\(\int e^{\text {arctanh}(a x)} (c-a^2 c x^2)^3 \, dx\) [921]

Optimal result

Integrand size = 20, 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 = 0.14 (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[E^ArcTanh[a*x]*(c - a^2*c*x^2)^3,x]
 

Output:

(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]]))/(3 
36*a)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6688, 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.

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

\(\Big \downarrow \) 6688

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

\(\Big \downarrow \) 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[E^ArcTanh[a*x]*(c - a^2*c*x^2)^3,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 6688

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] && IGtQ[(n + 1)/2, 0] &&  !I 
ntegerQ[p - n/2]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02

method	result	size
risch	\(-\frac {\left (48 x^{6} a^{6}+56 a^{5} x^{5}-144 a^{4} x^{4}-182 a^{3} x^{3}+144 a^{2} x^{2}+231 a x -48\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{336 a \sqrt {-a^{2} x^{2}+1}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{3}}{16 \sqrt {a^{2}}}\)	\(107\)
meijerg	\(-\frac {c^{3} \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (40 x^{6} a^{6}+48 a^{4} x^{4}+64 a^{2} x^{2}+128\right ) \sqrt {-a^{2} x^{2}+1}}{140}\right )}{2 a \sqrt {\pi }}-\frac {3 c^{3} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a \sqrt {\pi }}-\frac {3 c^{3} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a \sqrt {\pi }}-\frac {c^{3} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{3} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{3} \arcsin \left (a x \right )}{a}\)	\(419\)
default	\(-c^{3} \left (a^{6} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )+a^{7} \left (-\frac {x^{6} \sqrt {-a^{2} x^{2}+1}}{7 a^{2}}+\frac {-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}}{a^{2}}\right )-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {\sqrt {-a^{2} x^{2}+1}}{a}+3 a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )-3 a^{4} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-3 a^{5} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )+3 a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )\right )\)	\(495\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^3,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- 
48)*(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 = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {210 \, c^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (48 \, a^{6} c^{3} x^{6} + 56 \, a^{5} c^{3} x^{5} - 144 \, a^{4} c^{3} x^{4} - 182 \, a^{3} c^{3} x^{3} + 144 \, a^{2} c^{3} x^{2} + 231 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{336 \, a} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^3,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (90) = 180\).

Time = 0.96 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.24 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\begin {cases} \frac {5 c^{3} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{16 \sqrt {- a^{2}}} + \sqrt {- a^{2} x^{2} + 1} \left (\frac {a^{5} c^{3} x^{6}}{7} + \frac {a^{4} c^{3} x^{5}}{6} - \frac {3 a^{3} c^{3} x^{4}}{7} - \frac {13 a^{2} c^{3} x^{3}}{24} + \frac {3 a c^{3} x^{2}}{7} + \frac {11 c^{3} x}{16} - \frac {c^{3}}{7 a}\right ) & \text {for}\: a^{2} \neq 0 \\\begin {cases} c^{3} x & \text {for}\: a = 0 \\\frac {- \frac {a^{8} c^{3} x^{8}}{8} - \frac {a^{7} c^{3} x^{7}}{7} + \frac {a^{6} c^{3} x^{6}}{2} + \frac {3 a^{5} c^{3} x^{5}}{5} - \frac {3 a^{4} c^{3} x^{4}}{4} - a^{3} c^{3} x^{3} + \frac {a^{2} c^{3} x^{2}}{2} + a c^{3} x}{a} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((5*c**3*log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(16* 
sqrt(-a**2)) + sqrt(-a**2*x**2 + 1)*(a**5*c**3*x**6/7 + a**4*c**3*x**5/6 - 
 3*a**3*c**3*x**4/7 - 13*a**2*c**3*x**3/24 + 3*a*c**3*x**2/7 + 11*c**3*x/1 
6 - c**3/(7*a)), Ne(a**2, 0)), (Piecewise((c**3*x, Eq(a, 0)), ((-a**8*c**3 
*x**8/8 - a**7*c**3*x**7/7 + a**6*c**3*x**6/2 + 3*a**5*c**3*x**5/5 - 3*a** 
4*c**3*x**4/4 - a**3*c**3*x**3 + a**2*c**3*x**2/2 + a*c**3*x)/a, True)), T 
rue))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.56 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {1}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{3} x^{6} + \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} x^{5} - \frac {3}{7} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{4} - \frac {13}{24} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x^{3} + \frac {3}{7} \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{2} + \frac {11}{16} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {5 \, c^{3} \arcsin \left (a x\right )}{16 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{7 \, a} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.98 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {5 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, {\left | a \right |}} - \frac {1}{336} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {48 \, c^{3}}{a} - {\left (231 \, c^{3} + 2 \, {\left (72 \, a c^{3} - {\left (91 \, a^{2} c^{3} + 4 \, {\left (18 \, a^{3} c^{3} - {\left (6 \, a^{5} c^{3} x + 7 \, a^{4} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^3,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 = 23.72 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {5\,c^3\,x\,\sqrt {1-a^2\,x^2}}{16}+\frac {5\,c^3\,x\,{\left (1-a^2\,x^2\right )}^{3/2}}{24}+\frac {c^3\,x\,{\left (1-a^2\,x^2\right )}^{5/2}}{6}-\frac {c^3\,{\left (1-a^2\,x^2\right )}^{7/2}}{7\,a}-\frac {5\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{16\,a^2} \] Input:

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

Output:

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

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