Integrand size = 20, antiderivative size = 95 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\frac {64 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x}{15 \sqrt {c-a c x}}+\frac {16}{15} c \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}+\frac {2}{5} \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2} \] Output:
64/15*c^2*(1-1/a^2/x^2)^(1/2)*x/(-a*c*x+c)^(1/2)+16/15*c*(1-1/a^2/x^2)^(1/ 2)*x*(-a*c*x+c)^(1/2)+2/5*(1-1/a^2/x^2)^(1/2)*x*(-a*c*x+c)^(3/2)
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.63 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 c \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (43-14 a x+3 a^2 x^2\right )}{15 a \sqrt {1-\frac {1}{a x}}} \] Input:
Integrate[(c - a*c*x)^(3/2)/E^ArcCoth[a*x],x]
Output:
(-2*c*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(43 - 14*a*x + 3*a^2*x^2))/(15*a*S qrt[1 - 1/(a*x)])
Time = 0.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6727, 27, 100, 27, 87, 48}
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.
\(\displaystyle \int (c-a c x)^{3/2} e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \int \frac {\left (a-\frac {1}{x}\right )^2}{a^2 \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \int \frac {\left (a-\frac {1}{x}\right )^2}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \left (\frac {2}{5} \int -\frac {14 a-\frac {5}{x}}{2 \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{a^2 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \left (-\frac {1}{5} \int \frac {14 a-\frac {5}{x}}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{a^2 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \left (\frac {1}{5} \left (\frac {43}{3} \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}+\frac {28 a \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{a^2 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{5} \left (\frac {28 a \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}-\frac {86 \sqrt {\frac {1}{a x}+1}}{3 \sqrt {\frac {1}{x}}}\right )-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right ) (c-a c x)^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
Input:
Int[(c - a*c*x)^(3/2)/E^ArcCoth[a*x],x]
Output:
-(((((28*a*Sqrt[1 + 1/(a*x)])/(3*(x^(-1))^(3/2)) - (86*Sqrt[1 + 1/(a*x)])/ (3*Sqrt[x^(-1)]))/5 - (2*a^2*Sqrt[1 + 1/(a*x)])/(5*(x^(-1))^(5/2)))*(x^(-1 ))^(3/2)*(c - a*c*x)^(3/2))/(a^2*(1 - 1/(a*x))^(3/2)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.56
method | result | size |
risch | \(\frac {2 c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (3 a^{2} x^{2}-14 a x +43\right ) \left (a x +1\right )}{15 \sqrt {-c \left (a x -1\right )}\, a}\) | \(53\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (3 a^{2} x^{2}-14 a x +43\right ) \left (-a c x +c \right )^{\frac {3}{2}} \sqrt {\frac {a x -1}{a x +1}}}{15 a \left (a x -1\right )^{2}}\) | \(56\) |
orering | \(\frac {2 \left (a x +1\right ) \left (3 a^{2} x^{2}-14 a x +43\right ) \left (-a c x +c \right )^{\frac {3}{2}} \sqrt {\frac {a x -1}{a x +1}}}{15 a \left (a x -1\right )^{2}}\) | \(56\) |
default | \(-\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c \left (3 a^{2} x^{2}-14 a x +43\right )}{15 \left (a x -1\right ) a}\) | \(58\) |
Input:
int((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/15*c^2*((a*x-1)/(a*x+1))^(1/2)/(-c*(a*x-1))^(1/2)*(3*a^2*x^2-14*a*x+43)/ a*(a*x+1)
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (3 \, a^{3} c x^{3} - 11 \, a^{2} c x^{2} + 29 \, a c x + 43 \, c\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{2} x - a\right )}} \] Input:
integrate((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")
Output:
-2/15*(3*a^3*c*x^3 - 11*a^2*c*x^2 + 29*a*c*x + 43*c)*sqrt(-a*c*x + c)*sqrt ((a*x - 1)/(a*x + 1))/(a^2*x - a)
Timed out. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Timed out} \] Input:
integrate((-a*c*x+c)**(3/2)*((a*x-1)/(a*x+1))**(1/2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.76 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (3 \, a^{3} \sqrt {-c} c x^{3} - 11 \, a^{2} \sqrt {-c} c x^{2} + 29 \, a \sqrt {-c} c x + 43 \, \sqrt {-c} c\right )} {\left (a x - 1\right )}}{15 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \] Input:
integrate((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")
Output:
-2/15*(3*a^3*sqrt(-c)*c*x^3 - 11*a^2*sqrt(-c)*c*x^2 + 29*a*sqrt(-c)*c*x + 43*sqrt(-c)*c)*(a*x - 1)/((a^2*x - a)*sqrt(a*x + 1))
Exception generated. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(1/2),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
Time = 14.55 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (3\,a^2\,x^2-8\,a\,x+21\right )}{15\,a}-\frac {128\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{15\,a\,\left (a\,x-1\right )} \] Input:
int((c - a*c*x)^(3/2)*((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
- (2*c*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(3*a^2*x^2 - 8*a*x + 21))/(15*a) - (128*c*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2))/(15*a* (a*x - 1))
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.31 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\frac {2 \sqrt {c}\, \sqrt {a x +1}\, c i \left (3 a^{2} x^{2}-14 a x +43\right )}{15 a} \] Input:
int((-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1))^(1/2),x)
Output:
(2*sqrt(c)*sqrt(a*x + 1)*c*i*(3*a**2*x**2 - 14*a*x + 43))/(15*a)