Integrand size = 22, antiderivative size = 117 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \]
c^3*(1-1/a^2/x^2)^(3/2)*x/(c-c/a/x)^(3/2)-c^(3/2)*arctanh(c^(1/2)*(1-1/a^2 /x^2)^(1/2)/(c-c/a/x)^(1/2))/a+c^2*(1-1/a^2/x^2)^(1/2)/a/(c-c/a/x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.60 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\frac {c \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} (2+a x)-\text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{a \sqrt {1-\frac {1}{a x}}} \]
(c*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(2 + a*x) - ArcTanh[Sqrt[1 + 1/(a* x)]]))/(a*Sqrt[1 - 1/(a*x)])
Time = 0.40 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6731, 580, 576, 573, 219}
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.
\(\displaystyle \int \left (c-\frac {c}{a x}\right )^{3/2} e^{\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -c \int \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 580 |
\(\displaystyle -c \left (-\frac {c \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a}-\frac {c^2 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 576 |
\(\displaystyle -c \left (-\frac {c \left (\frac {\int \frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{c}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{2 a}-\frac {c^2 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 573 |
\(\displaystyle -c \left (-\frac {c \left (\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-2 \int \frac {1}{1-\frac {c}{x^2}}d\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{2 a}-\frac {c^2 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -c \left (-\frac {c \left (\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{\sqrt {c}}\right )}{2 a}-\frac {c^2 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}\right )\) |
-(c*(-((c^2*(1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2)) - (c*((2*Sqrt[ 1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)] - (2*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2* x^2)])/Sqrt[c - c/(a*x)]])/Sqrt[c]))/(2*a)))
3.5.41.3.1 Defintions of rubi rules used
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[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2*c Subst[Int[1/(a - c*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(e*x)^(m + 1))*(c + d*x)^n*((a + b*x^2)^p/(e*(n - m - 1 ))), x] - Simp[b*c*(n/(d^2*(n - m - 1))) Int[(e*x)^m*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c^2 + a* d^2, 0] && EqQ[n + p, 0] && GtQ[p, 0] && NeQ[m - n + 1, 0] && !IGtQ[m, 0] && !(IntegerQ[m + p] && LtQ[m + p + 2, 0]) && RationalQ[m]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 2)*((a + b*x^2)^(p + 1)/(b*e*(m + 1))), x] + Simp[d*((2*m + p + 3)/(e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p - 1, 0] && LtQ[m, -1] && IntegerQ [p + 1/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (-2 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x -4 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, a^{\frac {3}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(105\) |
risch | \(\frac {\left (a^{2} x^{2}+3 a x +2\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(151\) |
-1/2/((a*x-1)/(a*x+1))^(1/2)*(c*(a*x-1)/a/x)^(1/2)*c/a^(3/2)*(-2*a^(3/2)*x *((a*x+1)*x)^(1/2)+ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a *x-4*((a*x+1)*x)^(1/2)*a^(1/2))/((a*x+1)*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.68 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\left [\frac {{\left (a c x - c\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} c x^{2} + 3 \, a c x + 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {{\left (a c x - c\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} c x^{2} + 3 \, a c x + 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]
[1/4*((a*c*x - c)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3*a ^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(a^2*c*x^2 + 3*a*c*x + 2*c)*sqrt((a*x - 1)/(a*x + 1))*sq rt((a*c*x - c)/(a*x)))/(a^2*x - a), 1/2*((a*c*x - c)*sqrt(-c)*arctan(2*(a^ 2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2 *a^2*c*x^2 - a*c*x - c)) + 2*(a^2*c*x^2 + 3*a*c*x + 2*c)*sqrt((a*x - 1)/(a *x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^2*x - a)]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
Exception generated. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]