Integrand size = 25, antiderivative size = 176 \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{2 (1-a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}{8 (1+a x) \left (c-a^2 c x^2\right )^{5/2}}-\frac {3 a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \text {arctanh}(a x)}{8 \left (c-a^2 c x^2\right )^{5/2}} \] Output:
-1/8*a*(1-1/a^2/x^2)^(5/2)*x^5/(-a*x+1)^2/(-a^2*c*x^2+c)^(5/2)+1/2*a*(1-1/ a^2/x^2)^(5/2)*x^5/(-a*x+1)/(-a^2*c*x^2+c)^(5/2)-1/8*a*(1-1/a^2/x^2)^(5/2) *x^5/(a*x+1)/(-a^2*c*x^2+c)^(5/2)-3/8*a*(1-1/a^2/x^2)^(5/2)*x^5*arctanh(a* x)/(-a^2*c*x^2+c)^(5/2)
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.41 \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {a \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \left (\frac {2+a x-5 a^2 x^2}{(-1+a x)^2 (1+a x)}-3 \text {arctanh}(a x)\right )}{8 \left (c-a^2 c x^2\right )^{5/2}} \] Input:
Integrate[(E^ArcCoth[a*x]*x^3)/(c - a^2*c*x^2)^(5/2),x]
Output:
(a*(1 - 1/(a^2*x^2))^(5/2)*x^5*((2 + a*x - 5*a^2*x^2)/((-1 + a*x)^2*(1 + a *x)) - 3*ArcTanh[a*x]))/(8*(c - a^2*c*x^2)^(5/2))
Time = 0.90 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6746, 6747, 25, 99, 2009}
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.
\(\displaystyle \int \frac {x^3 e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle \frac {x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^2}dx}{\left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {a^5 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int -\frac {x^3}{(1-a x)^3 (a x+1)^2}dx}{\left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a^5 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \frac {x^3}{(1-a x)^3 (a x+1)^2}dx}{\left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {a^5 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \int \left (-\frac {1}{2 a^3 (a x-1)^2}-\frac {1}{8 a^3 (a x+1)^2}-\frac {1}{4 a^3 (a x-1)^3}-\frac {3}{8 a^3 \left (a^2 x^2-1\right )}\right )dx}{\left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \left (\frac {3 \text {arctanh}(a x)}{8 a^4}-\frac {1}{2 a^4 (1-a x)}+\frac {1}{8 a^4 (a x+1)}+\frac {1}{8 a^4 (1-a x)^2}\right )}{\left (c-a^2 c x^2\right )^{5/2}}\) |
Input:
Int[(E^ArcCoth[a*x]*x^3)/(c - a^2*c*x^2)^(5/2),x]
Output:
-((a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*(1/(8*a^4*(1 - a*x)^2) - 1/(2*a^4*(1 - a*x)) + 1/(8*a^4*(1 + a*x)) + (3*ArcTanh[a*x])/(8*a^4)))/(c - a^2*c*x^2)^( 5/2))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[c^p/a^(2*p) Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !Inte gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
Time = 0.15 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (3 \ln \left (a x +1\right ) x^{3} a^{3}-3 a^{3} \ln \left (a x -1\right ) x^{3}-3 \ln \left (a x +1\right ) x^{2} a^{2}+3 a^{2} \ln \left (a x -1\right ) x^{2}+10 a^{2} x^{2}-3 \ln \left (a x +1\right ) x a +3 a \ln \left (a x -1\right ) x -2 a x +3 \ln \left (a x +1\right )-3 \ln \left (a x -1\right )-4\right )}{16 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right ) \left (a^{2} x^{2}-1\right ) c^{3} a^{4} \left (a x +1\right )}\) | \(169\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*x^3/(-a^2*c*x^2+c)^(5/2),x,method=_RETURNVER BOSE)
Output:
1/16/((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*(-c*(a^2*x^2-1))^(1/2)*(3*ln(a*x+1)*x ^3*a^3-3*a^3*ln(a*x-1)*x^3-3*ln(a*x+1)*x^2*a^2+3*a^2*ln(a*x-1)*x^2+10*a^2* x^2-3*ln(a*x+1)*x*a+3*a*ln(a*x-1)*x-2*a*x+3*ln(a*x+1)-3*ln(a*x-1)-4)/(a^2* x^2-1)/c^3/a^4/(a*x+1)
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {3 \, {\left (a^{4} x^{3} - a^{3} x^{2} - a^{2} x + a\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c} \sqrt {-c} x + c}{a^{2} x^{2} - 1}\right ) - 2 \, {\left (5 \, a^{2} x^{2} - a x - 2\right )} \sqrt {-a^{2} c}}{16 \, {\left (a^{8} c^{3} x^{3} - a^{7} c^{3} x^{2} - a^{6} c^{3} x + a^{5} c^{3}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm= "fricas")
Output:
-1/16*(3*(a^4*x^3 - a^3*x^2 - a^2*x + a)*sqrt(-c)*log((a^2*c*x^2 + 2*sqrt( -a^2*c)*sqrt(-c)*x + c)/(a^2*x^2 - 1)) - 2*(5*a^2*x^2 - a*x - 2)*sqrt(-a^2 *c))/(a^8*c^3*x^3 - a^7*c^3*x^2 - a^6*c^3*x + a^5*c^3)
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*x**3/(-a**2*c*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*x^3/(-a^2*c*x^2+c)^(5/2),x, algorithm= "maxima")
Output:
integrate(x^3/((-a^2*c*x^2 + c)^(5/2)*sqrt((a*x - 1)/(a*x + 1))), x)
Exception generated. \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*x^3/(-a^2*c*x^2+c)^(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. \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int(x^3/((c - a^2*c*x^2)^(5/2)*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
int(x^3/((c - a^2*c*x^2)^(5/2)*((a*x - 1)/(a*x + 1))^(1/2)), x)
Time = 0.16 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c}\, i \left (-3 \,\mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a^{3} x^{3}+3 \,\mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a^{2} x^{2}+3 \,\mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a x -3 \,\mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right )-3 \,\mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a^{3} x^{3}+3 \,\mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a^{2} x^{2}+3 \,\mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a x -3 \,\mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right )+6 \,\mathrm {log}\left (\sqrt {-a x +1}\right ) a^{3} x^{3}-6 \,\mathrm {log}\left (\sqrt {-a x +1}\right ) a^{2} x^{2}-6 \,\mathrm {log}\left (\sqrt {-a x +1}\right ) a x +6 \,\mathrm {log}\left (\sqrt {-a x +1}\right )+5 a^{3} x^{3}-15 a^{2} x^{2}-3 a x +9\right )}{16 a^{4} c^{3} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*x^3/(-a^2*c*x^2+c)^(5/2),x)
Output:
(sqrt(c)*i*( - 3*log(sqrt( - a*x + 1) - sqrt(2))*a**3*x**3 + 3*log(sqrt( - a*x + 1) - sqrt(2))*a**2*x**2 + 3*log(sqrt( - a*x + 1) - sqrt(2))*a*x - 3 *log(sqrt( - a*x + 1) - sqrt(2)) - 3*log(sqrt( - a*x + 1) + sqrt(2))*a**3* x**3 + 3*log(sqrt( - a*x + 1) + sqrt(2))*a**2*x**2 + 3*log(sqrt( - a*x + 1 ) + sqrt(2))*a*x - 3*log(sqrt( - a*x + 1) + sqrt(2)) + 6*log(sqrt( - a*x + 1))*a**3*x**3 - 6*log(sqrt( - a*x + 1))*a**2*x**2 - 6*log(sqrt( - a*x + 1 ))*a*x + 6*log(sqrt( - a*x + 1)) + 5*a**3*x**3 - 15*a**2*x**2 - 3*a*x + 9) )/(16*a**4*c**3*(a**3*x**3 - a**2*x**2 - a*x + 1))