Integrand size = 20, antiderivative size = 137 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {44 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{7/2}}{63 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {214 \left (1+\frac {1}{a x}\right )^{5/2} (c-a c x)^{7/2}}{315 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^{7/2}}{9 \left (1-\frac {1}{a x}\right )^{7/2}} \] Output:
-44/63*(1+1/a/x)^(5/2)*(-a*c*x+c)^(7/2)/a/(1-1/a/x)^(7/2)+214/315*(1+1/a/x )^(5/2)*(-a*c*x+c)^(7/2)/a^2/(1-1/a/x)^(7/2)/x+2/9*(1+1/a/x)^(5/2)*x*(-a*c *x+c)^(7/2)/(1-1/a/x)^(7/2)
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.50 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 c^3 \sqrt {1+\frac {1}{a x}} (1+a x)^2 \sqrt {c-a c x} \left (107-110 a x+35 a^2 x^2\right )}{315 a \sqrt {1-\frac {1}{a x}}} \] Input:
Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^(7/2),x]
Output:
(-2*c^3*Sqrt[1 + 1/(a*x)]*(1 + a*x)^2*Sqrt[c - a*c*x]*(107 - 110*a*x + 35* a^2*x^2))/(315*a*Sqrt[1 - 1/(a*x)])
Time = 0.45 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87, 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.
\(\displaystyle \int (c-a c x)^{7/2} e^{3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \int \frac {\left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2}}{a^2 \left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \int \frac {\left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2}}{\left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}}{a^2 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (\frac {2}{9} \int -\frac {\left (22 a-\frac {9}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}{2 \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{5/2}}{9 \left (\frac {1}{x}\right )^{9/2}}\right )}{a^2 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {1}{9} \int \frac {\left (22 a-\frac {9}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2}}{\left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{5/2}}{9 \left (\frac {1}{x}\right )^{9/2}}\right )}{a^2 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (\frac {1}{9} \left (\frac {107}{7} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2}}{\left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}+\frac {44 a \left (\frac {1}{a x}+1\right )^{5/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{5/2}}{9 \left (\frac {1}{x}\right )^{9/2}}\right )}{a^2 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} \left (\frac {1}{9} \left (\frac {44 a \left (\frac {1}{a x}+1\right )^{5/2}}{7 \left (\frac {1}{x}\right )^{7/2}}-\frac {214 \left (\frac {1}{a x}+1\right )^{5/2}}{35 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{5/2}}{9 \left (\frac {1}{x}\right )^{9/2}}\right ) (c-a c x)^{7/2}}{a^2 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
Input:
Int[E^(3*ArcCoth[a*x])*(c - a*c*x)^(7/2),x]
Output:
-(((((44*a*(1 + 1/(a*x))^(5/2))/(7*(x^(-1))^(7/2)) - (214*(1 + 1/(a*x))^(5 /2))/(35*(x^(-1))^(5/2)))/9 - (2*a^2*(1 + 1/(a*x))^(5/2))/(9*(x^(-1))^(9/2 )))*(x^(-1))^(7/2)*(c - a*c*x)^(7/2))/(a^2*(1 - 1/(a*x))^(7/2)))
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 = 56, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (35 a^{2} x^{2}-110 a x +107\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(56\) |
orering | \(\frac {2 \left (a x +1\right ) \left (35 a^{2} x^{2}-110 a x +107\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(56\) |
default | \(-\frac {2 \left (a x -1\right ) \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{3} \left (35 a^{2} x^{2}-110 a x +107\right )}{315 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a}\) | \(58\) |
risch | \(\frac {2 c^{4} \left (a x -1\right ) \left (35 a^{4} x^{4}-40 a^{3} x^{3}-78 a^{2} x^{2}+104 a x +107\right )}{315 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(69\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(7/2),x,method=_RETURNVERBOSE)
Output:
2/315*(a*x+1)*(35*a^2*x^2-110*a*x+107)*(-a*c*x+c)^(7/2)/a/(a*x-1)^2/((a*x- 1)/(a*x+1))^(3/2)
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.69 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{5} c^{3} x^{5} - 5 \, a^{4} c^{3} x^{4} - 118 \, a^{3} c^{3} x^{3} + 26 \, a^{2} c^{3} x^{2} + 211 \, a c^{3} x + 107 \, c^{3}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{2} x - a\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(7/2),x, algorithm="fricas" )
Output:
-2/315*(35*a^5*c^3*x^5 - 5*a^4*c^3*x^4 - 118*a^3*c^3*x^3 + 26*a^2*c^3*x^2 + 211*a*c^3*x + 107*c^3)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - a)
Timed out. \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)*(-a*c*x+c)**(7/2),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.66 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{4} \sqrt {-c} c^{3} x^{4} - 110 \, a^{3} \sqrt {-c} c^{3} x^{3} + 72 \, a^{2} \sqrt {-c} c^{3} x^{2} + 110 \, a \sqrt {-c} c^{3} x - 107 \, \sqrt {-c} c^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}}{315 \, {\left (a x - 1\right )} a} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(7/2),x, algorithm="maxima" )
Output:
-2/315*(35*a^4*sqrt(-c)*c^3*x^4 - 110*a^3*sqrt(-c)*c^3*x^3 + 72*a^2*sqrt(- c)*c^3*x^2 + 110*a*sqrt(-c)*c^3*x - 107*sqrt(-c)*c^3)*(a*x + 1)^(3/2)/((a* x - 1)*a)
Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(7/2),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 = 13.96 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (35\,a^4\,x^4+30\,a^3\,x^3-88\,a^2\,x^2-62\,a\,x+149\right )}{315\,a}-\frac {512\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{315\,a\,\left (a\,x-1\right )} \] Input:
int((c - a*c*x)^(7/2)/((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
- (2*c^3*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(30*a^3*x^3 - 88*a^ 2*x^2 - 62*a*x + 35*a^4*x^4 + 149))/(315*a) - (512*c^3*(c - a*c*x)^(1/2)*( (a*x - 1)/(a*x + 1))^(1/2))/(315*a*(a*x - 1))
Time = 0.15 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.34 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 \sqrt {c}\, \sqrt {a x +1}\, c^{3} i \left (35 a^{4} x^{4}-40 a^{3} x^{3}-78 a^{2} x^{2}+104 a x +107\right )}{315 a} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(7/2),x)
Output:
(2*sqrt(c)*sqrt(a*x + 1)*c**3*i*(35*a**4*x**4 - 40*a**3*x**3 - 78*a**2*x** 2 + 104*a*x + 107))/(315*a)