Integrand size = 27, antiderivative size = 155 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^4 \, dx=-\frac {6 \sqrt {c-\frac {c}{a^2 x^2}} x}{5 a^4}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{4 a^3}-\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x^3}{5 a^2}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^4}{2 a}-\frac {1}{5} \sqrt {c-\frac {c}{a^2 x^2}} x^5-\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{4 a^5} \] Output:
-6/5*(c-c/a^2/x^2)^(1/2)*x/a^4-3/4*(c-c/a^2/x^2)^(1/2)*x^2/a^3-3/5*(c-c/a^ 2/x^2)^(1/2)*x^3/a^2-1/2*(c-c/a^2/x^2)^(1/2)*x^4/a-1/5*(c-c/a^2/x^2)^(1/2) *x^5-3/4*c^(1/2)*arctanh((c-c/a^2/x^2)^(1/2)/c^(1/2))/a^5
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.65 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^4 \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {-1+a^2 x^2} \left (24+15 a x+12 a^2 x^2+10 a^3 x^3+4 a^4 x^4\right )+15 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{20 a^4 \sqrt {-1+a^2 x^2}} \] Input:
Integrate[E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x^4,x]
Output:
-1/20*(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(24 + 15*a*x + 12*a^2*x ^2 + 10*a^3*x^3 + 4*a^4*x^4) + 15*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(a^4*Sqr t[-1 + a^2*x^2])
Time = 0.81 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6709, 541, 25, 27, 533, 27, 533, 27, 533, 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.
| \(\displaystyle \int x^4 e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {x^3 (a x+1)^2}{\sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {\int -\frac {a^2 x^3 (10 a x+9)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {\int \frac {a^2 x^3 (10 a x+9)}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5} \int \frac {x^3 (10 a x+9)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5} \left (\frac {\int \frac {6 a x^2 (6 a x+5)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {5 x^3 \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5} \left (\frac {3 \int \frac {x^2 (6 a x+5)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {5 x^3 \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\int \frac {3 a x (5 a x+4)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a}\right )}{2 a}-\frac {5 x^3 \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\int \frac {x (5 a x+4)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a}\right )}{2 a}-\frac {5 x^3 \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\int \frac {a (8 a x+5)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {5 x \sqrt {1-a^2 x^2}}{2 a}}{a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a}\right )}{2 a}-\frac {5 x^3 \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\int \frac {8 a x+5}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {5 x \sqrt {1-a^2 x^2}}{2 a}}{a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a}\right )}{2 a}-\frac {5 x^3 \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {8 \sqrt {1-a^2 x^2}}{a}}{2 a}-\frac {5 x \sqrt {1-a^2 x^2}}{2 a}}{a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a}\right )}{2 a}-\frac {5 x^3 \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {x \left (\frac {1}{5} \left (\frac {3 \left (\frac {\frac {\frac {5 \arcsin (a x)}{a}-\frac {8 \sqrt {1-a^2 x^2}}{a}}{2 a}-\frac {5 x \sqrt {1-a^2 x^2}}{2 a}}{a}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a}\right )}{2 a}-\frac {5 x^3 \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {1}{5} x^4 \sqrt {1-a^2 x^2}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x^4,x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-1/5*(x^4*Sqrt[1 - a^2*x^2]) + ((-5*x^3*Sqrt[1 - a^2*x^2])/(2*a) + (3*((-2*x^2*Sqrt[1 - a^2*x^2])/a + ((-5*x*Sqrt[1 - a^2* x^2])/(2*a) + ((-8*Sqrt[1 - a^2*x^2])/a + (5*ArcSin[a*x])/a)/(2*a))/a))/(2 *a))/5))/Sqrt[1 - a^2*x^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[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
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 = 0.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(-\frac {\left (4 a^{4} x^{4}+10 a^{3} x^{3}+12 a^{2} x^{2}+15 a x +24\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{20 a^{4}}-\frac {3 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{4 a^{3} \sqrt {a^{2} c}\, \left (a^{2} x^{2}-1\right )}\) | \(142\) |
| default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (4 x^{2} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{5}+10 x {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4}+16 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3}+25 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c x -25 c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )+40 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right )+40 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c \right )}{20 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{5} c}\) | \(220\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)*x^4,x,method=_RETURNVERBOSE )
Output:
-1/20*(4*a^4*x^4+10*a^3*x^3+12*a^2*x^2+15*a*x+24)/a^4*x*(c*(a^2*x^2-1)/a^2 /x^2)^(1/2)-3/4/a^3*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^ (1/2)*x*(c*(a^2*x^2-1))^(1/2)*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)
Time = 0.12 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.54 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^4 \, dx=\left [-\frac {2 \, {\left (4 \, a^{5} x^{5} + 10 \, a^{4} x^{4} + 12 \, a^{3} x^{3} + 15 \, a^{2} x^{2} + 24 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 15 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{40 \, a^{5}}, -\frac {{\left (4 \, a^{5} x^{5} + 10 \, a^{4} x^{4} + 12 \, a^{3} x^{3} + 15 \, a^{2} x^{2} + 24 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 15 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{20 \, a^{5}}\right ] \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)*x^4,x, algorithm="fri cas")
Output:
[-1/40*(2*(4*a^5*x^5 + 10*a^4*x^4 + 12*a^3*x^3 + 15*a^2*x^2 + 24*a*x)*sqrt ((a^2*c*x^2 - c)/(a^2*x^2)) - 15*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))/a^5, -1/20*((4*a^5*x^5 + 10*a^4*x ^4 + 12*a^3*x^3 + 15*a^2*x^2 + 24*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 1 5*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)))/a^5]
\[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^4 \, dx=- \int \frac {x^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x - 1}\, dx - \int \frac {a x^{5} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x - 1}\, dx \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2)*x**4,x)
Output:
-Integral(x**4*sqrt(c - c/(a**2*x**2))/(a*x - 1), x) - Integral(a*x**5*sqr t(c - c/(a**2*x**2))/(a*x - 1), x)
\[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^4 \, dx=\int { -\frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{4}}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)*x^4,x, algorithm="max ima")
Output:
-integrate((a*x + 1)^2*sqrt(c - c/(a^2*x^2))*x^4/(a^2*x^2 - 1), x)
Time = 0.14 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^4 \, dx=-\frac {1}{40} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left ({\left (2 \, {\left (x {\left (\frac {2 \, x \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {5 \, \mathrm {sgn}\left (x\right )}{a^{3}}\right )} + \frac {6 \, \mathrm {sgn}\left (x\right )}{a^{4}}\right )} x + \frac {15 \, \mathrm {sgn}\left (x\right )}{a^{5}}\right )} x + \frac {24 \, \mathrm {sgn}\left (x\right )}{a^{6}}\right )} - \frac {30 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{5} {\left | a \right |}} + \frac {3 \, {\left (5 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) - 16 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\left (x\right )}{a^{6} {\left | a \right |}}\right )} {\left | a \right |} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)*x^4,x, algorithm="gia c")
Output:
-1/40*(2*sqrt(a^2*c*x^2 - c)*((2*(x*(2*x*sgn(x)/a^2 + 5*sgn(x)/a^3) + 6*sg n(x)/a^4)*x + 15*sgn(x)/a^5)*x + 24*sgn(x)/a^6) - 30*sqrt(c)*log(abs(-sqrt (a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a^5*abs(a)) + 3*(5*a*sqrt(c)*log (abs(c)) - 16*sqrt(-c)*abs(a))*sgn(x)/(a^6*abs(a)))*abs(a)
Timed out. \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^4 \, dx=\int -\frac {x^4\,\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \] Input:
int(-(x^4*(c - c/(a^2*x^2))^(1/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
int(-(x^4*(c - c/(a^2*x^2))^(1/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)
Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.68 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^4 \, dx=\frac {\sqrt {c}\, \left (-4 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}-10 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}-12 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-15 \sqrt {a^{2} x^{2}-1}\, a x -24 \sqrt {a^{2} x^{2}-1}-15 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )\right )}{20 a^{5}} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)*x^4,x)
Output:
(sqrt(c)*( - 4*sqrt(a**2*x**2 - 1)*a**4*x**4 - 10*sqrt(a**2*x**2 - 1)*a**3 *x**3 - 12*sqrt(a**2*x**2 - 1)*a**2*x**2 - 15*sqrt(a**2*x**2 - 1)*a*x - 24 *sqrt(a**2*x**2 - 1) - 15*log(sqrt(a**2*x**2 - 1) + a*x)))/(20*a**5)