Integrand size = 24, antiderivative size = 216 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{4 c \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {15 \sqrt {1-\frac {1}{a^2 x^2}} x}{16 c^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {35 \sqrt {1-\frac {1}{a^2 x^2}} x}{16 c^3 \sqrt {c-\frac {c}{a x}}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{7/2}}-\frac {115 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )}{16 \sqrt {2} a c^{7/2}} \] Output:
-1/4*(1-1/a^2/x^2)^(1/2)*x/c/(c-c/a/x)^(5/2)-15/16*(1-1/a^2/x^2)^(1/2)*x/c ^2/(c-c/a/x)^(3/2)+35/16*(1-1/a^2/x^2)^(1/2)*x/c^3/(c-c/a/x)^(1/2)+5*arcta nh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))/a/c^(7/2)-115/32*arctanh(1 /2*c^(1/2)*(1-1/a^2/x^2)^(1/2)*2^(1/2)/(c-c/a/x)^(1/2))*2^(1/2)/a/c^(7/2)
Time = 0.16 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.62 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x \left (35-55 a x+16 a^2 x^2\right )+160 (-1+a x)^2 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-115 \sqrt {2} (-1+a x)^2 \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{32 a c^3 \sqrt {c-\frac {c}{a x}} (-1+a x)^2} \] Input:
Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(7/2)),x]
Output:
(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(35 - 55*a*x + 16*a^2*x^2) + 1 60*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]] - 115*Sqrt[2]*(-1 + a*x)^2*ArcT anh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(32*a*c^3*Sqrt[c - c/(a*x)]*(-1 + a*x)^2)
Time = 0.73 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.81, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6731, 585, 27, 114, 27, 168, 27, 168, 25, 174, 73, 221}
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.
| \(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {x^2}{\sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {\sqrt {1-\frac {1}{a x}} \int \frac {a^3 x^2}{\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \int \frac {x^2}{\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}-\frac {\int -\frac {5 \left (2 a+\frac {1}{x}\right ) x^2}{2 a \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{4 a}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {5 \int \frac {\left (2 a+\frac {1}{x}\right ) x^2}{\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{8 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {5 \left (\frac {3 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {\int -\frac {\left (14 a+\frac {9}{x}\right ) x^2}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 a}\right )}{8 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {5 \left (\frac {\int \frac {\left (14 a+\frac {9}{x}\right ) x^2}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{4 a}+\frac {3 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}\right )}{8 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {5 \left (\frac {-\frac {\int -\frac {\left (16 a+\frac {7}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-14 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {3 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}\right )}{8 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {5 \left (\frac {\frac {\int \frac {\left (16 a+\frac {7}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-14 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {3 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}\right )}{8 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {5 \left (\frac {\frac {23 \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+16 \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-14 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {3 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}\right )}{8 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {5 \left (\frac {\frac {46 a \int \frac {1}{2 a-\frac {a}{x^2}}d\sqrt {1+\frac {1}{a x}}+32 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}}{a}-14 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {3 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}\right )}{8 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {5 \left (\frac {\frac {23 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )-32 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a}-14 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {3 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}\right )}{8 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{4 a \left (a-\frac {1}{x}\right )^2}\right )}{c^3 \sqrt {c-\frac {c}{a x}}}\) |
Input:
Int[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(7/2)),x]
Output:
-((a^3*Sqrt[1 - 1/(a*x)]*((Sqrt[1 + 1/(a*x)]*x)/(4*a*(a - x^(-1))^2) + (5* ((3*Sqrt[1 + 1/(a*x)]*x)/(2*(a - x^(-1))) + (-14*Sqrt[1 + 1/(a*x)]*x + (-3 2*ArcTanh[Sqrt[1 + 1/(a*x)]] + 23*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2 ]])/a)/(4*a)))/(8*a^2)))/(c^3*Sqrt[c - c/(a*x)]))
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] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F racPart[n]) Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 0]
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 = 0.22 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.46
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {5 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{4} \sqrt {a^{2} c}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{4 a^{7} c \left (x -\frac {1}{a}\right )^{2}}-\frac {23 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{16 a^{6} c \left (x -\frac {1}{a}\right )}-\frac {115 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{64 a^{5} \sqrt {c}}\right ) a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) a c x}}{c^{3} x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(316\) |
| default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (64 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x^{2}-115 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x^{2}-220 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x +160 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {1}{a}}\, x^{2}+230 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x +140 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-320 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x +160 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-115 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{64 a^{\frac {3}{2}} c^{4} \left (a x -1\right )^{3} \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\) | \(371\) |
Input:
int(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/a*(a*x+1)/c^3*((a*x-1)/(a*x+1))^(1/2)/(c*(a*x-1)/a/x)^(1/2)+(5/2/a^4*ln( (1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2*c*x^2+a*c*x)^(1/2))/(a^2*c)^(1/2)-1/4 /a^7/c/(x-1/a)^2*((x-1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c)^(1/2)-23/16/a^6/c/(x- 1/a)*((x-1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c)^(1/2)-115/64/a^5/c^(1/2)*2^(1/2)* ln((4*c+3*(x-1/a)*a*c+2*2^(1/2)*c^(1/2)*((x-1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c )^(1/2))/(x-1/a)))/c^3*a^3*((a*x-1)/(a*x+1))^(1/2)/x/(c*(a*x-1)/a/x)^(1/2) *((a*x+1)*a*c*x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.09 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\left [\frac {115 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 160 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (16 \, a^{4} x^{4} - 39 \, a^{3} x^{3} - 20 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{128 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, \frac {115 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 160 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (16 \, a^{4} x^{4} - 39 \, a^{3} x^{3} - 20 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{64 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(7/2),x, algorithm="fricas")
Output:
[1/128*(115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-(17*a^3 *c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3*x^3 + 4*a^2*x^2 + a*x)* sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 160*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*lo g(-(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)) + 8*(16*a^4*x^ 4 - 39*a^3*x^3 - 20*a^2*x^2 + 35*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c* x - c)/(a*x)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4), 1/64*( 115*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(2*sqrt(2)*(a ^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/( 3*a^2*c*x^2 - 2*a*c*x - c)) - 160*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*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)) + 4*(16*a^4*x^4 - 39*a^3*x^3 - 20* a^2*x^2 + 35*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4* c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)]
Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(((a*x-1)/(a*x+1))**(1/2)/(c-c/a/x)**(7/2),x)
Output:
Timed out
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(7/2),x, algorithm="maxima")
Output:
integrate(sqrt((a*x - 1)/(a*x + 1))/(c - c/(a*x))^(7/2), x)
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(7/2),x, algorithm="giac")
Output:
integrate(sqrt((a*x - 1)/(a*x + 1))/(c - c/(a*x))^(7/2), x)
Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}} \,d x \] Input:
int(((a*x - 1)/(a*x + 1))^(1/2)/(c - c/(a*x))^(7/2),x)
Output:
int(((a*x - 1)/(a*x + 1))^(1/2)/(c - c/(a*x))^(7/2), x)
Time = 0.16 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.94 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (64 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-220 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +140 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+115 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a^{2} x^{2}-230 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a x +115 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right )-115 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a^{2} x^{2}+230 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a x -115 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right )-115 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a^{2} x^{2}+230 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a x -115 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right )+115 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a^{2} x^{2}-230 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a x +115 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right )+320 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a^{2} x^{2}-640 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a x +320 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )+90 a^{2} x^{2}-180 a x +90\right )}{64 a \,c^{4} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:
int(((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(7/2),x)
Output:
(sqrt(c)*(64*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 - 220*sqrt(x)*sqrt(a) *sqrt(a*x + 1)*a*x + 140*sqrt(x)*sqrt(a)*sqrt(a*x + 1) + 115*sqrt(2)*log(s qrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a**2*x**2 - 230*sqrt(2)*log( sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a*x + 115*sqrt(2)*log(sqrt( a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1) - 115*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a**2*x**2 + 230*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a*x - 115*sqrt(2)*log(sqrt(a*x + 1) + sq rt(x)*sqrt(a) - sqrt(2) + 1) - 115*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqr t(a) + sqrt(2) - 1)*a**2*x**2 + 230*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sq rt(a) + sqrt(2) - 1)*a*x - 115*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) - 1) + 115*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2 ) + 1)*a**2*x**2 - 230*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt( 2) + 1)*a*x + 115*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1) + 320*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a**2*x**2 - 640*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a*x + 320*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a)) + 90*a**2*x**2 - 180*a*x + 90))/(64*a*c**4*(a**2*x**2 - 2*a*x + 1))