\(\int e^{-3 \text {arctanh}(a x)} (c-a^2 c x^2)^3 \, dx\) [1277]

Optimal result

Integrand size = 22, antiderivative size = 130 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {9}{16} c^3 x \sqrt {1-a^2 x^2}+\frac {3}{8} c^3 x \left (1-a^2 x^2\right )^{3/2}+\frac {23 c^3 \left (1-a^2 x^2\right )^{5/2}}{35 a}-\frac {1}{2} c^3 x \left (1-a^2 x^2\right )^{5/2}+\frac {1}{7} a c^3 x^2 \left (1-a^2 x^2\right )^{5/2}+\frac {9 c^3 \arcsin (a x)}{16 a} \] Output:

9/16*c^3*x*(-a^2*x^2+1)^(1/2)+3/8*c^3*x*(-a^2*x^2+1)^(3/2)+23/35*c^3*(-a^2 
*x^2+1)^(5/2)/a-1/2*c^3*x*(-a^2*x^2+1)^(5/2)+1/7*a*c^3*x^2*(-a^2*x^2+1)^(5 
/2)+9/16*c^3*arcsin(a*x)/a
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.70 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^3 \left (\sqrt {1-a^2 x^2} \left (368+245 a x-656 a^2 x^2+350 a^3 x^3+208 a^4 x^4-280 a^5 x^5+80 a^6 x^6\right )-630 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{560 a} \] Input:

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

Output:

(c^3*(Sqrt[1 - a^2*x^2]*(368 + 245*a*x - 656*a^2*x^2 + 350*a^3*x^3 + 208*a 
^4*x^4 - 280*a^5*x^5 + 80*a^6*x^6) - 630*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/( 
560*a)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6689, 469, 469, 455, 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.

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

\(\Big \downarrow \) 6689

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

\(\Big \downarrow \) 469

\(\displaystyle c^3 \left (\frac {9}{7} \int (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}dx+\frac {(1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\)

\(\Big \downarrow \) 469

\(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \int (1-a x) \left (1-a^2 x^2\right )^{3/2}dx+\frac {(1-a x) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )+\frac {(1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\)

\(\Big \downarrow \) 455

\(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \left (\int \left (1-a^2 x^2\right )^{3/2}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}\right )+\frac {(1-a x) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )+\frac {(1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \left (\frac {3}{4} \int \sqrt {1-a^2 x^2}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {(1-a x) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )+\frac {(1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )+\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {(1-a x) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )+\frac {(1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle c^3 \left (\frac {9}{7} \left (\frac {7}{6} \left (\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )+\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {(1-a x) \left (1-a^2 x^2\right )^{5/2}}{6 a}\right )+\frac {(1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}{7 a}\right )\)

Input:

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

Output:

c^3*(((1 - a*x)^2*(1 - a^2*x^2)^(5/2))/(7*a) + (9*(((1 - a*x)*(1 - a^2*x^2 
)^(5/2))/(6*a) + (7*((x*(1 - a^2*x^2)^(3/2))/4 + (1 - a^2*x^2)^(5/2)/(5*a) 
 + (3*((x*Sqrt[1 - a^2*x^2])/2 + ArcSin[a*x]/(2*a)))/4))/6))/7)
 

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 469

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* 
((n + p)/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr 
eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* 
p + 1, 0] && IntegerQ[2*p]
 

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 = 0.49 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82

method	result	size
risch	\(-\frac {\left (80 x^{6} a^{6}-280 a^{5} x^{5}+208 a^{4} x^{4}+350 a^{3} x^{3}-656 a^{2} x^{2}+245 a x +368\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{560 a \sqrt {-a^{2} x^{2}+1}}+\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{16 \sqrt {a^{2}}}\)	\(107\)
default	\(-c^{3} \left (a^{3} \left (-\frac {x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{7 a^{2}}-\frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{35 a^{4}}\right )-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}-\frac {3 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{5 a}-3 a^{2} \left (-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{6 a^{2}}+\frac {\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}}{6 a^{2}}\right )\right )\)	\(210\)

Input:

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

Output:

-1/560*(80*a^6*x^6-280*a^5*x^5+208*a^4*x^4+350*a^3*x^3-656*a^2*x^2+245*a*x 
+368)*(a^2*x^2-1)/a/(-a^2*x^2+1)^(1/2)*c^3+9/16/(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 = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {630 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (80 \, a^{6} c^{3} x^{6} - 280 \, a^{5} c^{3} x^{5} + 208 \, a^{4} c^{3} x^{4} + 350 \, a^{3} c^{3} x^{3} - 656 \, a^{2} c^{3} x^{2} + 245 \, a c^{3} x + 368 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{560 \, a} \] Input:

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

Output:

-1/560*(630*c^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (80*a^6*c^3*x^6 - 
 280*a^5*c^3*x^5 + 208*a^4*c^3*x^4 + 350*a^3*c^3*x^3 - 656*a^2*c^3*x^2 + 2 
45*a*c^3*x + 368*c^3)*sqrt(-a^2*x^2 + 1))/a
 

Sympy [A] (verification not implemented)

Time = 1.86 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.90 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=a^{5} c^{3} \left (\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (\frac {x^{6}}{7} - \frac {x^{4}}{35 a^{2}} - \frac {4 x^{2}}{105 a^{4}} - \frac {8}{105 a^{6}}\right ) & \text {for}\: a^{2} \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{3} \left (\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (\frac {x^{5}}{6} - \frac {x^{3}}{24 a^{2}} - \frac {x}{16 a^{4}}\right ) + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{16 a^{4} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{5}}{5} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{3} \left (\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (\frac {x^{4}}{5} - \frac {x^{2}}{15 a^{2}} - \frac {2}{15 a^{4}}\right ) & \text {for}\: a^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{3} \left (\begin {cases} \left (\frac {x^{3}}{4} - \frac {x}{8 a^{2}}\right ) \sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) - 3 a c^{3} \left (\begin {cases} \left (\frac {x^{2}}{3} - \frac {1}{3 a^{2}}\right ) \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) \] Input:

integrate((-a**2*c*x**2+c)**3/(a*x+1)**3*(-a**2*x**2+1)**(3/2),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)) - 3*a**4*c**3* 
Piecewise((sqrt(-a**2*x**2 + 1)*(x**5/6 - x**3/(24*a**2) - x/(16*a**4)) + 
log(-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)) - 3*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* 
sqrt(-a**2)), Ne(a**2, 0)), (x, True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.83 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {1}{7} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a c^{3} x^{2} - \frac {1}{2} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{3} x + \frac {3}{8} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3} x + \frac {23 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{3}}{35 \, a} + \frac {9}{16} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {9 \, c^{3} \arcsin \left (a x\right )}{16 \, a} \] Input:

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

Output:

1/7*(-a^2*x^2 + 1)^(5/2)*a*c^3*x^2 - 1/2*(-a^2*x^2 + 1)^(5/2)*c^3*x + 3/8* 
(-a^2*x^2 + 1)^(3/2)*c^3*x + 23/35*(-a^2*x^2 + 1)^(5/2)*c^3/a + 9/16*sqrt( 
-a^2*x^2 + 1)*c^3*x + 9/16*c^3*arcsin(a*x)/a
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.78 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {9 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, {\left | a \right |}} + \frac {1}{560} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {368 \, c^{3}}{a} + {\left (245 \, c^{3} - 2 \, {\left (328 \, a c^{3} - {\left (175 \, a^{2} c^{3} + 4 \, {\left (26 \, a^{3} c^{3} + 5 \, {\left (2 \, a^{5} c^{3} x - 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

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

Output:

9/16*c^3*arcsin(a*x)*sgn(a)/abs(a) + 1/560*sqrt(-a^2*x^2 + 1)*(368*c^3/a + 
 (245*c^3 - 2*(328*a*c^3 - (175*a^2*c^3 + 4*(26*a^3*c^3 + 5*(2*a^5*c^3*x - 
 7*a^4*c^3)*x)*x)*x)*x)*x)
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.34 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {7\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {9\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,\sqrt {-a^2}}+\frac {23\,c^3\,\sqrt {1-a^2\,x^2}}{35\,a}-\frac {41\,a\,c^3\,x^2\,\sqrt {1-a^2\,x^2}}{35}+\frac {5\,a^2\,c^3\,x^3\,\sqrt {1-a^2\,x^2}}{8}+\frac {13\,a^3\,c^3\,x^4\,\sqrt {1-a^2\,x^2}}{35}-\frac {a^4\,c^3\,x^5\,\sqrt {1-a^2\,x^2}}{2}+\frac {a^5\,c^3\,x^6\,\sqrt {1-a^2\,x^2}}{7} \] Input:

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

Output:

(7*c^3*x*(1 - a^2*x^2)^(1/2))/16 + (9*c^3*asinh(x*(-a^2)^(1/2)))/(16*(-a^2 
)^(1/2)) + (23*c^3*(1 - a^2*x^2)^(1/2))/(35*a) - (41*a*c^3*x^2*(1 - a^2*x^ 
2)^(1/2))/35 + (5*a^2*c^3*x^3*(1 - a^2*x^2)^(1/2))/8 + (13*a^3*c^3*x^4*(1 
- a^2*x^2)^(1/2))/35 - (a^4*c^3*x^5*(1 - a^2*x^2)^(1/2))/2 + (a^5*c^3*x^6* 
(1 - a^2*x^2)^(1/2))/7
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.07 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^{3} \left (315 \mathit {asin} \left (a x \right )+80 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-280 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+208 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+350 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-656 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+245 \sqrt {-a^{2} x^{2}+1}\, a x +368 \sqrt {-a^{2} x^{2}+1}-368\right )}{560 a} \] Input:

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

Output:

(c**3*(315*asin(a*x) + 80*sqrt( - a**2*x**2 + 1)*a**6*x**6 - 280*sqrt( - a 
**2*x**2 + 1)*a**5*x**5 + 208*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 350*sqrt( 
 - a**2*x**2 + 1)*a**3*x**3 - 656*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 245*s 
qrt( - a**2*x**2 + 1)*a*x + 368*sqrt( - a**2*x**2 + 1) - 368))/(560*a)