Integrand size = 27, antiderivative size = 150 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx=-\frac {1}{2} a^3 c^2 (6+a x) \sqrt {c-a^2 c x^2}-\frac {a c (3+a x) \left (c-a^2 c x^2\right )^{3/2}}{3 x^2}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 x^3}-\frac {1}{2} a^3 c^{5/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+3 a^3 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \] Output:
-1/2*a^3*c^2*(a*x+6)*(-a^2*c*x^2+c)^(1/2)-1/3*a*c*(a*x+3)*(-a^2*c*x^2+c)^( 3/2)/x^2-1/3*(-a^2*c*x^2+c)^(5/2)/x^3-1/2*a^3*c^(5/2)*arctan(a*c^(1/2)*x/( -a^2*c*x^2+c)^(1/2))+3*a^3*c^(5/2)*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))
Time = 0.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx=-\frac {c^2 \sqrt {c-a^2 c x^2} \left (2+6 a x-2 a^2 x^2+12 a^3 x^3+3 a^4 x^4\right )}{6 x^3}+\frac {1}{2} a^3 c^{5/2} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )-3 a^3 c^{5/2} \log (x)+3 a^3 c^{5/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \] Input:
Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2))/x^4,x]
Output:
-1/6*(c^2*Sqrt[c - a^2*c*x^2]*(2 + 6*a*x - 2*a^2*x^2 + 12*a^3*x^3 + 3*a^4* x^4))/x^3 + (a^3*c^(5/2)*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a ^2*x^2))])/2 - 3*a^3*c^(5/2)*Log[x] + 3*a^3*c^(5/2)*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]
Time = 0.51 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6701, 540, 25, 27, 537, 27, 535, 538, 224, 216, 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.
\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle c \int \frac {(a x+1)^2 \left (c-a^2 c x^2\right )^{3/2}}{x^4}dx\) |
\(\Big \downarrow \) 540 |
\(\displaystyle c \left (-\frac {\int -\frac {a c (a x+6) \left (c-a^2 c x^2\right )^{3/2}}{x^3}dx}{3 c}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c \left (\frac {\int \frac {a c (a x+6) \left (c-a^2 c x^2\right )^{3/2}}{x^3}dx}{3 c}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c \left (\frac {1}{3} a \int \frac {(a x+6) \left (c-a^2 c x^2\right )^{3/2}}{x^3}dx-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 537 |
\(\displaystyle c \left (\frac {1}{3} a \left (\frac {3}{2} a^2 c \int -\frac {2 (a x+3) \sqrt {c-a^2 c x^2}}{x}dx-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c \left (\frac {1}{3} a \left (-3 a^2 c \int \frac {(a x+3) \sqrt {c-a^2 c x^2}}{x}dx-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 535 |
\(\displaystyle c \left (\frac {1}{3} a \left (-3 a^2 c \left (\frac {1}{2} c \int \frac {a x+6}{x \sqrt {c-a^2 c x^2}}dx+\frac {1}{2} (a x+6) \sqrt {c-a^2 c x^2}\right )-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle c \left (\frac {1}{3} a \left (-3 a^2 c \left (\frac {1}{2} c \left (a \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+6 \int \frac {1}{x \sqrt {c-a^2 c x^2}}dx\right )+\frac {1}{2} (a x+6) \sqrt {c-a^2 c x^2}\right )-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle c \left (\frac {1}{3} a \left (-3 a^2 c \left (\frac {1}{2} c \left (6 \int \frac {1}{x \sqrt {c-a^2 c x^2}}dx+a \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}\right )+\frac {1}{2} (a x+6) \sqrt {c-a^2 c x^2}\right )-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle c \left (\frac {1}{3} a \left (-3 a^2 c \left (\frac {1}{2} c \left (6 \int \frac {1}{x \sqrt {c-a^2 c x^2}}dx+\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}\right )+\frac {1}{2} (a x+6) \sqrt {c-a^2 c x^2}\right )-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c \left (\frac {1}{3} a \left (-3 a^2 c \left (\frac {1}{2} c \left (3 \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2+\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}\right )+\frac {1}{2} (a x+6) \sqrt {c-a^2 c x^2}\right )-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c \left (\frac {1}{3} a \left (-3 a^2 c \left (\frac {1}{2} c \left (\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}-\frac {6 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}}{a^2 c}\right )+\frac {1}{2} (a x+6) \sqrt {c-a^2 c x^2}\right )-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c \left (\frac {1}{3} a \left (-3 a^2 c \left (\frac {1}{2} c \left (\frac {\arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}-\frac {6 \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}\right )+\frac {1}{2} (a x+6) \sqrt {c-a^2 c x^2}\right )-\frac {(a x+3) \left (c-a^2 c x^2\right )^{3/2}}{x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{3 c x^3}\right )\) |
Input:
Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2))/x^4,x]
Output:
c*(-1/3*(c - a^2*c*x^2)^(5/2)/(c*x^3) + (a*(-(((3 + a*x)*(c - a^2*c*x^2)^( 3/2))/x^2) - 3*a^2*c*(((6 + a*x)*Sqrt[c - a^2*c*x^2])/2 + (c*(ArcTan[(a*Sq rt[c]*x)/Sqrt[c - a^2*c*x^2]]/Sqrt[c] - (6*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqr t[c]])/Sqrt[c]))/2)))/3)
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p + 1) Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), x] - Simp[2*b*(p/((m + 1)*(m + 2))) Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) *x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && GtQ[p, 0] && !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^(n/2) Int[x^m*(c + d*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/2, 0]
Time = 0.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\frac {\left (a^{4} x^{4}-3 a^{3} x^{3}-2 a^{2} x^{2}+3 a x +1\right ) c^{3}}{3 x^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\left (-\frac {a^{4} x \sqrt {-a^{2} c \,x^{2}+c}}{2 c}-\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}+\frac {3 a^{3} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}-\frac {2 a^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{3}\) | \(172\) |
default | \(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{3 c \,x^{3}}+\frac {2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{c x}-6 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )\right )}{3}+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}}-\frac {5 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )\right )}{2}\right )+2 a^{3} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )\right )-2 a^{3} \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{5}-a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )\right )\) | \(549\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^4,x,method=_RETURNVERBOS E)
Output:
-1/3*(a^4*x^4-3*a^3*x^3-2*a^2*x^2+3*a*x+1)/x^3/(-c*(a^2*x^2-1))^(1/2)*c^3+ (-1/2*a^4*x/c*(-a^2*c*x^2+c)^(1/2)-1/2*a^4/(a^2*c)^(1/2)*arctan((a^2*c)^(1 /2)*x/(-a^2*c*x^2+c)^(1/2))+3*a^3/c^(1/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c) ^(1/2))/x)-2*a^3/c*(-c*(a^2*x^2-1))^(1/2))*c^3
Time = 0.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.12 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx=\left [\frac {3 \, a^{3} c^{\frac {5}{2}} x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + 9 \, a^{3} c^{\frac {5}{2}} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - {\left (3 \, a^{4} c^{2} x^{4} + 12 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 6 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{6 \, x^{3}}, -\frac {36 \, a^{3} \sqrt {-c} c^{2} x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) - 3 \, a^{3} \sqrt {-c} c^{2} x^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (3 \, a^{4} c^{2} x^{4} + 12 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 6 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{12 \, x^{3}}\right ] \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^4,x, algorithm="fr icas")
Output:
[1/6*(3*a^3*c^(5/2)*x^3*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 9*a^3*c^(5/2)*x^3*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c ) - 2*c)/x^2) - (3*a^4*c^2*x^4 + 12*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + 6*a*c^2* x + 2*c^2)*sqrt(-a^2*c*x^2 + c))/x^3, -1/12*(36*a^3*sqrt(-c)*c^2*x^3*arcta n(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/c) - 3*a^3*sqrt(-c)*c^2*x^3*log(2*a^2*c*x^ 2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(3*a^4*c^2*x^4 + 12*a^3*c ^2*x^3 - 2*a^2*c^2*x^2 + 6*a*c^2*x + 2*c^2)*sqrt(-a^2*c*x^2 + c))/x^3]
Result contains complex when optimal does not.
Time = 8.18 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.12 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx=- a^{4} c^{2} \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {- a^{2} c x^{2} + c}}{2} & \text {for}\: a^{2} c \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{2} \left (\begin {cases} i \sqrt {c} \sqrt {a^{2} x^{2} - 1} - \sqrt {c} \log {\left (a x \right )} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} + i \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {c} \sqrt {- a^{2} x^{2} + 1} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} - \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right ) + 2 a c^{2} \left (\begin {cases} \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(5/2)/x**4,x)
Output:
-a**4*c**2*Piecewise((c*Piecewise((log(-2*a**2*c*x + 2*sqrt(-a**2*c)*sqrt( -a**2*c*x**2 + c))/sqrt(-a**2*c), Ne(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), True))/2 + x*sqrt(-a**2*c*x**2 + c)/2, Ne(a**2*c, 0)), (sqrt(c)*x, True)) - 2*a**3*c**2*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c)*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1) , (sqrt(c)*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(s qrt(-a**2*x**2 + 1) + 1), True)) + 2*a*c**2*Piecewise((a**2*sqrt(c)*acosh( 1/(a*x))/2 + a*sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3* sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/ (a*x))/2 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) + c**2*Piecew ise((a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2 *x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2* x**2))/3 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))
\[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx=\int { -\frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{4}} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^4,x, algorithm="ma xima")
Output:
-integrate((-a^2*c*x^2 + c)^(5/2)*(a*x + 1)^2/((a^2*x^2 - 1)*x^4), x)
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (124) = 248\).
Time = 0.15 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.95 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx=-\frac {6 \, a^{3} c^{3} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {a^{4} \sqrt {-c} c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{2 \, {\left | a \right |}} - \frac {1}{2} \, {\left (a^{4} c^{2} x + 4 \, a^{3} c^{2}\right )} \sqrt {-a^{2} c x^{2} + c} - \frac {2 \, {\left (3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{3} c^{3} {\left | a \right |} + 3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{4} \sqrt {-c} c^{3} - 3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{3} c^{5} {\left | a \right |} + a^{4} \sqrt {-c} c^{5}\right )}}{3 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{3} {\left | a \right |}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^4,x, algorithm="gi ac")
Output:
-6*a^3*c^3*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt( -c) - 1/2*a^4*sqrt(-c)*c^2*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)) )/abs(a) - 1/2*(a^4*c^2*x + 4*a^3*c^2)*sqrt(-a^2*c*x^2 + c) - 2/3*(3*(sqrt (-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^5*a^3*c^3*abs(a) + 3*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^4*a^4*sqrt(-c)*c^3 - 3*(sqrt(-a^2*c)*x - sqrt(-a^2*c *x^2 + c))*a^3*c^5*abs(a) + a^4*sqrt(-c)*c^5)/(((sqrt(-a^2*c)*x - sqrt(-a^ 2*c*x^2 + c))^2 - c)^3*abs(a))
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{x^4\,\left (a^2\,x^2-1\right )} \,d x \] Input:
int(-((c - a^2*c*x^2)^(5/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)),x)
Output:
-int(((c - a^2*c*x^2)^(5/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)), x)
Time = 0.16 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.88 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^4} \, dx=\frac {\sqrt {c}\, c^{2} \left (-3 \mathit {asin} \left (a x \right ) a^{3} x^{3}-3 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-12 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-6 \sqrt {-a^{2} x^{2}+1}\, a x -2 \sqrt {-a^{2} x^{2}+1}-18 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{3} x^{3}+15 a^{3} x^{3}\right )}{6 x^{3}} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^4,x)
Output:
(sqrt(c)*c**2*( - 3*asin(a*x)*a**3*x**3 - 3*sqrt( - a**2*x**2 + 1)*a**4*x* *4 - 12*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 2*sqrt( - a**2*x**2 + 1)*a**2*x **2 - 6*sqrt( - a**2*x**2 + 1)*a*x - 2*sqrt( - a**2*x**2 + 1) - 18*log(tan (asin(a*x)/2))*a**3*x**3 + 15*a**3*x**3))/(6*x**3)