Integrand size = 18, antiderivative size = 136 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx=-\frac {32 c^3 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {32 c^3 \sqrt {1-a^2 x^2}}{a}+\frac {67}{8} c^3 x \sqrt {1-a^2 x^2}+\frac {1}{4} a^2 c^3 x^3 \sqrt {1-a^2 x^2}+\frac {2 c^3 \left (1-a^2 x^2\right )^{3/2}}{a}-\frac {315 c^3 \arcsin (a x)}{8 a} \] Output:
-32*c^3*(-a*x+1)/a/(-a^2*x^2+1)^(1/2)-32*c^3*(-a^2*x^2+1)^(1/2)/a+67/8*c^3 *x*(-a^2*x^2+1)^(1/2)+1/4*a^2*c^3*x^3*(-a^2*x^2+1)^(1/2)+2*c^3*(-a^2*x^2+1 )^(3/2)/a-315/8*c^3*arcsin(a*x)/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.33 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx=-\frac {c^3 (1-a x)^{11/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {11}{2},\frac {13}{2},\frac {1}{2} (1-a x)\right )}{11 \sqrt {2} a} \] Input:
Integrate[(c - a*c*x)^3/E^(3*ArcTanh[a*x]),x]
Output:
-1/11*(c^3*(1 - a*x)^(11/2)*Hypergeometric2F1[3/2, 11/2, 13/2, (1 - a*x)/2 ])/(Sqrt[2]*a)
Time = 0.83 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {6677, 27, 462, 2346, 25, 2346, 27, 2346, 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.
| \(\displaystyle \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx\) |
| \(\Big \downarrow \) 6677 |
| \(\displaystyle \frac {\int \frac {c^6 (1-a x)^6}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^3 \int \frac {(1-a x)^6}{\left (1-a^2 x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 462 |
| \(\displaystyle c^3 \left (-\int \frac {a^4 x^4-6 a^3 x^3+16 a^2 x^2-26 a x+31}{\sqrt {1-a^2 x^2}}dx-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle c^3 \left (\frac {\int -\frac {-24 x^3 a^5+67 x^2 a^4-104 x a^3+124 a^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c^3 \left (-\frac {\int \frac {-24 x^3 a^5+67 x^2 a^4-104 x a^3+124 a^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle c^3 \left (-\frac {8 a^3 x^2 \sqrt {1-a^2 x^2}-\frac {\int -\frac {3 \left (67 x^2 a^6-120 x a^5+124 a^4\right )}{\sqrt {1-a^2 x^2}}dx}{3 a^2}}{4 a^2}-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^3 \left (-\frac {\frac {\int \frac {67 x^2 a^6-120 x a^5+124 a^4}{\sqrt {1-a^2 x^2}}dx}{a^2}+8 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle c^3 \left (-\frac {\frac {-\frac {\int -\frac {15 a^6 (21-16 a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {67}{2} a^4 x \sqrt {1-a^2 x^2}}{a^2}+8 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^3 \left (-\frac {\frac {\frac {15}{2} a^4 \int \frac {21-16 a x}{\sqrt {1-a^2 x^2}}dx-\frac {67}{2} a^4 x \sqrt {1-a^2 x^2}}{a^2}+8 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c^3 \left (-\frac {\frac {\frac {15}{2} a^4 \left (21 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {16 \sqrt {1-a^2 x^2}}{a}\right )-\frac {67}{2} a^4 x \sqrt {1-a^2 x^2}}{a^2}+8 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle c^3 \left (-\frac {32 (1-a x)}{a \sqrt {1-a^2 x^2}}+\frac {1}{4} a^2 x^3 \sqrt {1-a^2 x^2}-\frac {\frac {\frac {15}{2} a^4 \left (\frac {16 \sqrt {1-a^2 x^2}}{a}+\frac {21 \arcsin (a x)}{a}\right )-\frac {67}{2} a^4 x \sqrt {1-a^2 x^2}}{a^2}+8 a^3 x^2 \sqrt {1-a^2 x^2}}{4 a^2}\right )\) |
Input:
Int[(c - a*c*x)^3/E^(3*ArcTanh[a*x]),x]
Output:
c^3*((-32*(1 - a*x))/(a*Sqrt[1 - a^2*x^2]) + (a^2*x^3*Sqrt[1 - a^2*x^2])/4 - (8*a^3*x^2*Sqrt[1 - a^2*x^2] + ((-67*a^4*x*Sqrt[1 - a^2*x^2])/2 + (15*a ^4*((16*Sqrt[1 - a^2*x^2])/a + (21*ArcSin[a*x])/a))/2)/a^2)/(4*a^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp [(-2^(n - 1))*d*c^(n - 2)*((c + d*x)/(b*Sqrt[a + b*x^2])), x] + Simp[d^2/b Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(n - 1)*c^(n - 1) - (c + d*x)^(n - 1))/(c - d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[n, 2]
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]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[c^n Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(-\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+67 a x -240\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{8 a \sqrt {-a^{2} x^{2}+1}}+\left (-\frac {315 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}-\frac {32 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a^{2} \left (x +\frac {1}{a}\right )}\right ) c^{3}\) | \(120\) |
| default | \(-c^{3} \left (\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 {8 \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )}{a^{3}}+\frac {\frac {12 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+36 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{a^{2}}-\frac {6 \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{a}\right )\) | \(521\) |
Input:
int((-a*c*x+c)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/8*(2*a^3*x^3-16*a^2*x^2+67*a*x-240)*(a^2*x^2-1)/a/(-a^2*x^2+1)^(1/2)*c^ 3+(-315/8/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-32/a^2/(x+1 /a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))*c^3
Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx=-\frac {496 \, a c^{3} x + 496 \, c^{3} - 630 \, {\left (a c^{3} x + c^{3}\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{4} c^{3} x^{4} - 14 \, a^{3} c^{3} x^{3} + 51 \, a^{2} c^{3} x^{2} - 173 \, a c^{3} x - 496 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, {\left (a^{2} x + a\right )}} \] Input:
integrate((-a*c*x+c)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")
Output:
-1/8*(496*a*c^3*x + 496*c^3 - 630*(a*c^3*x + c^3)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^4*c^3*x^4 - 14*a^3*c^3*x^3 + 51*a^2*c^3*x^2 - 173*a* c^3*x - 496*c^3)*sqrt(-a^2*x^2 + 1))/(a^2*x + a)
\[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx=- c^{3} \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {3 a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \left (- \frac {2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx\right ) \] Input:
integrate((-a*c*x+c)**3/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)
Output:
-c**3*(Integral(-sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1 ), x) + Integral(3*a*x*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a *x + 1), x) + Integral(-2*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a* *2*x**2 + 3*a*x + 1), x) + Integral(-2*a**3*x**3*sqrt(-a**2*x**2 + 1)/(a** 3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(3*a**4*x**4*sqrt(-a**2*x* *2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-a**5*x**5*sq rt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.71 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx=-\frac {1}{4} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3} x + 3 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{3} x - \frac {3}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{3} x + \frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac {6 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a^{2} x + a} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{3}}{a} - \frac {3 i \, c^{3} \arcsin \left (a x + 2\right )}{a} - \frac {339 \, c^{3} \arcsin \left (a x\right )}{8 \, a} - \frac {48 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2} x + a} + \frac {6 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{3}}{a} - \frac {18 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a} \] Input:
integrate((-a*c*x+c)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")
Output:
-1/4*(-a^2*x^2 + 1)^(3/2)*c^3*x + 3*sqrt(a^2*x^2 + 4*a*x + 3)*c^3*x - 3/8* sqrt(-a^2*x^2 + 1)*c^3*x + 8*(-a^2*x^2 + 1)^(3/2)*c^3/(a^3*x^2 + 2*a^2*x + a) - 6*(-a^2*x^2 + 1)^(3/2)*c^3/(a^2*x + a) + 2*(-a^2*x^2 + 1)^(3/2)*c^3/ a - 3*I*c^3*arcsin(a*x + 2)/a - 339/8*c^3*arcsin(a*x)/a - 48*sqrt(-a^2*x^2 + 1)*c^3/(a^2*x + a) + 6*sqrt(a^2*x^2 + 4*a*x + 3)*c^3/a - 18*sqrt(-a^2*x ^2 + 1)*c^3/a
Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.76 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx=-\frac {315 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} - \frac {1}{8} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {240 \, c^{3}}{a} - {\left (67 \, c^{3} + 2 \, {\left (a^{2} c^{3} x - 8 \, a c^{3}\right )} x\right )} x\right )} + \frac {64 \, c^{3}}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \] Input:
integrate((-a*c*x+c)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
-315/8*c^3*arcsin(a*x)*sgn(a)/abs(a) - 1/8*sqrt(-a^2*x^2 + 1)*(240*c^3/a - (67*c^3 + 2*(a^2*c^3*x - 8*a*c^3)*x)*x) + 64*c^3/(((sqrt(-a^2*x^2 + 1)*ab s(a) + a)/(a^2*x) + 1)*abs(a))
Time = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.22 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx=\frac {32\,c^3\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {315\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {4\,a^3\,c^3}{{\left (-a^2\right )}^{3/2}}-\frac {67\,c^3\,x\,\sqrt {-a^2}}{8}-\frac {26\,a\,c^3}{\sqrt {-a^2}}+\frac {c^3\,x^3\,{\left (-a^2\right )}^{3/2}}{4}+\frac {2\,a^5\,c^3\,x^2}{{\left (-a^2\right )}^{3/2}}\right )}{\sqrt {-a^2}} \] Input:
int(((1 - a^2*x^2)^(3/2)*(c - a*c*x)^3)/(a*x + 1)^3,x)
Output:
(32*c^3*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/ 2)) - (315*c^3*asinh(x*(-a^2)^(1/2)))/(8*(-a^2)^(1/2)) - ((1 - a^2*x^2)^(1 /2)*((4*a^3*c^3)/(-a^2)^(3/2) - (67*c^3*x*(-a^2)^(1/2))/8 - (26*a*c^3)/(-a ^2)^(1/2) + (c^3*x^3*(-a^2)^(3/2))/4 + (2*a^5*c^3*x^2)/(-a^2)^(3/2)))/(-a^ 2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.33 \[ \int e^{-3 \text {arctanh}(a x)} (c-a c x)^3 \, dx=\frac {c^{3} \left (-315 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+315 \mathit {asin} \left (a x \right ) a x +315 \mathit {asin} \left (a x \right )-2 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+14 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-51 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+173 \sqrt {-a^{2} x^{2}+1}\, a x +646 \sqrt {-a^{2} x^{2}+1}-2 a^{5} x^{5}+16 a^{4} x^{4}-65 a^{3} x^{3}+224 a^{2} x^{2}+173 a x -646\right )}{8 a \left (\sqrt {-a^{2} x^{2}+1}-a x -1\right )} \] Input:
int((-a*c*x+c)^3/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
Output:
(c**3*( - 315*sqrt( - a**2*x**2 + 1)*asin(a*x) + 315*asin(a*x)*a*x + 315*a sin(a*x) - 2*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 14*sqrt( - a**2*x**2 + 1)* a**3*x**3 - 51*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 173*sqrt( - a**2*x**2 + 1)*a*x + 646*sqrt( - a**2*x**2 + 1) - 2*a**5*x**5 + 16*a**4*x**4 - 65*a**3 *x**3 + 224*a**2*x**2 + 173*a*x - 646))/(8*a*(sqrt( - a**2*x**2 + 1) - a*x - 1))