\(\int e^{-\text {arctanh}(a x)} (c-a^2 c x^2)^3 \, dx\) [1230]

Optimal result

Integrand size = 22, antiderivative size = 105 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {5}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {5}{24} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {1}{6} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{7 a}+\frac {5 c^3 \arcsin (a x)}{16 a} \] Output:

5/16*c^3*x*(-a^2*x^2+1)^(1/2)+5/24*c^3*x*(-a^2*x^2+1)^(3/2)+1/6*c^3*x*(-a^ 
2*x^2+1)^(5/2)+1/7*c^3*(-a^2*x^2+1)^(7/2)/a+5/16*c^3*arcsin(a*x)/a
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {c^3 \left (\sqrt {1-a^2 x^2} \left (-48-231 a x+144 a^2 x^2+182 a^3 x^3-144 a^4 x^4-56 a^5 x^5+48 a^6 x^6\right )+210 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{336 a} \] Input:

Integrate[(c - a^2*c*x^2)^3/E^ArcTanh[a*x],x]
 

Output:

-1/336*(c^3*(Sqrt[1 - a^2*x^2]*(-48 - 231*a*x + 144*a^2*x^2 + 182*a^3*x^3 
- 144*a^4*x^4 - 56*a^5*x^5 + 48*a^6*x^6) + 210*ArcSin[Sqrt[1 - a*x]/Sqrt[2 
]]))/a
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6689, 455, 211, 211, 211, 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.

\(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx\)

\(\Big \downarrow \) 6689

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

Input:

Int[(c - a^2*c*x^2)^3/E^ArcTanh[a*x],x]
 

Output:

c^3*((x*(1 - a^2*x^2)^(5/2))/6 + (1 - a^2*x^2)^(7/2)/(7*a) + (5*((x*(1 - a 
^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - a^2*x^2])/2 + ArcSin[a*x]/(2*a)))/4))/6 
)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 6689

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02

method	result	size
risch	\(\frac {\left (48 x^{6} a^{6}-56 a^{5} x^{5}-144 a^{4} x^{4}+182 a^{3} x^{3}+144 a^{2} x^{2}-231 a x -48\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{336 a \sqrt {-a^{2} x^{2}+1}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{16 \sqrt {a^{2}}}\)	\(107\)
default	\(-c^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}+a^{5} \left (-\frac {x^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{7 a^{2}}+\frac {-\frac {4 x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{35 a^{2}}-\frac {8 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{105 a^{4}}}{a^{2}}\right )-\frac {x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-2 a^{3} \left (-\frac {x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5 a^{2}}-\frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{15 a^{4}}\right )-a^{4} \left (-\frac {x^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{6 a^{2}}+\frac {-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{2}}+\frac {\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}}{4 a^{2}}}{2 a^{2}}\right )+2 a^{2} \left (-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{2}}+\frac {\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}}{4 a^{2}}\right )\right )\)	\(347\)

Input:

int((-a^2*c*x^2+c)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/336*(48*a^6*x^6-56*a^5*x^5-144*a^4*x^4+182*a^3*x^3+144*a^2*x^2-231*a*x-4 
8)*(a^2*x^2-1)/a/(-a^2*x^2+1)^(1/2)*c^3+5/16/(a^2)^(1/2)*arctan((a^2)^(1/2 
)*x/(-a^2*x^2+1)^(1/2))*c^3
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {210 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (48 \, a^{6} c^{3} x^{6} - 56 \, a^{5} c^{3} x^{5} - 144 \, a^{4} c^{3} x^{4} + 182 \, a^{3} c^{3} x^{3} + 144 \, a^{2} c^{3} x^{2} - 231 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{336 \, a} \] Input:

integrate((-a^2*c*x^2+c)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas 
")
 

Output:

-1/336*(210*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (48*a^6*c^3*x^6 - 
 56*a^5*c^3*x^5 - 144*a^4*c^3*x^4 + 182*a^3*c^3*x^3 + 144*a^2*c^3*x^2 - 23 
1*a*c^3*x - 48*c^3)*sqrt(-a^2*x^2 + 1))/a
 

Sympy [A] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.56 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- a^{5} c^{3} \left (\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (\frac {x^{6}}{7} - \frac {x^{4}}{35 a^{2}} - \frac {4 x^{2}}{105 a^{4}} - \frac {8}{105 a^{6}}\right ) & \text {for}\: a^{2} \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) + a^{4} c^{3} \left (\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (\frac {x^{5}}{6} - \frac {x^{3}}{24 a^{2}} - \frac {x}{16 a^{4}}\right ) + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{16 a^{4} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{5}}{5} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (\frac {x^{4}}{5} - \frac {x^{2}}{15 a^{2}} - \frac {2}{15 a^{4}}\right ) & \text {for}\: a^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - 2 a^{2} c^{3} \left (\begin {cases} \left (\frac {x^{3}}{4} - \frac {x}{8 a^{2}}\right ) \sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) - a c^{3} \left (\begin {cases} \left (\frac {x^{2}}{3} - \frac {1}{3 a^{2}}\right ) \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) \] Input:

integrate((-a**2*c*x**2+c)**3/(a*x+1)*(-a**2*x**2+1)**(1/2),x)
 

Output:

-a**5*c**3*Piecewise((sqrt(-a**2*x**2 + 1)*(x**6/7 - x**4/(35*a**2) - 4*x* 
*2/(105*a**4) - 8/(105*a**6)), Ne(a**2, 0)), (x**6/6, True)) + a**4*c**3*P 
iecewise((sqrt(-a**2*x**2 + 1)*(x**5/6 - x**3/(24*a**2) - x/(16*a**4)) + l 
og(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(16*a**4*sqrt(-a**2)), 
Ne(a**2, 0)), (x**5/5, True)) + 2*a**3*c**3*Piecewise((sqrt(-a**2*x**2 + 1 
)*(x**4/5 - x**2/(15*a**2) - 2/(15*a**4)), Ne(a**2, 0)), (x**4/4, True)) - 
 2*a**2*c**3*Piecewise(((x**3/4 - x/(8*a**2))*sqrt(-a**2*x**2 + 1) + log(- 
2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(8*a**2*sqrt(-a**2)), Ne(a* 
*2, 0)), (x**3/3, True)) - a*c**3*Piecewise(((x**2/3 - 1/(3*a**2))*sqrt(-a 
**2*x**2 + 1), Ne(a**2, 0)), (x**2/2, True)) + c**3*Piecewise((x*sqrt(-a** 
2*x**2 + 1)/2 + log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(2*sqr 
t(-a**2)), Ne(a**2, 0)), (x, True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.30 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {1}{7} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{3} c^{3} x^{4} - \frac {1}{6} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} c^{3} x^{3} - \frac {2}{7} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a c^{3} x^{2} + \frac {3}{8} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3} x + \frac {5}{16} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{7 \, a} + \frac {5 \, c^{3} \arcsin \left (a x\right )}{16 \, a} \] Input:

integrate((-a^2*c*x^2+c)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima 
")
 

Output:

1/7*(-a^2*x^2 + 1)^(3/2)*a^3*c^3*x^4 - 1/6*(-a^2*x^2 + 1)^(3/2)*a^2*c^3*x^ 
3 - 2/7*(-a^2*x^2 + 1)^(3/2)*a*c^3*x^2 + 3/8*(-a^2*x^2 + 1)^(3/2)*c^3*x + 
5/16*sqrt(-a^2*x^2 + 1)*c^3*x + 1/7*(-a^2*x^2 + 1)^(3/2)*c^3/a + 5/16*c^3* 
arcsin(a*x)/a
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {5 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, {\left | a \right |}} + \frac {1}{336} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {48 \, c^{3}}{a} + {\left (231 \, c^{3} - 2 \, {\left (72 \, a c^{3} + {\left (91 \, a^{2} c^{3} - 4 \, {\left (18 \, a^{3} c^{3} - {\left (6 \, a^{5} c^{3} x - 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

integrate((-a^2*c*x^2+c)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

5/16*c^3*arcsin(a*x)*sgn(a)/abs(a) + 1/336*sqrt(-a^2*x^2 + 1)*(48*c^3/a + 
(231*c^3 - 2*(72*a*c^3 + (91*a^2*c^3 - 4*(18*a^3*c^3 - (6*a^5*c^3*x - 7*a^ 
4*c^3)*x)*x)*x)*x)*x)
 

Mupad [B] (verification not implemented)

Time = 22.99 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.66 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {11\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {5\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,\sqrt {-a^2}}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{7\,a}-\frac {3\,a\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{7}-\frac {13\,a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{24}+\frac {3\,a^3\,c^3\,x^4\,\sqrt {1-a^2\,x^2}}{7}+\frac {a^4\,c^3\,x^5\,\sqrt {1-a^2\,x^2}}{6}-\frac {a^5\,c^3\,x^6\,\sqrt {1-a^2\,x^2}}{7} \] Input:

int(((c - a^2*c*x^2)^3*(1 - a^2*x^2)^(1/2))/(a*x + 1),x)
 

Output:

(11*c^3*x*(1 - a^2*x^2)^(1/2))/16 + (5*c^3*asinh(x*(-a^2)^(1/2)))/(16*(-a^ 
2)^(1/2)) + (c^3*(1 - a^2*x^2)^(1/2))/(7*a) - (3*a*c^3*x^2*(1 - a^2*x^2)^( 
1/2))/7 - (13*a^2*c^3*x^3*(1 - a^2*x^2)^(1/2))/24 + (3*a^3*c^3*x^4*(1 - a^ 
2*x^2)^(1/2))/7 + (a^4*c^3*x^5*(1 - a^2*x^2)^(1/2))/6 - (a^5*c^3*x^6*(1 - 
a^2*x^2)^(1/2))/7
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.32 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^{3} \left (105 \mathit {asin} \left (a x \right )-48 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+56 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+144 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-182 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-144 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+231 \sqrt {-a^{2} x^{2}+1}\, a x +48 \sqrt {-a^{2} x^{2}+1}-48\right )}{336 a} \] Input:

int((-a^2*c*x^2+c)^3/(a*x+1)*(-a^2*x^2+1)^(1/2),x)
 

Output:

(c**3*(105*asin(a*x) - 48*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 56*sqrt( - a* 
*2*x**2 + 1)*a**5*x**5 + 144*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 182*sqrt( 
- a**2*x**2 + 1)*a**3*x**3 - 144*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 231*sq 
rt( - a**2*x**2 + 1)*a*x + 48*sqrt( - a**2*x**2 + 1) - 48))/(336*a)