2*x^2 + 1)*x^3/(a^2*c) - 2*sqrt(-a^2*x^2 + 1)/(a^6*c*x - a^5*c 
) + 2/3*sqrt(-a^2*x^2 + 1)*x^2/(a^3*c) + 11/8*sqrt(-a^2*x^2 + 1)*x/(a^4*c) 
 - 27/8*arcsin(a*x)/(a^5*c) + 10/3*sqrt(-a^2*x^2 + 1)/(a^5*c)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 24.17 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.14 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {10\,\sqrt {1-a^2\,x^2}}{3\,a^5\,c}-\frac {2\,\sqrt {1-a^2\,x^2}}{\sqrt {-a^2}\,\left (a^3\,c\,\sqrt {-a^2}-a^4\,c\,x\,\sqrt {-a^2}\right )}+\frac {11\,x\,\sqrt {1-a^2\,x^2}}{8\,a^4\,c}-\frac {27\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^4\,c\,\sqrt {-a^2}}+\frac {x^3\,\sqrt {1-a^2\,x^2}}{4\,a^2\,c}+\frac {2\,x^2\,\sqrt {1-a^2\,x^2}}{3\,a^3\,c} \] Input:

int((x^4*(a*x + 1))/((1 - a^2*x^2)^(1/2)*(c - a*c*x)),x)
 

Output:

(10*(1 - a^2*x^2)^(1/2))/(3*a^5*c) - (2*(1 - a^2*x^2)^(1/2))/((-a^2)^(1/2) 
*(a^3*c*(-a^2)^(1/2) - a^4*c*x*(-a^2)^(1/2))) + (11*x*(1 - a^2*x^2)^(1/2)) 
/(8*a^4*c) - (27*asinh(x*(-a^2)^(1/2)))/(8*a^4*c*(-a^2)^(1/2)) + (x^3*(1 - 
 a^2*x^2)^(1/2))/(4*a^2*c) + (2*x^2*(1 - a^2*x^2)^(1/2))/(3*a^3*c)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a c x} \, dx=\frac {-81 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-81 \mathit {asin} \left (a x \right ) a x +81 \mathit {asin} \left (a x \right )+6 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+10 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+17 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+47 \sqrt {-a^{2} x^{2}+1}\, a x -162 \sqrt {-a^{2} x^{2}+1}-6 a^{5} x^{5}-16 a^{4} x^{4}-27 a^{3} x^{3}-64 a^{2} x^{2}+47 a x +162}{24 a^{5} c \left (\sqrt {-a^{2} x^{2}+1}+a x -1\right )} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c),x)
 

Output:

( - 81*sqrt( - a**2*x**2 + 1)*asin(a*x) - 81*asin(a*x)*a*x + 81*asin(a*x) 
+ 6*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 10*sqrt( - a**2*x**2 + 1)*a**3*x**3 
 + 17*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 47*sqrt( - a**2*x**2 + 1)*a*x - 1 
62*sqrt( - a**2*x**2 + 1) - 6*a**5*x**5 - 16*a**4*x**4 - 27*a**3*x**3 - 64 
*a**2*x**2 + 47*a*x + 162)/(24*a**5*c*(sqrt( - a**2*x**2 + 1) + a*x - 1))