Integrand size = 25, antiderivative size = 113 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {22 a^3 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {12 a^3 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{35 \sqrt {c-\frac {c}{a x}}}-\frac {2}{7} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}} \] Output:
-22/105*a^3*c^2*(1-1/a^2/x^2)^(3/2)/(c-c/a/x)^(3/2)+12/35*a^3*c*(1-1/a^2/x ^2)^(3/2)/(c-c/a/x)^(1/2)-2/7*a^3*(1-1/a^2/x^2)^(3/2)*(c-c/a/x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (15+3 a x-4 a^2 x^2+8 a^3 x^3\right )}{105 x^2 (-1+a x)} \] Input:
Integrate[(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)])/x^4,x]
Output:
(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(15 + 3*a*x - 4*a^2*x^2 + 8* a^3*x^3))/(105*x^2*(-1 + a*x))
Time = 0.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6733, 581, 27, 672, 458}
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.
\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a x}} e^{\coth ^{-1}(a x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 6733 |
\(\displaystyle -c \int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}} x^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 581 |
\(\displaystyle -c \left (\frac {2 a^2 \int \frac {c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {6}{x}\right )}{2 a \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{7 c^2}+\frac {2 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{7 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (\frac {1}{7} a \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (a+\frac {6}{x}\right )}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}+\frac {2 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{7 c}\right )\) |
\(\Big \downarrow \) 672 |
\(\displaystyle -c \left (\frac {1}{7} a \left (\frac {11}{5} a \int \frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-\frac {12 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-\frac {c}{a x}}}\right )+\frac {2 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{7 c}\right )\) |
\(\Big \downarrow \) 458 |
\(\displaystyle -c \left (\frac {1}{7} a \left (\frac {22 a^2 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{15 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {12 a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-\frac {c}{a x}}}\right )+\frac {2 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}}}{7 c}\right )\) |
Input:
Int[(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)])/x^4,x]
Output:
-(c*((a*((22*a^2*c*(1 - 1/(a^2*x^2))^(3/2))/(15*(c - c/(a*x))^(3/2)) - (12 *a^2*(1 - 1/(a^2*x^2))^(3/2))/(5*Sqrt[c - c/(a*x)])))/7 + (2*a^3*(1 - 1/(a ^2*x^2))^(3/2)*Sqrt[c - c/(a*x)])/(7*c)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c , d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b*x^ 2)^p*ExpandToSum[d^m*(m + n + 2*p + 1)*x^m - (m + n + 2*p + 1)*(c + d*x)^m + c*(c + d*x)^(m - 2)*(c*(m + n - 1) + c*(m + n + 2*p + 1) + 2*d*(m + n + p )*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] & & IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[2*p] || ILtQ[m + n, 0] )
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)) Int[(d + e*x )^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^ 2 + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_S ymbol] :> Simp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && Int egerQ[(n - 1)/2] && IntegerQ[m] && IntegerQ[2*p]
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.47
method | result | size |
orering | \(-\frac {2 \left (8 a^{2} x^{2}-12 a x +15\right ) \left (a x +1\right ) \sqrt {c -\frac {c}{a x}}}{105 x^{3} \sqrt {\frac {a x -1}{a x +1}}}\) | \(53\) |
gosper | \(-\frac {2 \left (a x +1\right ) \left (8 a^{2} x^{2}-12 a x +15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{105 x^{3} \sqrt {\frac {a x -1}{a x +1}}}\) | \(55\) |
default | \(-\frac {2 \left (a x +1\right ) \left (8 a^{2} x^{2}-12 a x +15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{105 x^{3} \sqrt {\frac {a x -1}{a x +1}}}\) | \(55\) |
risch | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (8 a^{4} x^{4}+4 a^{3} x^{3}-a^{2} x^{2}+18 a x +15\right )}{105 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x^{3}}\) | \(73\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2)/x^4,x,method=_RETURNVERBOSE)
Output:
-2/105*(8*a^2*x^2-12*a*x+15)/x^3*(a*x+1)/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x) ^(1/2)
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {2 \, {\left (8 \, a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} + 18 \, a x + 15\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a x^{4} - x^{3}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2)/x^4,x, algorithm="fric as")
Output:
-2/105*(8*a^4*x^4 + 4*a^3*x^3 - a^2*x^2 + 18*a*x + 15)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(a*x^4 - x^3)
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(c-c/a/x)**(1/2)/x**4,x)
Output:
Timed out
\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{x^{4} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2)/x^4,x, algorithm="maxi ma")
Output:
integrate(sqrt(c - c/(a*x))/(x^4*sqrt((a*x - 1)/(a*x + 1))), x)
\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{x^{4} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2)/x^4,x, algorithm="giac ")
Output:
integrate(sqrt(c - c/(a*x))/(x^4*sqrt((a*x - 1)/(a*x + 1))), x)
Time = 13.69 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (8\,a^3\,x^3+12\,a^2\,x^2+11\,a\,x+29\right )\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3}-\frac {88\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3\,\left (a\,x-1\right )} \] Input:
int((c - c/(a*x))^(1/2)/(x^4*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
- (2*((a*x - 1)/(a*x + 1))^(1/2)*(11*a*x + 12*a^2*x^2 + 8*a^3*x^3 + 29)*(( c*(a*x - 1))/(a*x))^(1/2))/(105*x^3) - (88*((a*x - 1)/(a*x + 1))^(1/2)*((c *(a*x - 1))/(a*x))^(1/2))/(105*x^3*(a*x - 1))
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c}\, \left (-8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}+4 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-3 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -15 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+8 a^{4} x^{4}\right )}{105 a \,x^{4}} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2)/x^4,x)
Output:
(2*sqrt(c)*( - 8*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**3*x**3 + 4*sqrt(x)*sqrt( a)*sqrt(a*x + 1)*a**2*x**2 - 3*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 15*sqrt (x)*sqrt(a)*sqrt(a*x + 1) + 8*a**4*x**4))/(105*a*x**4)