Integrand size = 22, antiderivative size = 159 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{a c^3}+\frac {8 \sqrt {1-a^2 x^2}}{a c^3 (1-a x)}-\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^6}+\frac {4 \left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{5/2}}{a c^3 (1-a x)^4}-\frac {6 \arcsin (a x)}{a c^3} \] Output:
2*(-a^2*x^2+1)^(1/2)/a/c^3+8*(-a^2*x^2+1)^(1/2)/a/c^3/(-a*x+1)-1/7*(-a^2*x ^2+1)^(5/2)/a/c^3/(-a*x+1)^6+4/7*(-a^2*x^2+1)^(5/2)/a/c^3/(-a*x+1)^5-(-a^2 *x^2+1)^(5/2)/a/c^3/(-a*x+1)^4-6*arcsin(a*x)/a/c^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.50 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {(1+a x)^{5/2} \left (-10+18 a x-7 a^2 x^2\right )+56 \sqrt {2} (-1+a x)^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1-a x)\right )}{7 a c^3 (1-a x)^{7/2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])/(c - c/(a*x))^3,x]
Output:
-1/7*((1 + a*x)^(5/2)*(-10 + 18*a*x - 7*a^2*x^2) + 56*Sqrt[2]*(-1 + a*x)^2 *Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(a*c^3*(1 - a*x)^(7/2))
Time = 0.62 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6681, 6678, 570, 529, 2166, 27, 669, 462, 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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx\) |
| \(\Big \downarrow \) 6681 |
| \(\displaystyle -\frac {a^3 \int \frac {e^{3 \text {arctanh}(a x)} x^3}{(1-a x)^3}dx}{c^3}\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle -\frac {a^3 \int \frac {x^3 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^6}dx}{c^3}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {a^3 \int \frac {x^3 (a x+1)^6}{\left (1-a^2 x^2\right )^{9/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle -\frac {a^3 \left (\frac {(a x+1)^6}{7 a^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{7} \int \frac {(a x+1)^5 \left (\frac {7 x^2}{a}+\frac {7 x}{a^2}+\frac {6}{a^3}\right )}{\left (1-a^2 x^2\right )^{7/2}}dx\right )}{c^3}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {35 (a x+1)^4 (a x+2)}{a^3 \left (1-a^2 x^2\right )^{5/2}}dx-\frac {4 (a x+1)^5}{a^4 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^6}{7 a^4 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{7} \left (\frac {7 \int \frac {(a x+1)^4 (a x+2)}{\left (1-a^2 x^2\right )^{5/2}}dx}{a^3}-\frac {4 (a x+1)^5}{a^4 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^6}{7 a^4 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\) |
| \(\Big \downarrow \) 669 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{7} \left (\frac {7 \left (\frac {(a x+1)^4}{a \left (1-a^2 x^2\right )^{3/2}}-2 \int \frac {(a x+1)^3}{\left (1-a^2 x^2\right )^{3/2}}dx\right )}{a^3}-\frac {4 (a x+1)^5}{a^4 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^6}{7 a^4 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\) |
| \(\Big \downarrow \) 462 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{7} \left (\frac {7 \left (\frac {(a x+1)^4}{a \left (1-a^2 x^2\right )^{3/2}}-2 \left (\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}-\int \frac {a x+3}{\sqrt {1-a^2 x^2}}dx\right )\right )}{a^3}-\frac {4 (a x+1)^5}{a^4 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^6}{7 a^4 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{7} \left (\frac {7 \left (\frac {(a x+1)^4}{a \left (1-a^2 x^2\right )^{3/2}}-2 \left (-3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a}\right )\right )}{a^3}-\frac {4 (a x+1)^5}{a^4 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^6}{7 a^4 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {a^3 \left (\frac {(a x+1)^6}{7 a^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {1}{7} \left (\frac {7 \left (\frac {(a x+1)^4}{a \left (1-a^2 x^2\right )^{3/2}}-2 \left (\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a}-\frac {3 \arcsin (a x)}{a}\right )\right )}{a^3}-\frac {4 (a x+1)^5}{a^4 \left (1-a^2 x^2\right )^{5/2}}\right )\right )}{c^3}\) |
Input:
Int[E^(3*ArcTanh[a*x])/(c - c/(a*x))^3,x]
Output:
-((a^3*((1 + a*x)^6/(7*a^4*(1 - a^2*x^2)^(7/2)) + ((-4*(1 + a*x)^5)/(a^4*( 1 - a^2*x^2)^(5/2)) + (7*((1 + a*x)^4/(a*(1 - a^2*x^2)^(3/2)) - 2*((4*(1 + a*x))/(a*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/a - (3*ArcSin[a*x])/a)))/ a^3)/7))/c^3)
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[(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]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^( p_), x_Symbol] :> Simp[(d*g + e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d* (p + 1))), x] - Simp[e*((m*(d*g + e*f) + 2*e*f*(p + 1))/(2*c*d*(p + 1))) Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
Time = 0.35 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.47
| method | result | size |
| risch | \(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{3}}-\frac {\left (\frac {6 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {4 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a^{8} \left (x -\frac {1}{a}\right )^{4}}+\frac {20 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a^{7} \left (x -\frac {1}{a}\right )^{3}}+\frac {45 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a^{6} \left (x -\frac {1}{a}\right )^{2}}+\frac {88 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a^{5} \left (x -\frac {1}{a}\right )}\right ) a^{3}}{c^{3}}\) | \(234\) |
| default | \(\frac {a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {20}{a^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {38 x}{a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {36}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {108 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{5}}+\frac {\frac {6 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {6 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}}{a}+\frac {\frac {8}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {32 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}}{a^{6}}+\frac {\frac {22}{a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {22 \left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right )}{a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{4}}\right )}{c^{3}}\) | \(537\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^3,x,method=_RETURNVERBOSE)
Output:
-1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^3-(6/a^3/(a^2)^(1/2)*arctan((a^2)^(1 /2)*x/(-a^2*x^2+1)^(1/2))+4/7/a^8/(x-1/a)^4*(-(x-1/a)^2*a^2-2*a*(x-1/a))^( 1/2)+20/7/a^7/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+45/7/a^6/(x-1/a )^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+88/7/a^5/(x-1/a)*(-(x-1/a)^2*a^2-2* a*(x-1/a))^(1/2))*a^3/c^3
Time = 0.09 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {66 \, a^{4} x^{4} - 264 \, a^{3} x^{3} + 396 \, a^{2} x^{2} - 264 \, a x + 84 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (7 \, a^{4} x^{4} - 116 \, a^{3} x^{3} + 261 \, a^{2} x^{2} - 222 \, a x + 66\right )} \sqrt {-a^{2} x^{2} + 1} + 66}{7 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^3,x, algorithm="fricas")
Output:
1/7*(66*a^4*x^4 - 264*a^3*x^3 + 396*a^2*x^2 - 264*a*x + 84*(a^4*x^4 - 4*a^ 3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (7 *a^4*x^4 - 116*a^3*x^3 + 261*a^2*x^2 - 222*a*x + 66)*sqrt(-a^2*x^2 + 1) + 66)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {a^{3} \left (\int \frac {x^{3}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{4}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{5}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{6}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{3}} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)**3,x)
Output:
a**3*(Integral(x**3/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a **2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2* x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integ ral(3*a*x**4/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x** 2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a **2*x**5/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x* *6/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2 *a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x *sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{3}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^3,x, algorithm="maxima")
Output:
integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a*x))^3), x)
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^3,x, algorithm="giac")
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
Time = 15.43 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.14 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {\sqrt {1-a^2\,x^2}}{a\,c^3}-\frac {6\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^3\,\sqrt {-a^2}}-\frac {8\,a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^3\,x^2-2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {31\,a\,\sqrt {1-a^2\,x^2}}{5\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^3\,x^4-4\,a^5\,c^3\,x^3+6\,a^4\,c^3\,x^2-4\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {88\,\sqrt {1-a^2\,x^2}}{7\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {20\,\sqrt {1-a^2\,x^2}}{7\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )} \] Input:
int((a*x + 1)^3/((c - c/(a*x))^3*(1 - a^2*x^2)^(3/2)),x)
Output:
(1 - a^2*x^2)^(1/2)/(a*c^3) - (6*asinh(x*(-a^2)^(1/2)))/(c^3*(-a^2)^(1/2)) - (8*a^3*(1 - a^2*x^2)^(1/2))/(35*(a^4*c^3 - 2*a^5*c^3*x + a^6*c^3*x^2)) - (31*a*(1 - a^2*x^2)^(1/2))/(5*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) - ( 4*a*(1 - a^2*x^2)^(1/2))/(7*(a^2*c^3 - 4*a^3*c^3*x + 6*a^4*c^3*x^2 - 4*a^5 *c^3*x^3 + a^6*c^3*x^4)) + (88*(1 - a^2*x^2)^(1/2))/(7*(-a^2)^(1/2)*(c^3*x *(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a)) + (20*(1 - a^2*x^2)^(1/2))/(7*(-a^2 )^(1/2)*(3*c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^ (1/2) - 3*a*c^3*x^2*(-a^2)^(1/2)))
Time = 0.15 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.55 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {21 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{4} x^{4}-84 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{3} x^{3}+126 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{2} x^{2}-84 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a x +21 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right )+7 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-116 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+261 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-222 \sqrt {-a^{2} x^{2}+1}\, a x +66 \sqrt {-a^{2} x^{2}+1}}{7 a \,c^{3} \left (a^{4} x^{4}-4 a^{3} x^{3}+6 a^{2} x^{2}-4 a x +1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^3,x)
Output:
(21*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2* a**3*x**3 - 2*a*x))*a**4*x**4 - 84*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x** 2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x))*a**3*x**3 + 126*atan((2 *sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x))*a**2*x**2 - 84*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x))*a*x + 21*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x)) + 7*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 116*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 261*s qrt( - a**2*x**2 + 1)*a**2*x**2 - 222*sqrt( - a**2*x**2 + 1)*a*x + 66*sqrt ( - a**2*x**2 + 1))/(7*a*c**3*(a**4*x**4 - 4*a**3*x**3 + 6*a**2*x**2 - 4*a *x + 1))