Integrand size = 23, antiderivative size = 216 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {\sqrt {1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {13 a \sqrt {1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac {19 a^2 \sqrt {1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac {45 a^3 (c-a c x)^{3/2} \text {arctanh}\left (\sqrt {1+a x}\right )}{8 c (1-a x)^{3/2}}+\frac {4 \sqrt {2} a^3 (c-a c x)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )}{c (1-a x)^{3/2}} \]
-45/8*a^3*(-a*c*x+c)^(3/2)*arctanh((a*x+1)^(1/2))/c/(-a*x+1)^(3/2)+4*a^3*( -a*c*x+c)^(3/2)*arctanh(1/2*(a*x+1)^(1/2)*2^(1/2))*2^(1/2)/c/(-a*x+1)^(3/2 )-1/3*(-a*c*x+c)^(3/2)*(a*x+1)^(1/2)/c/x^3/(-a*x+1)^(3/2)-13/12*a*(-a*c*x+ c)^(3/2)*(a*x+1)^(1/2)/c/x^2/(-a*x+1)^(3/2)-19/8*a^2*(-a*c*x+c)^(3/2)*(a*x +1)^(1/2)/c/x/(-a*x+1)^(3/2)
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.46 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {\sqrt {c-a c x} \left (\sqrt {1+a x} \left (8+26 a x+57 a^2 x^2\right )+135 a^3 x^3 \text {arctanh}\left (\sqrt {1+a x}\right )-96 \sqrt {2} a^3 x^3 \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{24 x^3 \sqrt {1-a x}} \]
-1/24*(Sqrt[c - a*c*x]*(Sqrt[1 + a*x]*(8 + 26*a*x + 57*a^2*x^2) + 135*a^3* x^3*ArcTanh[Sqrt[1 + a*x]] - 96*Sqrt[2]*a^3*x^3*ArcTanh[Sqrt[1 + a*x]/Sqrt [2]]))/(x^3*Sqrt[1 - a*x])
Time = 0.36 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.56, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6680, 37, 109, 27, 168, 27, 168, 27, 174, 73, 219, 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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx\) |
| \(\Big \downarrow \) 6680 |
| \(\displaystyle \int \frac {(a x+1)^{3/2} \sqrt {c-a c x}}{x^4 (1-a x)^{3/2}}dx\) |
| \(\Big \downarrow \) 37 |
| \(\displaystyle \frac {\sqrt {c-a c x} \int \frac {(a x+1)^{3/2}}{x^4 (1-a x)}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{3} \int -\frac {a (11 a x+13)}{2 x^3 (1-a x) \sqrt {a x+1}}dx-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \int \frac {11 a x+13}{x^3 (1-a x) \sqrt {a x+1}}dx-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (-\frac {1}{2} \int -\frac {3 a (13 a x+19)}{2 x^2 (1-a x) \sqrt {a x+1}}dx-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \int \frac {13 a x+19}{x^2 (1-a x) \sqrt {a x+1}}dx-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (-\int -\frac {a (19 a x+45)}{2 x (1-a x) \sqrt {a x+1}}dx-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {19 a x+45}{x (1-a x) \sqrt {a x+1}}dx-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \left (45 \int \frac {1}{x \sqrt {a x+1}}dx+64 a \int \frac {1}{(1-a x) \sqrt {a x+1}}dx\right )-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \left (128 \int \frac {1}{1-a x}d\sqrt {a x+1}+\frac {90 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}\right )-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \left (\frac {90 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}+64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )\right )-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {1}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \left (64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )-90 \text {arctanh}\left (\sqrt {a x+1}\right )\right )-\frac {19 \sqrt {a x+1}}{x}\right )-\frac {13 \sqrt {a x+1}}{2 x^2}\right )-\frac {\sqrt {a x+1}}{3 x^3}\right ) \sqrt {c-a c x}}{\sqrt {1-a x}}\) |
(Sqrt[c - a*c*x]*(-1/3*Sqrt[1 + a*x]/x^3 + (a*((-13*Sqrt[1 + a*x])/(2*x^2) + (3*a*((-19*Sqrt[1 + a*x])/x + (a*(-90*ArcTanh[Sqrt[1 + a*x]] + 64*Sqrt[ 2]*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/2))/4))/6))/Sqrt[1 - a*x]
3.5.9.3.1 Defintions of rubi rules used
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] :> 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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[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 = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (-96 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}+135 c \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{3} x^{3}+57 a^{2} x^{2} \sqrt {c \left (a x +1\right )}\, \sqrt {c}+26 a x \sqrt {c \left (a x +1\right )}\, \sqrt {c}+8 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{24 \sqrt {c}\, \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x^{3}}\) | \(151\) |
| risch | \(\frac {\left (57 a^{3} x^{3}+83 a^{2} x^{2}+34 a x +8\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{24 x^{3} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {a^{3} \left (\frac {64 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {90 \,\operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{16 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(187\) |
1/24*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(-96*2^(1/2)*arctanh(1/2*(c*(a* x+1))^(1/2)*2^(1/2)/c^(1/2))*a^3*c*x^3+135*c*arctanh((c*(a*x+1))^(1/2)/c^( 1/2))*a^3*x^3+57*a^2*x^2*(c*(a*x+1))^(1/2)*c^(1/2)+26*a*x*(c*(a*x+1))^(1/2 )*c^(1/2)+8*(c*(a*x+1))^(1/2)*c^(1/2))/c^(1/2)/(a*x-1)/(c*(a*x+1))^(1/2)/x ^3
Time = 0.28 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.88 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (57 \, a^{2} x^{2} + 26 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{48 \, {\left (a x^{4} - x^{3}\right )}}, \frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (57 \, a^{2} x^{2} + 26 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \]
[1/48*(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2* a*x + 1)) + 135*(a^4*x^4 - a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sq rt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*(57*a^2* x^2 + 26*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3), 1/24 *(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1 )*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 135*(a^4*x^4 - a^3*x^3)*sqr t(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (57*a^2*x^2 + 26*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^4 - x^3)]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{x^{4} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int { \frac {\sqrt {-a c x + c} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{x^4\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]