\(\int e^{-\text {arctanh}(a x)} (c-a^2 c x^2)^4 \, dx\) [1214]

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {35}{128} c^4 x \sqrt {1-a^2 x^2}+\frac {35}{192} c^4 x \left (1-a^2 x^2\right )^{3/2}+\frac {7}{48} c^4 x \left (1-a^2 x^2\right )^{5/2}+\frac {1}{8} c^4 x \left (1-a^2 x^2\right )^{7/2}+\frac {c^4 \left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {35 c^4 \arcsin (a x)}{128 a} \] Output:

35/128*c^4*x*(-a^2*x^2+1)^(1/2)+35/192*c^4*x*(-a^2*x^2+1)^(3/2)+7/48*c^4*x 
*(-a^2*x^2+1)^(5/2)+1/8*c^4*x*(-a^2*x^2+1)^(7/2)+1/9*c^4*(-a^2*x^2+1)^(9/2 
)/a+35/128*c^4*arcsin(a*x)/a
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (128+837 a x-512 a^2 x^2-978 a^3 x^3+768 a^4 x^4+600 a^5 x^5-512 a^6 x^6-144 a^7 x^7+128 a^8 x^8\right )-630 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{1152 a} \] Input:

Integrate[(c - a^2*c*x^2)^4/E^ArcTanh[a*x],x]
 

Output:

(c^4*(Sqrt[1 - a^2*x^2]*(128 + 837*a*x - 512*a^2*x^2 - 978*a^3*x^3 + 768*a 
^4*x^4 + 600*a^5*x^5 - 512*a^6*x^6 - 144*a^7*x^7 + 128*a^8*x^8) - 630*ArcS 
in[Sqrt[1 - a*x]/Sqrt[2]]))/(1152*a)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6689, 455, 211, 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.

\(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx\)

\(\Big \downarrow \) 6689

\(\displaystyle c^4 \int (1-a x) \left (1-a^2 x^2\right )^{7/2}dx\)

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 211

\(\displaystyle c^4 \left (\frac {7}{8} \int \left (1-a^2 x^2\right )^{5/2}dx+\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle c^4 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (1-a^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2}\right )+\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle c^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-a^2 x^2}dx+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2}\right )+\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle c^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2}\right )+\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle c^4 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-a^2 x^2\right )^{5/2}\right )+\frac {\left (1-a^2 x^2\right )^{9/2}}{9 a}+\frac {1}{8} x \left (1-a^2 x^2\right )^{7/2}\right )\)

Input:

Int[(c - a^2*c*x^2)^4/E^ArcTanh[a*x],x]
 

Output:

c^4*((x*(1 - a^2*x^2)^(7/2))/8 + (1 - a^2*x^2)^(9/2)/(9*a) + (7*((x*(1 - a 
^2*x^2)^(5/2))/6 + (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))/8)
 

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 = 0.55 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {\left (128 a^{8} x^{8}-144 a^{7} x^{7}-512 x^{6} a^{6}+600 a^{5} x^{5}+768 a^{4} x^{4}-978 a^{3} x^{3}-512 a^{2} x^{2}+837 a x +128\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{1152 a \sqrt {-a^{2} x^{2}+1}}+\frac {35 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{128 \sqrt {a^{2}}}\)	\(123\)
default	\(c^{4} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+a^{7} \left (-\frac {x^{6} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{9 a^{2}}+\frac {-\frac {2 x^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{21 a^{2}}+\frac {2 \left (-\frac {4 x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{35 a^{2}}-\frac {8 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{105 a^{4}}\right )}{3 a^{2}}}{a^{2}}\right )+\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}+3 a^{3} \left (-\frac {x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5 a^{2}}-\frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{15 a^{4}}\right )+3 a^{4} \left (-\frac {x^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{6 a^{2}}+\frac {-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{2}}+\frac {\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}}{4 a^{2}}}{2 a^{2}}\right )-3 a^{5} \left (-\frac {x^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{7 a^{2}}+\frac {-\frac {4 x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{35 a^{2}}-\frac {8 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{105 a^{4}}}{a^{2}}\right )-a^{6} \left (-\frac {x^{5} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{8 a^{2}}+\frac {-\frac {5 x^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{48 a^{2}}+\frac {5 \left (-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{2}}+\frac {\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}}{4 a^{2}}\right )}{16 a^{2}}}{a^{2}}\right )-3 a^{2} \left (-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{2}}+\frac {\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}}{4 a^{2}}\right )\right )\)	\(565\)

Input:

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

Output:

-1/1152*(128*a^8*x^8-144*a^7*x^7-512*a^6*x^6+600*a^5*x^5+768*a^4*x^4-978*a 
^3*x^3-512*a^2*x^2+837*a*x+128)*(a^2*x^2-1)/a/(-a^2*x^2+1)^(1/2)*c^4+35/12 
8/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))*c^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.08 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {630 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (128 \, a^{8} c^{4} x^{8} - 144 \, a^{7} c^{4} x^{7} - 512 \, a^{6} c^{4} x^{6} + 600 \, a^{5} c^{4} x^{5} + 768 \, a^{4} c^{4} x^{4} - 978 \, a^{3} c^{4} x^{3} - 512 \, a^{2} c^{4} x^{2} + 837 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{1152 \, a} \] Input:

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

Output:

-1/1152*(630*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (128*a^8*c^4*x^8 
 - 144*a^7*c^4*x^7 - 512*a^6*c^4*x^6 + 600*a^5*c^4*x^5 + 768*a^4*c^4*x^4 - 
 978*a^3*c^4*x^3 - 512*a^2*c^4*x^2 + 837*a*c^4*x + 128*c^4)*sqrt(-a^2*x^2 
+ 1))/a
 

Sympy [A] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 547, normalized size of antiderivative = 4.31 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx =\text {Too large to display} \] Input:

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

Output:

a**7*c**4*Piecewise((sqrt(-a**2*x**2 + 1)*(x**8/9 - x**6/(63*a**2) - 2*x** 
4/(105*a**4) - 8*x**2/(315*a**6) - 16/(315*a**8)), Ne(a**2, 0)), (x**8/8, 
True)) - a**6*c**4*Piecewise((sqrt(-a**2*x**2 + 1)*(x**7/8 - x**5/(48*a**2 
) - 5*x**3/(192*a**4) - 5*x/(128*a**6)) + 5*log(-2*a**2*x + 2*sqrt(-a**2)* 
sqrt(-a**2*x**2 + 1))/(128*a**6*sqrt(-a**2)), Ne(a**2, 0)), (x**7/7, True) 
) - 3*a**5*c**4*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**4*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)) + 3*a**3*c**4*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, T 
rue)) - 3*a**2*c**4*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**4*Piecewise(((x**2/3 - 1/(3*a**2))* 
sqrt(-a**2*x**2 + 1), Ne(a**2, 0)), (x**2/2, True)) + c**4*Piecewise((x*sq 
rt(-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 = 182, normalized size of antiderivative = 1.43 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=-\frac {1}{9} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{5} c^{4} x^{6} + \frac {1}{8} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4} c^{4} x^{5} + \frac {1}{3} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{3} c^{4} x^{4} - \frac {19}{48} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} c^{4} x^{3} - \frac {1}{3} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a c^{4} x^{2} + \frac {29}{64} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{4} x + \frac {35}{128} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{4}}{9 \, a} + \frac {35 \, c^{4} \arcsin \left (a x\right )}{128 \, a} \] Input:

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

Output:

-1/9*(-a^2*x^2 + 1)^(3/2)*a^5*c^4*x^6 + 1/8*(-a^2*x^2 + 1)^(3/2)*a^4*c^4*x 
^5 + 1/3*(-a^2*x^2 + 1)^(3/2)*a^3*c^4*x^4 - 19/48*(-a^2*x^2 + 1)^(3/2)*a^2 
*c^4*x^3 - 1/3*(-a^2*x^2 + 1)^(3/2)*a*c^4*x^2 + 29/64*(-a^2*x^2 + 1)^(3/2) 
*c^4*x + 35/128*sqrt(-a^2*x^2 + 1)*c^4*x + 1/9*(-a^2*x^2 + 1)^(3/2)*c^4/a 
+ 35/128*c^4*arcsin(a*x)/a
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {35 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{128 \, {\left | a \right |}} + \frac {1}{1152} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {128 \, c^{4}}{a} + {\left (837 \, c^{4} - 2 \, {\left (256 \, a c^{4} + {\left (489 \, a^{2} c^{4} - 4 \, {\left (96 \, a^{3} c^{4} + {\left (75 \, a^{4} c^{4} - 2 \, {\left (32 \, a^{5} c^{4} - {\left (8 \, a^{7} c^{4} x - 9 \, a^{6} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

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

Output:

35/128*c^4*arcsin(a*x)*sgn(a)/abs(a) + 1/1152*sqrt(-a^2*x^2 + 1)*(128*c^4/ 
a + (837*c^4 - 2*(256*a*c^4 + (489*a^2*c^4 - 4*(96*a^3*c^4 + (75*a^4*c^4 - 
 2*(32*a^5*c^4 - (8*a^7*c^4*x - 9*a^6*c^4)*x)*x)*x)*x)*x)*x)*x)
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.73 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {93\,c^4\,x\,\sqrt {1-a^2\,x^2}}{128}+\frac {35\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{128\,\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{9\,a}-\frac {4\,a\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{9}-\frac {163\,a^2\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{192}+\frac {2\,a^3\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{3}+\frac {25\,a^4\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{48}-\frac {4\,a^5\,c^4\,x^6\,\sqrt {1-a^2\,x^2}}{9}-\frac {a^6\,c^4\,x^7\,\sqrt {1-a^2\,x^2}}{8}+\frac {a^7\,c^4\,x^8\,\sqrt {1-a^2\,x^2}}{9} \] Input:

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

Output:

(93*c^4*x*(1 - a^2*x^2)^(1/2))/128 + (35*c^4*asinh(x*(-a^2)^(1/2)))/(128*( 
-a^2)^(1/2)) + (c^4*(1 - a^2*x^2)^(1/2))/(9*a) - (4*a*c^4*x^2*(1 - a^2*x^2 
)^(1/2))/9 - (163*a^2*c^4*x^3*(1 - a^2*x^2)^(1/2))/192 + (2*a^3*c^4*x^4*(1 
 - a^2*x^2)^(1/2))/3 + (25*a^4*c^4*x^5*(1 - a^2*x^2)^(1/2))/48 - (4*a^5*c^ 
4*x^6*(1 - a^2*x^2)^(1/2))/9 - (a^6*c^4*x^7*(1 - a^2*x^2)^(1/2))/8 + (a^7* 
c^4*x^8*(1 - a^2*x^2)^(1/2))/9
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.39 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^4 \, dx=\frac {c^{4} \left (315 \mathit {asin} \left (a x \right )+128 \sqrt {-a^{2} x^{2}+1}\, a^{8} x^{8}-144 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}-512 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+600 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+768 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-978 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-512 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+837 \sqrt {-a^{2} x^{2}+1}\, a x +128 \sqrt {-a^{2} x^{2}+1}-128\right )}{1152 a} \] Input:

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

Output:

(c**4*(315*asin(a*x) + 128*sqrt( - a**2*x**2 + 1)*a**8*x**8 - 144*sqrt( - 
a**2*x**2 + 1)*a**7*x**7 - 512*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 600*sqrt 
( - a**2*x**2 + 1)*a**5*x**5 + 768*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 978* 
sqrt( - a**2*x**2 + 1)*a**3*x**3 - 512*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 
837*sqrt( - a**2*x**2 + 1)*a*x + 128*sqrt( - a**2*x**2 + 1) - 128))/(1152* 
a)