\(\int \frac {e^{\text {arctanh}(a x)} x^7}{(c-a^2 c x^2)^3} \, dx\) [929]

Optimal result

Integrand size = 23, antiderivative size = 184 \[ \int \frac {e^{\text {arctanh}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1+a x}{5 a^8 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{a^8 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {16 x}{15 a^7 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {3}{a^8 c^3 \sqrt {1-a^2 x^2}}+\frac {58 x}{15 a^7 c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a^8 c^3}+\frac {x \sqrt {1-a^2 x^2}}{2 a^7 c^3}-\frac {7 \arcsin (a x)}{2 a^8 c^3} \] Output:

1/5*(a*x+1)/a^8/c^3/(-a^2*x^2+1)^(5/2)-1/a^8/c^3/(-a^2*x^2+1)^(3/2)-16/15* 
x/a^7/c^3/(-a^2*x^2+1)^(3/2)+3/a^8/c^3/(-a^2*x^2+1)^(1/2)+58/15*x/a^7/c^3/ 
(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)/a^8/c^3+1/2*x*(-a^2*x^2+1)^(1/2)/a^7 
/c^3-7/2*arcsin(a*x)/a^8/c^3
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.63 \[ \int \frac {e^{\text {arctanh}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {96+9 a x-249 a^2 x^2+4 a^3 x^3+176 a^4 x^4-15 a^5 x^5-15 a^6 x^6-105 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \arcsin (a x)}{30 a^8 c^3 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \] Input:

Integrate[(E^ArcTanh[a*x]*x^7)/(c - a^2*c*x^2)^3,x]
 

Output:

(96 + 9*a*x - 249*a^2*x^2 + 4*a^3*x^3 + 176*a^4*x^4 - 15*a^5*x^5 - 15*a^6* 
x^6 - 105*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(30*a^8*c^ 
3*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6698, 529, 2345, 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.

\(\displaystyle \int \frac {x^7 e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\frac {15 x^4}{a^3}+\frac {15 x^3}{a^4}+\frac {30 x^2}{a^5}+\frac {30 x}{a^6}+\frac {13}{a^7}}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {16 a x+15}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {58 a x+45}{a^8 \sqrt {1-a^2 x^2}}-\int \frac {15 \left (\frac {x^2}{a^5}+\frac {x}{a^6}+\frac {3}{a^7}\right )}{\sqrt {1-a^2 x^2}}dx\right )-\frac {16 a x+15}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {58 a x+45}{a^8 \sqrt {1-a^2 x^2}}-15 \int \frac {\frac {x^2}{a^5}+\frac {x}{a^6}+\frac {3}{a^7}}{\sqrt {1-a^2 x^2}}dx\right )-\frac {16 a x+15}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {58 a x+45}{a^8 \sqrt {1-a^2 x^2}}-15 \left (-\frac {\int -\frac {2 a x+7}{a^5 \sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^7}\right )\right )-\frac {16 a x+15}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {58 a x+45}{a^8 \sqrt {1-a^2 x^2}}-15 \left (\frac {\int \frac {2 a x+7}{a^5 \sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^7}\right )\right )-\frac {16 a x+15}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {58 a x+45}{a^8 \sqrt {1-a^2 x^2}}-15 \left (\frac {\int \frac {2 a x+7}{\sqrt {1-a^2 x^2}}dx}{2 a^7}-\frac {x \sqrt {1-a^2 x^2}}{2 a^7}\right )\right )-\frac {16 a x+15}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {58 a x+45}{a^8 \sqrt {1-a^2 x^2}}-15 \left (\frac {7 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{a}}{2 a^7}-\frac {x \sqrt {1-a^2 x^2}}{2 a^7}\right )\right )-\frac {16 a x+15}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {a x+1}{5 a^8 \left (1-a^2 x^2\right )^{5/2}}+\frac {1}{5} \left (\frac {1}{3} \left (\frac {58 a x+45}{a^8 \sqrt {1-a^2 x^2}}-15 \left (\frac {\frac {7 \arcsin (a x)}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a}}{2 a^7}-\frac {x \sqrt {1-a^2 x^2}}{2 a^7}\right )\right )-\frac {16 a x+15}{3 a^8 \left (1-a^2 x^2\right )^{3/2}}\right )}{c^3}\)

Input:

Int[(E^ArcTanh[a*x]*x^7)/(c - a^2*c*x^2)^3,x]
 

Output:

((1 + a*x)/(5*a^8*(1 - a^2*x^2)^(5/2)) + (-1/3*(15 + 16*a*x)/(a^8*(1 - a^2 
*x^2)^(3/2)) + ((45 + 58*a*x)/(a^8*Sqrt[1 - a^2*x^2]) - 15*(-1/2*(x*Sqrt[1 
 - a^2*x^2])/a^7 + ((-2*Sqrt[1 - a^2*x^2])/a + (7*ArcSin[a*x])/a)/(2*a^7)) 
)/3)/5)/c^3
 

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 = 0.33 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.44

method	result	size
risch	\(-\frac {\left (a x +2\right ) \left (a^{2} x^{2}-1\right )}{2 a^{8} \sqrt {-a^{2} x^{2}+1}\, c^{3}}-\frac {\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{7} \sqrt {a^{2}}}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{20 a^{11} \left (x -\frac {1}{a}\right )^{3}}+\frac {7 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 a^{10} \left (x -\frac {1}{a}\right )^{2}}+\frac {773 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{240 a^{9} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{24 a^{10} \left (x +\frac {1}{a}\right )^{2}}+\frac {31 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{48 a^{9} \left (x +\frac {1}{a}\right )}}{c^{3}}\)	\(265\)
default	\(-\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{8}}+\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{5}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{7} \sqrt {a^{2}}}+\frac {11 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a^{9} \left (x +\frac {1}{a}\right )}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{10}}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{2 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{2 \left (x -\frac {1}{a}\right )}}{a^{9}}+\frac {59 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a^{9} \left (x -\frac {1}{a}\right )}+\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a^{9}}}{c^{3}}\)	\(471\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOS 
E)
 

Output:

-1/2*(a*x+2)*(a^2*x^2-1)/a^8/(-a^2*x^2+1)^(1/2)/c^3-(7/2/a^7/(a^2)^(1/2)*a 
rctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/20/a^11/(x-1/a)^3*(-(x-1/a)^2*a^ 
2-2*a*(x-1/a))^(1/2)+7/15/a^10/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2 
)+773/240/a^9/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-1/24/a^10/(x+1/a) 
^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+31/48/a^9/(x+1/a)*(-a^2*(x+1/a)^2+2* 
a*(x+1/a))^(1/2))/c^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\text {arctanh}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {96 \, a^{5} x^{5} - 96 \, a^{4} x^{4} - 192 \, a^{3} x^{3} + 192 \, a^{2} x^{2} + 96 \, a x + 210 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{6} x^{6} + 15 \, a^{5} x^{5} - 176 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 249 \, a^{2} x^{2} - 9 \, a x - 96\right )} \sqrt {-a^{2} x^{2} + 1} - 96}{30 \, {\left (a^{13} c^{3} x^{5} - a^{12} c^{3} x^{4} - 2 \, a^{11} c^{3} x^{3} + 2 \, a^{10} c^{3} x^{2} + a^{9} c^{3} x - a^{8} c^{3}\right )}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x, algorithm="fr 
icas")
 

Output:

1/30*(96*a^5*x^5 - 96*a^4*x^4 - 192*a^3*x^3 + 192*a^2*x^2 + 96*a*x + 210*( 
a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 
 + 1) - 1)/(a*x)) + (15*a^6*x^6 + 15*a^5*x^5 - 176*a^4*x^4 - 4*a^3*x^3 + 2 
49*a^2*x^2 - 9*a*x - 96)*sqrt(-a^2*x^2 + 1) - 96)/(a^13*c^3*x^5 - a^12*c^3 
*x^4 - 2*a^11*c^3*x^3 + 2*a^10*c^3*x^2 + a^9*c^3*x - a^8*c^3)
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{7}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{8}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**7/(-a**2*c*x**2+c)**3,x)
 

Output:

(Integral(x**7/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x 
**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + 
Integral(a*x**8/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2* 
x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/ 
c**3
 

Maxima [F]

\[ \int \frac {e^{\text {arctanh}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {{\left (a x + 1\right )} x^{7}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x, algorithm="ma 
xima")
 

Output:

-a*integrate(x^8/((a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*sqrt 
(a*x + 1)*sqrt(-a*x + 1)), x) + 1/5*(5*sqrt(-a^2*x^2 + 1)/c^3 + (5*a^2*x^2 
 + 15*(a^2*x^2 - 1)^2 - 4)/((-a^2*x^2 + 1)^(5/2)*c^3))/a^8
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,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 = 14.38 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.91 \[ \int \frac {e^{\text {arctanh}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^{10}\,c^3\,x^2+2\,a^9\,c^3\,x+a^8\,c^3\right )}-\frac {7\,\sqrt {1-a^2\,x^2}}{15\,\left (a^{10}\,c^3\,x^2-2\,a^9\,c^3\,x+a^8\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^6\,c^3\,\sqrt {-a^2}+3\,a^8\,c^3\,x^2\,\sqrt {-a^2}-a^9\,c^3\,x^3\,\sqrt {-a^2}-3\,a^7\,c^3\,x\,\sqrt {-a^2}\right )}+\frac {31\,\sqrt {1-a^2\,x^2}}{48\,\left (a^6\,c^3\,\sqrt {-a^2}+a^7\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {773\,\sqrt {1-a^2\,x^2}}{240\,\left (a^6\,c^3\,\sqrt {-a^2}-a^7\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{a^8\,c^3}+\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^7\,c^3}-\frac {7\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^7\,c^3\,\sqrt {-a^2}} \] Input:

int((x^7*(a*x + 1))/((c - a^2*c*x^2)^3*(1 - a^2*x^2)^(1/2)),x)
 

Output:

(1 - a^2*x^2)^(1/2)/(24*(a^8*c^3 + 2*a^9*c^3*x + a^10*c^3*x^2)) - (7*(1 - 
a^2*x^2)^(1/2))/(15*(a^8*c^3 - 2*a^9*c^3*x + a^10*c^3*x^2)) - (1 - a^2*x^2 
)^(1/2)/(20*(-a^2)^(1/2)*(a^6*c^3*(-a^2)^(1/2) + 3*a^8*c^3*x^2*(-a^2)^(1/2 
) - a^9*c^3*x^3*(-a^2)^(1/2) - 3*a^7*c^3*x*(-a^2)^(1/2))) + (31*(1 - a^2*x 
^2)^(1/2))/(48*(a^6*c^3*(-a^2)^(1/2) + a^7*c^3*x*(-a^2)^(1/2))*(-a^2)^(1/2 
)) - (773*(1 - a^2*x^2)^(1/2))/(240*(a^6*c^3*(-a^2)^(1/2) - a^7*c^3*x*(-a^ 
2)^(1/2))*(-a^2)^(1/2)) + (1 - a^2*x^2)^(1/2)/(a^8*c^3) + (x*(1 - a^2*x^2) 
^(1/2))/(2*a^7*c^3) - (7*asinh(x*(-a^2)^(1/2)))/(2*a^7*c^3*(-a^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\text {arctanh}(a x)} x^7}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{3} x^{3}+105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}+105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-9 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+9 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+9 \sqrt {-a^{2} x^{2}+1}\, a x -9 \sqrt {-a^{2} x^{2}+1}-15 a^{6} x^{6}-15 a^{5} x^{5}+176 a^{4} x^{4}+4 a^{3} x^{3}-249 a^{2} x^{2}+9 a x +96}{30 \sqrt {-a^{2} x^{2}+1}\, a^{8} c^{3} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^7/(-a^2*c*x^2+c)^3,x)
 

Output:

( - 105*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**3*x**3 + 105*sqrt( - a**2*x**2 
 + 1)*asin(a*x)*a**2*x**2 + 105*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 105 
*sqrt( - a**2*x**2 + 1)*asin(a*x) - 9*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 9 
*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 9*sqrt( - a**2*x**2 + 1)*a*x - 9*sqrt( 
 - a**2*x**2 + 1) - 15*a**6*x**6 - 15*a**5*x**5 + 176*a**4*x**4 + 4*a**3*x 
**3 - 249*a**2*x**2 + 9*a*x + 96)/(30*sqrt( - a**2*x**2 + 1)*a**8*c**3*(a* 
*3*x**3 - a**2*x**2 - a*x + 1))