Integrand size = 24, antiderivative size = 118 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {3 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \] Output:
-3*c^2*(1-1/a^2/x^2)^(1/2)/a/(c-c/a/x)^(1/2)+c^3*(1-1/a^2/x^2)^(3/2)*x/(c- c/a/x)^(3/2)+3*c^(3/2)*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2) )/a
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.56 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} \left (c-\frac {c}{a x}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1+\frac {1}{a x}\right )}{5 a \left (1-\frac {1}{a x}\right )^{3/2}} \] Input:
Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(3/2),x]
Output:
(-2*(1 + 1/(a*x))^(5/2)*(c - c/(a*x))^(3/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + 1/(a*x)])/(5*a*(1 - 1/(a*x))^(3/2))
Time = 0.66 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6731, 575, 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.
\(\displaystyle \int \left (c-\frac {c}{a x}\right )^{3/2} e^{3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -c^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (c-\frac {c}{a x}\right )^{3/2}}d\frac {1}{x}\) |
\(\Big \downarrow \) 575 |
\(\displaystyle -c^3 \left (\frac {3 \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a c}-\frac {x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 576 |
\(\displaystyle -c^3 \left (\frac {3 \left (\frac {\int \frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{c}+\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{2 a c}-\frac {x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 573 |
\(\displaystyle -c^3 \left (\frac {3 \left (\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-2 \int \frac {1}{1-\frac {c}{x^2}}d\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{2 a c}-\frac {x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -c^3 \left (\frac {3 \left (\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{\sqrt {c}}\right )}{2 a c}-\frac {x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{\left (c-\frac {c}{a x}\right )^{3/2}}\right )\) |
Input:
Int[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(3/2),x]
Output:
-(c^3*(-(((1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2)) + (3*((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*c)))
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*(m + 1))), x] + Simp[b*(n/(d*e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^(n + 1)*(a + b*x^ 2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0] && GtQ[p, 0] && LtQ[m, -1] && !(IntegerQ[m + p] && LeQ[m + p + 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^(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 = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (2 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}+3 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x -4 \sqrt {x \left (a x +1\right )}\, \sqrt {a}\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) a^{\frac {3}{2}} \sqrt {x \left (a x +1\right )}}\) | \(118\) |
risch | \(\frac {\left (a^{2} x^{2}-a x -2\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}+\frac {3 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(151\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*c/a^(3/2 )*(2*a^(3/2)*x*(x*(a*x+1))^(1/2)+3*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a *x+1)/a^(1/2))*a*x-4*(x*(a*x+1))^(1/2)*a^(1/2))/(x*(a*x+1))^(1/2)
Time = 0.15 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.67 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\left [\frac {3 \, {\left (a c x - c\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} c x^{2} - a c x - 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, -\frac {3 \, {\left (a c x - c\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (a^{2} c x^{2} - a c x - 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(3/2),x, algorithm="fricas")
Output:
[1/4*(3*(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 - 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*(3*(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 - 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. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**(3/2),x)
Output:
Timed out
\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(3/2),x, algorithm="maxima")
Output:
integrate((c - c/(a*x))^(3/2)/((a*x - 1)/(a*x + 1))^(3/2), x)
Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(3/2),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. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:
int((c - c/(a*x))^(3/2)/((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
int((c - c/(a*x))^(3/2)/((a*x - 1)/(a*x + 1))^(3/2), x)
Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.50 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\frac {\sqrt {c}\, c \left (4 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+12 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a x -9 a x \right )}{4 a^{2} x} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(3/2),x)
Output:
(sqrt(c)*c*(4*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 8*sqrt(x)*sqrt(a)*sqrt(a *x + 1) + 12*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a*x - 9*a*x))/(4*a**2*x)