Integrand size = 20, antiderivative size = 195 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {184 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {16 (c-a c x)^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}} \]
-8/5*(-a*c*x+c)^(3/2)/a/(1-1/a/x)^(3/2)/(1+1/a/x)^(1/2)+184/5*(-a*c*x+c)^( 3/2)/a^3/(1-1/a/x)^(3/2)/x^2/(1+1/a/x)^(1/2)+16*(-a*c*x+c)^(3/2)/a^2/(1-1/ a/x)^(3/2)/x/(1+1/a/x)^(1/2)+2/5*(a-1/x)^3*x*(-a*c*x+c)^(3/2)/a^3/(1-1/a/x )^(3/2)/(1+1/a/x)^(1/2)
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.29 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 c \sqrt {c-a c x} \left (91+43 a x-7 a^2 x^2+a^3 x^3\right )}{5 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6727, 27, 105, 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.
\(\displaystyle \int (c-a c x)^{3/2} e^{-3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \int \frac {\left (a-\frac {1}{x}\right )^3}{a^3 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \int \frac {\left (a-\frac {1}{x}\right )^3}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{a^3 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \left (-\frac {12}{5} \int \frac {\left (a-\frac {1}{x}\right )^2}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \left (-\frac {12}{5} \left (\frac {2}{3} \int -\frac {10 a-\frac {3}{x}}{2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \left (-\frac {12}{5} \left (-\frac {1}{3} \int \frac {10 a-\frac {3}{x}}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2} \left (-\frac {12}{5} \left (\frac {1}{3} \left (23 \int \frac {1}{\left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}d\frac {1}{x}+\frac {20 a}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{3/2} \left (-\frac {12}{5} \left (\frac {1}{3} \left (\frac {20 a}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}+\frac {46 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right ) (c-a c x)^{3/2}}{a^3 \left (1-\frac {1}{a x}\right )^{3/2}}\) |
-((((-12*(((20*a)/(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]) + (46*Sqrt[x^(-1)])/Sqr t[1 + 1/(a*x)])/3 - (2*a^2)/(3*Sqrt[1 + 1/(a*x)]*(x^(-1))^(3/2))))/5 - (2* (a - x^(-1))^3)/(5*Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2)))*(x^(-1))^(3/2)*(c - a*c*x)^(3/2))/(a^3*(1 - 1/(a*x))^(3/2)))
3.3.73.3.1 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[((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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -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 = 0.48 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.32
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (a^{3} x^{3}-7 a^{2} x^{2}+43 a x +91\right ) \left (-a c x +c \right )^{\frac {3}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{5 a \left (a x -1\right )^{3}}\) | \(63\) |
default | \(-\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c \left (a^{3} x^{3}-7 a^{2} x^{2}+43 a x +91\right )}{5 \left (a x -1\right )^{2} a}\) | \(65\) |
risch | \(\frac {2 \left (a^{2} x^{2}-8 a x +51\right ) \left (a x +1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}}{5 a \sqrt {-c \left (a x -1\right )}}+\frac {16 c^{2} \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\) | \(86\) |
2/5*(a*x+1)*(a^3*x^3-7*a^2*x^2+43*a*x+91)*(-a*c*x+c)^(3/2)*((a*x-1)/(a*x+1 ))^(3/2)/a/(a*x-1)^3
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.32 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (a^{3} c x^{3} - 7 \, a^{2} c x^{2} + 43 \, a c x + 91 \, c\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{2} x - a\right )}} \]
-2/5*(a^3*c*x^3 - 7*a^2*c*x^2 + 43*a*c*x + 91*c)*sqrt(-a*c*x + c)*sqrt((a* x - 1)/(a*x + 1))/(a^2*x - a)
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.48 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (a^{4} \sqrt {-c} c x^{4} - 6 \, a^{3} \sqrt {-c} c x^{3} + 36 \, a^{2} \sqrt {-c} c x^{2} + 134 \, a \sqrt {-c} c x + 91 \, \sqrt {-c} c\right )} {\left (a x - 1\right )}^{2}}{5 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
-2/5*(a^4*sqrt(-c)*c*x^4 - 6*a^3*sqrt(-c)*c*x^3 + 36*a^2*sqrt(-c)*c*x^2 + 134*a*sqrt(-c)*c*x + 91*sqrt(-c)*c)*(a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*( a*x + 1)^(3/2))
Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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
Time = 4.42 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.42 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (a^2\,x^2-6\,a\,x+37\right )}{5\,a}-\frac {256\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{5\,a\,\left (a\,x-1\right )} \]