Integrand size = 23, antiderivative size = 151 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx=\frac {a^2 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5 a^2}{2 c^2 \sqrt {1-a^2 x^2}}-\frac {1}{2 c^2 x^2 \sqrt {1-a^2 x^2}}-\frac {a}{c^2 x \sqrt {1-a^2 x^2}}+\frac {8 a^3 x}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {5 a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 c^2} \] Output:
1/3*a^2*(a*x+1)/c^2/(-a^2*x^2+1)^(3/2)+5/2*a^2/c^2/(-a^2*x^2+1)^(1/2)-1/2/ c^2/x^2/(-a^2*x^2+1)^(1/2)-a/c^2/x/(-a^2*x^2+1)^(1/2)+8/3*a^3*x/c^2/(-a^2* x^2+1)^(1/2)-5/2*a^2*arctanh((-a^2*x^2+1)^(1/2))/c^2
Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx=\frac {3+3 a x-23 a^2 x^2-a^3 x^3+16 a^4 x^4-15 a^2 x^2 (-1+a x) \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{6 c^2 x^2 (-1+a x) \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^2),x]
Output:
(3 + 3*a*x - 23*a^2*x^2 - a^3*x^3 + 16*a^4*x^4 - 15*a^2*x^2*(-1 + a*x)*Sqr t[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(6*c^2*x^2*(-1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.92 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6698, 532, 25, 2336, 27, 2338, 25, 27, 534, 243, 73, 221}
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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {a x+1}{x^3 \left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int -\frac {2 a^3 x^3+3 a^2 x^2+3 a x+3}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{3} \int \frac {2 a^3 x^3+3 a^2 x^2+3 a x+3}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 2336 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}-\int -\frac {3 \left (2 a^2 x^2+a x+1\right )}{x^3 \sqrt {1-a^2 x^2}}dx\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \int \frac {2 a^2 x^2+a x+1}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \left (-\frac {1}{2} \int -\frac {a (5 a x+2)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{2} \int \frac {a (5 a x+2)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{2} a \int \frac {5 a x+2}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{2} a \left (5 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{2} a \left (\frac {5}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{2} a \left (-\frac {5 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \left (\frac {1}{2} a \left (-5 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )-\frac {\sqrt {1-a^2 x^2}}{2 x^2}\right )+\frac {a^2 (5 a x+6)}{\sqrt {1-a^2 x^2}}\right )+\frac {a^2 (a x+1)}{3 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
Input:
Int[E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^2),x]
Output:
((a^2*(1 + a*x))/(3*(1 - a^2*x^2)^(3/2)) + ((a^2*(6 + 5*a*x))/Sqrt[1 - a^2 *x^2] + 3*(-1/2*Sqrt[1 - a^2*x^2]/x^2 + (a*((-2*Sqrt[1 - a^2*x^2])/x - 5*a *ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/3)/c^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*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)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
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]
Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {5 a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {a \sqrt {-a^{2} x^{2}+1}}{x}+\frac {a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{2}-\frac {7 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 \left (x -\frac {1}{a}\right )}+\frac {a \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 x +\frac {4}{a}}}{c^{2}}\) | \(214\) |
risch | \(\frac {2 a^{3} x^{3}+a^{2} x^{2}-2 a x -1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {a^{2} \left (\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}-5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {7 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{2 a \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{2 a \left (x +\frac {1}{a}\right )}\right )}{2 c^{2}}\) | \(227\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^2,x,method=_RETURNVERBOS E)
Output:
1/c^2*(-1/2*(-a^2*x^2+1)^(1/2)/x^2-5/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))-a *(-a^2*x^2+1)^(1/2)/x+1/2*a*(1/3/a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^ (1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))-7/4*a/(x-1/a)*(-(x-1 /a)^2*a^2-2*a*(x-1/a))^(1/2)+1/4*a/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1 /2))
Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.13 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx=\frac {14 \, a^{5} x^{5} - 14 \, a^{4} x^{4} - 14 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 15 \, {\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (16 \, a^{4} x^{4} - a^{3} x^{3} - 23 \, a^{2} x^{2} + 3 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{3} c^{2} x^{5} - a^{2} c^{2} x^{4} - a c^{2} x^{3} + c^{2} x^{2}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^2,x, algorithm="fr icas")
Output:
1/6*(14*a^5*x^5 - 14*a^4*x^4 - 14*a^3*x^3 + 14*a^2*x^2 + 15*(a^5*x^5 - a^4 *x^4 - a^3*x^3 + a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (16*a^4*x^4 - a^3*x^3 - 23*a^2*x^2 + 3*a*x + 3)*sqrt(-a^2*x^2 + 1))/(a^3*c^2*x^5 - a^2*c ^2*x^4 - a*c^2*x^3 + c^2*x^2)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {a}{a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a**2*c*x**2+c)**2,x)
Output:
(Integral(a/(a**4*x**6*sqrt(-a**2*x**2 + 1) - 2*a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**7*sqrt(-a**2*x **2 + 1) - 2*a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c**2
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^2,x, algorithm="ma xima")
Output:
integrate((a*x + 1)/((a^2*c*x^2 - c)^2*sqrt(-a^2*x^2 + 1)*x^3), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^2,x, algorithm="gi ac")
Output:
integrate((a*x + 1)/((a^2*c*x^2 - c)^2*sqrt(-a^2*x^2 + 1)*x^3), x)
Time = 24.31 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.46 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx=\frac {a^4\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^2\,x^2}-\frac {a\,\sqrt {1-a^2\,x^2}}{c^2\,x}-\frac {a^3\,\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}+\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {23\,a^3\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{2\,c^2} \] Input:
int((a*x + 1)/(x^3*(c - a^2*c*x^2)^2*(1 - a^2*x^2)^(1/2)),x)
Output:
(a^2*atan((1 - a^2*x^2)^(1/2)*1i)*5i)/(2*c^2) + (a^4*(1 - a^2*x^2)^(1/2))/ (6*(a^2*c^2 - 2*a^3*c^2*x + a^4*c^2*x^2)) - (1 - a^2*x^2)^(1/2)/(2*c^2*x^2 ) - (a*(1 - a^2*x^2)^(1/2))/(c^2*x) - (a^3*(1 - a^2*x^2)^(1/2))/(4*(-a^2)^ (1/2)*(c^2*x*(-a^2)^(1/2) + (c^2*(-a^2)^(1/2))/a)) + (23*a^3*(1 - a^2*x^2) ^(1/2))/(12*(-a^2)^(1/2)*(c^2*x*(-a^2)^(1/2) - (c^2*(-a^2)^(1/2))/a))
Time = 0.15 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.94 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx=\frac {-32 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+2 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+46 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-6 \sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}+15 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{5} x^{5}-15 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{4} x^{4}-15 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{3} x^{3}+15 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{2} x^{2}-15 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{5} x^{5}+15 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{4} x^{4}+15 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{3} x^{3}-15 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{2} x^{2}}{12 c^{2} x^{2} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^2,x)
Output:
( - 32*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 2*sqrt( - a**2*x**2 + 1)*a**3*x* *3 + 46*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 6*sqrt( - a**2*x**2 + 1)*a*x - 6*sqrt( - a**2*x**2 + 1) + 15*log(sqrt( - a**2*x**2 + 1) - 1)*a**5*x**5 - 15*log(sqrt( - a**2*x**2 + 1) - 1)*a**4*x**4 - 15*log(sqrt( - a**2*x**2 + 1) - 1)*a**3*x**3 + 15*log(sqrt( - a**2*x**2 + 1) - 1)*a**2*x**2 - 15*log( sqrt( - a**2*x**2 + 1) + 1)*a**5*x**5 + 15*log(sqrt( - a**2*x**2 + 1) + 1) *a**4*x**4 + 15*log(sqrt( - a**2*x**2 + 1) + 1)*a**3*x**3 - 15*log(sqrt( - a**2*x**2 + 1) + 1)*a**2*x**2)/(12*c**2*x**2*(a**3*x**3 - a**2*x**2 - a*x + 1))