Integrand size = 23, antiderivative size = 231 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {1115 a^4 \sqrt {c-a c x}}{64 \sqrt {1-a x} \sqrt {1+a x}}-\frac {\sqrt {c-a c x}}{4 x^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {25 a \sqrt {c-a c x}}{24 x^3 \sqrt {1-a x} \sqrt {1+a x}}-\frac {223 a^2 \sqrt {c-a c x}}{96 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {1115 a^3 \sqrt {c-a c x}}{192 x \sqrt {1-a x} \sqrt {1+a x}}-\frac {1115 a^4 \sqrt {c-a c x} \text {arctanh}\left (\sqrt {1+a x}\right )}{64 \sqrt {1-a x}} \] Output:
1115/64*a^4*(-a*c*x+c)^(1/2)/(-a*x+1)^(1/2)/(a*x+1)^(1/2)-1/4*(-a*c*x+c)^( 1/2)/x^4/(-a*x+1)^(1/2)/(a*x+1)^(1/2)+25/24*a*(-a*c*x+c)^(1/2)/x^3/(-a*x+1 )^(1/2)/(a*x+1)^(1/2)-223/96*a^2*(-a*c*x+c)^(1/2)/x^2/(-a*x+1)^(1/2)/(a*x+ 1)^(1/2)+1115/192*a^3*(-a*c*x+c)^(1/2)/x/(-a*x+1)^(1/2)/(a*x+1)^(1/2)-1115 /64*a^4*(-a*c*x+c)^(1/2)*arctanh((a*x+1)^(1/2))/(-a*x+1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.28 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {c \sqrt {1-a x} \left (-6+25 a x+223 a^4 x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},3,\frac {1}{2},1+a x\right )\right )}{24 x^4 \sqrt {1+a x} \sqrt {c-a c x}} \] Input:
Integrate[Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x^5),x]
Output:
(c*Sqrt[1 - a*x]*(-6 + 25*a*x + 223*a^4*x^4*Hypergeometric2F1[-1/2, 3, 1/2 , 1 + a*x]))/(24*x^4*Sqrt[1 + a*x]*Sqrt[c - a*c*x])
Time = 0.35 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.57, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6680, 37, 100, 27, 87, 52, 52, 61, 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.
| \(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6680 |
| \(\displaystyle \int \frac {(1-a x)^{3/2} \sqrt {c-a c x}}{x^5 (a x+1)^{3/2}}dx\) |
| \(\Big \downarrow \) 37 |
| \(\displaystyle \frac {\sqrt {c-a c x} \int \frac {(1-a x)^2}{x^5 (a x+1)^{3/2}}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{4} \int -\frac {a (25-8 a x)}{2 x^4 (a x+1)^{3/2}}dx-\frac {1}{4 x^4 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{8} a \int \frac {25-8 a x}{x^4 (a x+1)^{3/2}}dx-\frac {1}{4 x^4 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{8} a \left (-\frac {223}{6} a \int \frac {1}{x^3 (a x+1)^{3/2}}dx-\frac {25}{3 x^3 \sqrt {a x+1}}\right )-\frac {1}{4 x^4 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{8} a \left (-\frac {223}{6} a \left (-\frac {5}{4} a \int \frac {1}{x^2 (a x+1)^{3/2}}dx-\frac {1}{2 x^2 \sqrt {a x+1}}\right )-\frac {25}{3 x^3 \sqrt {a x+1}}\right )-\frac {1}{4 x^4 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{8} a \left (-\frac {223}{6} a \left (-\frac {5}{4} a \left (-\frac {3}{2} a \int \frac {1}{x (a x+1)^{3/2}}dx-\frac {1}{x \sqrt {a x+1}}\right )-\frac {1}{2 x^2 \sqrt {a x+1}}\right )-\frac {25}{3 x^3 \sqrt {a x+1}}\right )-\frac {1}{4 x^4 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{8} a \left (-\frac {223}{6} a \left (-\frac {5}{4} a \left (-\frac {3}{2} a \left (\int \frac {1}{x \sqrt {a x+1}}dx+\frac {2}{\sqrt {a x+1}}\right )-\frac {1}{x \sqrt {a x+1}}\right )-\frac {1}{2 x^2 \sqrt {a x+1}}\right )-\frac {25}{3 x^3 \sqrt {a x+1}}\right )-\frac {1}{4 x^4 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{8} a \left (-\frac {223}{6} a \left (-\frac {5}{4} a \left (-\frac {3}{2} a \left (\frac {2 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}+\frac {2}{\sqrt {a x+1}}\right )-\frac {1}{x \sqrt {a x+1}}\right )-\frac {1}{2 x^2 \sqrt {a x+1}}\right )-\frac {25}{3 x^3 \sqrt {a x+1}}\right )-\frac {1}{4 x^4 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (-\frac {1}{8} a \left (-\frac {223}{6} a \left (-\frac {5}{4} a \left (-\frac {3}{2} a \left (\frac {2}{\sqrt {a x+1}}-2 \text {arctanh}\left (\sqrt {a x+1}\right )\right )-\frac {1}{x \sqrt {a x+1}}\right )-\frac {1}{2 x^2 \sqrt {a x+1}}\right )-\frac {25}{3 x^3 \sqrt {a x+1}}\right )-\frac {1}{4 x^4 \sqrt {a x+1}}\right ) \sqrt {c-a c x}}{\sqrt {1-a x}}\) |
Input:
Int[Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x^5),x]
Output:
(Sqrt[c - a*c*x]*(-1/4*1/(x^4*Sqrt[1 + a*x]) - (a*(-25/(3*x^3*Sqrt[1 + a*x ]) - (223*a*(-1/2*1/(x^2*Sqrt[1 + a*x]) - (5*a*(-(1/(x*Sqrt[1 + a*x])) - ( 3*a*(2/Sqrt[1 + a*x] - 2*ArcTanh[Sqrt[1 + a*x]]))/2))/4))/6))/8))/Sqrt[1 - a*x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c , d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.53
| method | result | size |
| default | \(\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (3345 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{4} x^{4} \sqrt {c \left (a x +1\right )}-3345 a^{4} x^{4} \sqrt {c}-1115 a^{3} x^{3} \sqrt {c}+446 \sqrt {c}\, a^{2} x^{2}-200 \sqrt {c}\, a x +48 \sqrt {c}\right )}{192 \sqrt {c}\, \left (a x -1\right ) \left (a x +1\right ) x^{4}}\) | \(122\) |
| risch | \(-\frac {\left (1809 a^{4} x^{4}+1115 a^{3} x^{3}-446 a^{2} x^{2}+200 a x -48\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{192 x^{4} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {a^{4} \left (\frac {1024}{\sqrt {a c x +c}}-\frac {2230 \,\operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{128 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(180\) |
Input:
int((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x,method=_RETURNVERB OSE)
Output:
1/192*(-c*(a*x-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(3345*arctanh((c*(a*x+1))^(1/2 )/c^(1/2))*a^4*x^4*(c*(a*x+1))^(1/2)-3345*a^4*x^4*c^(1/2)-1115*a^3*x^3*c^( 1/2)+446*c^(1/2)*a^2*x^2-200*c^(1/2)*a*x+48*c^(1/2))/c^(1/2)/(a*x-1)/(a*x+ 1)/x^4
Time = 0.12 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {3345 \, {\left (a^{6} x^{6} - a^{4} x^{4}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, {\left (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{384 \, {\left (a^{2} x^{6} - x^{4}\right )}}, -\frac {3345 \, {\left (a^{6} x^{6} - a^{4} x^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + {\left (3345 \, a^{4} x^{4} + 1115 \, a^{3} x^{3} - 446 \, a^{2} x^{2} + 200 \, a x - 48\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{192 \, {\left (a^{2} x^{6} - x^{4}\right )}}\right ] \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm=" fricas")
Output:
[1/384*(3345*(a^6*x^6 - a^4*x^4)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt( -a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) - 2*(3345*a^4*x ^4 + 1115*a^3*x^3 - 446*a^2*x^2 + 200*a*x - 48)*sqrt(-a^2*x^2 + 1)*sqrt(-a *c*x + c))/(a^2*x^6 - x^4), -1/192*(3345*(a^6*x^6 - a^4*x^4)*sqrt(-c)*arct an(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + (3345*a^4*x ^4 + 1115*a^3*x^3 - 446*a^2*x^2 + 200*a*x - 48)*sqrt(-a^2*x^2 + 1)*sqrt(-a *c*x + c))/(a^2*x^6 - x^4)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{5} \left (a x + 1\right )^{3}}\, dx \] Input:
integrate((-a*c*x+c)**(1/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/x**5,x)
Output:
Integral(sqrt(-c*(a*x - 1))*(-(a*x - 1)*(a*x + 1))**(3/2)/(x**5*(a*x + 1)* *3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {-a c x + c}}{{\left (a x + 1\right )}^{3} x^{5}} \,d x } \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x, algorithm=" maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*sqrt(-a*c*x + c)/((a*x + 1)^3*x^5), x)
Exception generated. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,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. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {c-a\,c\,x}}{x^5\,{\left (a\,x+1\right )}^3} \,d x \] Input:
int(((1 - a^2*x^2)^(3/2)*(c - a*c*x)^(1/2))/(x^5*(a*x + 1)^3),x)
Output:
int(((1 - a^2*x^2)^(3/2)*(c - a*c*x)^(1/2))/(x^5*(a*x + 1)^3), x)
Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.39 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c}\, \left (3345 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}-1\right ) a^{4} x^{4}-3345 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}+1\right ) a^{4} x^{4}+6690 a^{4} x^{4}+2230 a^{3} x^{3}-892 a^{2} x^{2}+400 a x -96\right )}{384 \sqrt {a x +1}\, x^{4}} \] Input:
int((-a*c*x+c)^(1/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/x^5,x)
Output:
(sqrt(c)*(3345*sqrt(a*x + 1)*log(sqrt(a*x + 1) - 1)*a**4*x**4 - 3345*sqrt( a*x + 1)*log(sqrt(a*x + 1) + 1)*a**4*x**4 + 6690*a**4*x**4 + 2230*a**3*x** 3 - 892*a**2*x**2 + 400*a*x - 96))/(384*sqrt(a*x + 1)*x**4)