Integrand size = 24, antiderivative size = 187 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=-\frac {2}{3 a c \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {2}{a c^2 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2}{5 c \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (a+\frac {1}{x}\right )}+\frac {13}{15 a^2 c \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {41}{15 a^2 c^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{c^3}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a c^{5/2}} \] Output:
-2/3/a/c/(c-c/a^2/x^2)^(3/2)-2/a/c^2/(c-c/a^2/x^2)^(1/2)-2/5/c/(c-c/a^2/x^ 2)^(3/2)/(a+1/x)+13/15/a^2/c/(c-c/a^2/x^2)^(3/2)/x+41/15/a^2/c^2/(c-c/a^2/ x^2)^(1/2)/x-(c-c/a^2/x^2)^(1/2)*x/c^3+2*arctanh((c-c/a^2/x^2)^(1/2)/c^(1/ 2))/a/c^(5/2)
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {56+82 a x-32 a^2 x^2-76 a^3 x^3-15 a^4 x^4+30 (1+a x)^2 \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{15 a^2 c^2 \sqrt {c-\frac {c}{a^2 x^2}} x (1+a x)^2} \] Input:
Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(5/2)),x]
Output:
(56 + 82*a*x - 32*a^2*x^2 - 76*a^3*x^3 - 15*a^4*x^4 + 30*(1 + a*x)^2*Sqrt[ -1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(15*a^2*c^2*Sqrt[c - c/(a^2*x ^2)]*x*(1 + a*x)^2)
Time = 0.59 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6709, 570, 529, 25, 2166, 2345, 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.
\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6709 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \int \frac {x^5}{(a x+1)^2 \left (1-a^2 x^2\right )^{3/2}}dx}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 570 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \int \frac {x^5 (1-a x)^2}{\left (1-a^2 x^2\right )^{7/2}}dx}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 529 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(1-a x)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int -\frac {(1-a x) \left (\frac {5 x^4}{a}-\frac {5 x^3}{a^2}+\frac {5 x^2}{a^3}-\frac {5 x}{a^4}+\frac {2}{a^5}\right )}{\left (1-a^2 x^2\right )^{5/2}}dx\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \int \frac {(1-a x) \left (\frac {5 x^4}{a}-\frac {5 x^3}{a^2}+\frac {5 x^2}{a^3}-\frac {5 x}{a^4}+\frac {2}{a^5}\right )}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {(1-a x)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 2166 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \left (-\frac {1}{3} \int \frac {-\frac {15 x^3}{a^2}+\frac {30 x^2}{a^3}-\frac {45 x}{a^4}+\frac {16}{a^5}}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {22 (1-a x)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(1-a x)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {15 (2-a x)}{a^5 \sqrt {1-a^2 x^2}}dx+\frac {2 (30-23 a x)}{a^6 \sqrt {1-a^2 x^2}}\right )-\frac {22 (1-a x)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(1-a x)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {15 \int \frac {2-a x}{\sqrt {1-a^2 x^2}}dx}{a^5}+\frac {2 (30-23 a x)}{a^6 \sqrt {1-a^2 x^2}}\right )-\frac {22 (1-a x)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(1-a x)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {15 \left (2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^5}+\frac {2 (30-23 a x)}{a^6 \sqrt {1-a^2 x^2}}\right )-\frac {22 (1-a x)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(1-a x)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{5/2} \left (\frac {(1-a x)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}+\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (30-23 a x)}{a^6 \sqrt {1-a^2 x^2}}+\frac {15 \left (\frac {\sqrt {1-a^2 x^2}}{a}+\frac {2 \arcsin (a x)}{a}\right )}{a^5}\right )-\frac {22 (1-a x)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )\right )}{x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}\) |
Input:
Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(5/2)),x]
Output:
((1 - a^2*x^2)^(5/2)*((1 - a*x)^2/(5*a^6*(1 - a^2*x^2)^(5/2)) + ((-22*(1 - a*x))/(3*a^6*(1 - a^2*x^2)^(3/2)) + ((2*(30 - 23*a*x))/(a^6*Sqrt[1 - a^2* x^2]) + (15*(Sqrt[1 - a^2*x^2]/a + (2*ArcSin[a*x])/a))/a^5)/3)/5))/((c - c /(a^2*x^2))^(5/2)*x^5)
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[(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[(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[(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]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
Time = 0.20 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.54
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a^{2} c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}-\frac {\left (-\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{5} \sqrt {a^{2} c}}+\frac {383 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{120 a^{7} c \left (x +\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +2 \left (x -\frac {1}{a}\right ) a c}}{8 a^{7} c \left (x -\frac {1}{a}\right )}+\frac {\sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{10 a^{9} c \left (x +\frac {1}{a}\right )^{3}}-\frac {41 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{60 a^{8} c \left (x +\frac {1}{a}\right )^{2}}\right ) a^{4} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c^{2} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}\) | \(288\) |
default | \(\frac {\left (-15 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} c^{\frac {5}{2}} a^{5} x^{5}-16 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}} a^{4} x^{4}-45 x^{4} c^{\frac {5}{2}} a^{4} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}}-16 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}} a^{3} x^{3}+60 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} c^{\frac {5}{2}} a^{3} x^{3}+30 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{4} c x +24 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}} a^{2} x^{2}+90 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} c^{\frac {5}{2}} a^{2} x^{2}+30 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a^{3} c +24 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}} a x -50 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} c^{\frac {5}{2}} a x -6 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} c^{\frac {5}{2}}-50 c^{\frac {5}{2}} \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}}\right ) \left (a x -1\right )}{15 \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{5} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}} a^{6} c^{\frac {5}{2}}}\) | \(462\) |
Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/a^2*(a^2*x^2-1)/c^2/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x-(-2/a^5*ln(a^2*c*x/ (a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)+383/120/a^7/c/(x+1/a)*((x +1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2)-1/8/a^7/c/(x-1/a)*((x-1/a)^2*a^2*c+2*(x -1/a)*a*c)^(1/2)+1/10/a^9/c/(x+1/a)^3*((x+1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2 )-41/60/a^8/c/(x+1/a)^2*((x+1/a)^2*a^2*c-2*(x+1/a)*a*c)^(1/2))*a^4/c^2*(c* (a^2*x^2-1))^(1/2)/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x
Time = 0.11 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.88 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (15 \, a^{5} x^{5} + 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} - 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}}, -\frac {30 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (15 \, a^{5} x^{5} + 76 \, a^{4} x^{4} + 32 \, a^{3} x^{3} - 82 \, a^{2} x^{2} - 56 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{15 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x, algorithm="frica s")
Output:
[1/15*(15*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*sqrt(c)*log(2*a^2*c*x^2 + 2*a^ 2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (15*a^5*x^5 + 76*a^4* x^4 + 32*a^3*x^3 - 82*a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/( a^5*c^3*x^4 + 2*a^4*c^3*x^3 - 2*a^2*c^3*x - a*c^3), -1/15*(30*(a^4*x^4 + 2 *a^3*x^3 - 2*a*x - 1)*sqrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c )/(a^2*x^2))/(a^2*c*x^2 - c)) + (15*a^5*x^5 + 76*a^4*x^4 + 32*a^3*x^3 - 82 *a^2*x^2 - 56*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^5*c^3*x^4 + 2*a^4*c ^3*x^3 - 2*a^2*c^3*x - a*c^3)]
\[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=- \int \frac {a x}{a c^{2} x \sqrt {c - \frac {c}{a^{2} x^{2}}} + c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {2 c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} - \frac {2 c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac {c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{3} x^{3}} + \frac {c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{4} x^{4}}}\, dx - \int \left (- \frac {1}{a c^{2} x \sqrt {c - \frac {c}{a^{2} x^{2}}} + c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {2 c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} - \frac {2 c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac {c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{3} x^{3}} + \frac {c^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{4} x^{4}}}\right )\, dx \] Input:
integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a**2/x**2)**(5/2),x)
Output:
-Integral(a*x/(a*c**2*x*sqrt(c - c/(a**2*x**2)) + c**2*sqrt(c - c/(a**2*x* *2)) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a*x) - 2*c**2*sqrt(c - c/(a**2*x**2 ))/(a**2*x**2) + c**2*sqrt(c - c/(a**2*x**2))/(a**3*x**3) + c**2*sqrt(c - c/(a**2*x**2))/(a**4*x**4)), x) - Integral(-1/(a*c**2*x*sqrt(c - c/(a**2*x **2)) + c**2*sqrt(c - c/(a**2*x**2)) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a*x ) - 2*c**2*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + c**2*sqrt(c - c/(a**2*x** 2))/(a**3*x**3) + c**2*sqrt(c - c/(a**2*x**2))/(a**4*x**4)), x)
\[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\int { -\frac {a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x, algorithm="maxim a")
Output:
-integrate((a^2*x^2 - 1)/((a*x + 1)^2*(c - c/(a^2*x^2))^(5/2)), x)
Exception generated. \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),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
Timed out. \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=-\int \frac {a^2\,x^2-1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,{\left (a\,x+1\right )}^2} \,d x \] Input:
int(-(a^2*x^2 - 1)/((c - c/(a^2*x^2))^(5/2)*(a*x + 1)^2),x)
Output:
-int((a^2*x^2 - 1)/((c - c/(a^2*x^2))^(5/2)*(a*x + 1)^2), x)
Time = 0.16 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (-30 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-152 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-64 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+164 \sqrt {a^{2} x^{2}-1}\, a x +112 \sqrt {a^{2} x^{2}-1}+60 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{4} x^{4}+120 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{3} x^{3}-120 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a x -60 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )+47 a^{4} x^{4}+94 a^{3} x^{3}-94 a x -47\right )}{30 a \,c^{3} \left (a^{4} x^{4}+2 a^{3} x^{3}-2 a x -1\right )} \] Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a^2/x^2)^(5/2),x)
Output:
(sqrt(c)*( - 30*sqrt(a**2*x**2 - 1)*a**4*x**4 - 152*sqrt(a**2*x**2 - 1)*a* *3*x**3 - 64*sqrt(a**2*x**2 - 1)*a**2*x**2 + 164*sqrt(a**2*x**2 - 1)*a*x + 112*sqrt(a**2*x**2 - 1) + 60*log(sqrt(a**2*x**2 - 1) + a*x)*a**4*x**4 + 1 20*log(sqrt(a**2*x**2 - 1) + a*x)*a**3*x**3 - 120*log(sqrt(a**2*x**2 - 1) + a*x)*a*x - 60*log(sqrt(a**2*x**2 - 1) + a*x) + 47*a**4*x**4 + 94*a**3*x* *3 - 94*a*x - 47))/(30*a*c**3*(a**4*x**4 + 2*a**3*x**3 - 2*a*x - 1))