\(\int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a^2 c x^2)^2} \, dx\) [919]

Optimal result

Integrand size = 23, antiderivative size = 138 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {1+a x}{3 a^6 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{a^6 c^2 \sqrt {1-a^2 x^2}}-\frac {7 x}{3 a^5 c^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^6 c^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^5 c^2}+\frac {5 \arcsin (a x)}{2 a^6 c^2} \] Output:

1/3*(a*x+1)/a^6/c^2/(-a^2*x^2+1)^(3/2)-2/a^6/c^2/(-a^2*x^2+1)^(1/2)-7/3*x/ 
a^5/c^2/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/a^6/c^2-1/2*x*(-a^2*x^2+1)^( 
1/2)/a^5/c^2+5/2*arcsin(a*x)/a^6/c^2
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {16-a x-23 a^2 x^2+3 a^3 x^3+3 a^4 x^4+15 (-1+a x) \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^6 c^2 (-1+a x) \sqrt {1-a^2 x^2}} \] Input:

Integrate[(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^2,x]
 

Output:

(16 - a*x - 23*a^2*x^2 + 3*a^3*x^3 + 3*a^4*x^4 + 15*(-1 + a*x)*Sqrt[1 - a^ 
2*x^2]*ArcSin[a*x])/(6*a^6*c^2*(-1 + a*x)*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6698, 529, 2345, 27, 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.

\(\displaystyle \int \frac {x^5 e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\)

\(\Big \downarrow \) 6698

\(\displaystyle \frac {\int \frac {x^5 (a x+1)}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\)

\(\Big \downarrow \) 529

\(\displaystyle \frac {\frac {a x+1}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {\frac {3 x^4}{a}+\frac {3 x^3}{a^2}+\frac {3 x^2}{a^3}+\frac {3 x}{a^4}+\frac {1}{a^5}}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\frac {1}{3} \left (\int \frac {3 \left (\frac {x^2}{a^3}+\frac {x}{a^4}+\frac {2}{a^5}\right )}{\sqrt {1-a^2 x^2}}dx-\frac {7 a x+6}{a^6 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \left (3 \int \frac {\frac {x^2}{a^3}+\frac {x}{a^4}+\frac {2}{a^5}}{\sqrt {1-a^2 x^2}}dx-\frac {7 a x+6}{a^6 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {1}{3} \left (3 \left (-\frac {\int -\frac {2 a x+5}{a^3 \sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^5}\right )-\frac {7 a x+6}{a^6 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {\int \frac {2 a x+5}{a^3 \sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^5}\right )-\frac {7 a x+6}{a^6 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {\int \frac {2 a x+5}{\sqrt {1-a^2 x^2}}dx}{2 a^5}-\frac {x \sqrt {1-a^2 x^2}}{2 a^5}\right )-\frac {7 a x+6}{a^6 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{a}}{2 a^5}-\frac {x \sqrt {1-a^2 x^2}}{2 a^5}\right )-\frac {7 a x+6}{a^6 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {a x+1}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{3} \left (3 \left (\frac {\frac {5 \arcsin (a x)}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a}}{2 a^5}-\frac {x \sqrt {1-a^2 x^2}}{2 a^5}\right )-\frac {7 a x+6}{a^6 \sqrt {1-a^2 x^2}}\right )}{c^2}\)

Input:

Int[(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^2,x]
 

Output:

((1 + a*x)/(3*a^6*(1 - a^2*x^2)^(3/2)) + (-((6 + 7*a*x)/(a^6*Sqrt[1 - a^2* 
x^2])) + 3*(-1/2*(x*Sqrt[1 - a^2*x^2])/a^5 + ((-2*Sqrt[1 - a^2*x^2])/a + ( 
5*ArcSin[a*x])/a)/(2*a^5)))/3)/c^2
 

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 529

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

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

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 6698

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

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.37

method	result	size
risch	\(\frac {\left (a x +2\right ) \left (a^{2} x^{2}-1\right )}{2 a^{6} \sqrt {-a^{2} x^{2}+1}\, c^{2}}-\frac {-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{6 a^{8} \left (x -\frac {1}{a}\right )^{2}}-\frac {25 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 a^{7} \left (x -\frac {1}{a}\right )}-\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{5} \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{7} \left (x +\frac {1}{a}\right )}}{c^{2}}\)	\(189\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{6}}+\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{3}}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{5} \sqrt {a^{2}}}+\frac {9 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{7} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{7} \left (x +\frac {1}{a}\right )}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{2 a^{7}}}{c^{2}}\)	\(267\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x,method=_RETURNVERBOS 
E)
 

Output:

1/2*(a*x+2)*(a^2*x^2-1)/a^6/(-a^2*x^2+1)^(1/2)/c^2-(-1/6/a^8/(x-1/a)^2*(-( 
x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-25/12/a^7/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1 
/a))^(1/2)-5/2/a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/ 
4/a^7/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))/c^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {16 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 16 \, a x + 30 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 23 \, a^{2} x^{2} - a x + 16\right )} \sqrt {-a^{2} x^{2} + 1} + 16}{6 \, {\left (a^{9} c^{2} x^{3} - a^{8} c^{2} x^{2} - a^{7} c^{2} x + a^{6} c^{2}\right )}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x, algorithm="fr 
icas")
 

Output:

-1/6*(16*a^3*x^3 - 16*a^2*x^2 - 16*a*x + 30*(a^3*x^3 - a^2*x^2 - a*x + 1)* 
arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^4*x^4 + 3*a^3*x^3 - 23*a^2*x 
^2 - a*x + 16)*sqrt(-a^2*x^2 + 1) + 16)/(a^9*c^2*x^3 - a^8*c^2*x^2 - a^7*c 
^2*x + a^6*c^2)
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{6}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a**2*c*x**2+c)**2,x)
 

Output:

(Integral(x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x* 
*2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**6/(a**4*x**4*sqrt(-a** 
2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) 
)/c**2
 

Maxima [F]

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

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x, algorithm="ma 
xima")
 

Output:

a*integrate(x^6/((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*sqrt(a*x + 1)*sqrt(-a 
*x + 1)), x) - 1/3*(3*sqrt(-a^2*x^2 + 1)/c^2 - (6*a^2*x^2 - 5)/((-a^2*x^2 
+ 1)^(3/2)*c^2))/a^6
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x, algorithm="gi 
ac")
 

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 = 0.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\sqrt {1-a^2\,x^2}}{6\,\left (a^8\,c^2\,x^2-2\,a^7\,c^2\,x+a^6\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{4\,\left (a^4\,c^2\,\sqrt {-a^2}+a^5\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {25\,\sqrt {1-a^2\,x^2}}{12\,\left (a^4\,c^2\,\sqrt {-a^2}-a^5\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^6\,c^2}-\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^5\,c^2}+\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^5\,c^2\,\sqrt {-a^2}} \] Input:

int((x^5*(a*x + 1))/((c - a^2*c*x^2)^2*(1 - a^2*x^2)^(1/2)),x)
 

Output:

(1 - a^2*x^2)^(1/2)/(6*(a^6*c^2 - 2*a^7*c^2*x + a^8*c^2*x^2)) - (1 - a^2*x 
^2)^(1/2)/(4*(a^4*c^2*(-a^2)^(1/2) + a^5*c^2*x*(-a^2)^(1/2))*(-a^2)^(1/2)) 
 + (25*(1 - a^2*x^2)^(1/2))/(12*(a^4*c^2*(-a^2)^(1/2) - a^5*c^2*x*(-a^2)^( 
1/2))*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a^6*c^2) - (x*(1 - a^2*x^2)^(1/ 
2))/(2*a^5*c^2) + (5*asinh(x*(-a^2)^(1/2)))/(2*a^5*c^2*(-a^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-\sqrt {-a^{2} x^{2}+1}\, a x +\sqrt {-a^{2} x^{2}+1}+3 a^{4} x^{4}+3 a^{3} x^{3}-23 a^{2} x^{2}-a x +16}{6 \sqrt {-a^{2} x^{2}+1}\, a^{6} c^{2} \left (a x -1\right )} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^2,x)
 

Output:

(15*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 15*sqrt( - a**2*x**2 + 1)*asin( 
a*x) - sqrt( - a**2*x**2 + 1)*a*x + sqrt( - a**2*x**2 + 1) + 3*a**4*x**4 + 
 3*a**3*x**3 - 23*a**2*x**2 - a*x + 16)/(6*sqrt( - a**2*x**2 + 1)*a**6*c** 
2*(a*x - 1))