Integrand size = 22, antiderivative size = 61 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {3 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{2 a} \] Output:
3/2*c^3*(1-1/a^2/x^2)^(1/2)/a^2/x+c^3*(1-1/a^2/x^2)^(3/2)*x+3/2*c^3*arccsc (a*x)/a
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (1+2 a^2 x^2\right )+3 a x \arcsin \left (\frac {1}{a x}\right )\right )}{2 a^2 x} \] Input:
Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^3,x]
Output:
(c^3*(Sqrt[1 - 1/(a^2*x^2)]*(1 + 2*a^2*x^2) + 3*a*x*ArcSin[1/(a*x)]))/(2*a ^2*x)
Time = 0.43 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6731, 247, 211, 223}
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.
\(\displaystyle \int \left (c-\frac {c}{a x}\right )^3 e^{3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -c^3 \int \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -c^3 \left (-\frac {3 \int \sqrt {1-\frac {1}{a^2 x^2}}d\frac {1}{x}}{a^2}-x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -c^3 \left (-\frac {3 \left (\frac {1}{2} \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 x}\right )}{a^2}-x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -c^3 \left (-\frac {3 \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+\frac {1}{2} a \arcsin \left (\frac {1}{a x}\right )\right )}{a^2}-x \left (1-\frac {1}{a^2 x^2}\right )^{3/2}\right )\) |
Input:
Int[E^(3*ArcCoth[a*x])*(c - c/(a*x))^3,x]
Output:
-(c^3*(-((1 - 1/(a^2*x^2))^(3/2)*x) - (3*(Sqrt[1 - 1/(a^2*x^2)]/(2*x) + (a *ArcSin[1/(a*x)])/2))/a^2))
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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 = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.72
method | result | size |
default | \(-\frac {\left (a x -1\right )^{2} c^{3} \left (-3 a^{2} x^{2} \sqrt {a^{2} x^{2}-1}-3 a^{2} x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}}\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{2}}\) | \(105\) |
risch | \(\frac {\left (a x -1\right ) c^{3}}{2 x^{2} a^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}+\frac {3 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}\right ) c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(110\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(a*x-1)^2*c^3*(-3*a^2*x^2*(a^2*x^2-1)^(1/2)-3*a^2*x^2*arctan(1/(a^2*x ^2-1)^(1/2))+(a^2*x^2-1)^(3/2))/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)/((a*x-1)*( a*x+1))^(1/2)/a^3/x^2
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.39 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {6 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (2 \, a^{3} c^{3} x^{3} + 2 \, a^{2} c^{3} x^{2} + a c^{3} x + c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^3,x, algorithm="fricas")
Output:
-1/2*(6*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - (2*a^3*c^3*x^3 + 2 *a^2*c^3*x^2 + a*c^3*x + c^3)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*x^2)
\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (\int \frac {3 a}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {3 a^{2}}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{3}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {1}{\frac {a x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx\right )}{a^{3}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**3,x)
Output:
c**3*(Integral(3*a/(a*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x **2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(-3*a**2/(a *x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(a**3/(a*x*sqrt(a*x/(a*x + 1) - 1/( a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + I ntegral(-1/(a*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - x**3*sqrt (a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x))/a**3
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (53) = 106\).
Time = 0.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.48 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-{\left (\frac {3 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {3 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^3,x, algorithm="maxima")
Output:
-(3*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - (3*c^3*((a*x - 1)/(a*x + 1 ))^(5/2) + 2*c^3*((a*x - 1)/(a*x + 1))^(3/2) + 3*c^3*sqrt((a*x - 1)/(a*x + 1)))/((a*x - 1)*a^2/(a*x + 1) - (a*x - 1)^2*a^2/(a*x + 1)^2 - (a*x - 1)^3 *a^2/(a*x + 1)^3 + a^2))*a
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {3 \, c^{3} \arctan \left (\sqrt {a^{2} x^{2} - 1}\right ) - 2 \, \sqrt {a^{2} x^{2} - 1} c^{3} - \frac {\sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2} x^{2}}}{2 \, a \mathrm {sgn}\left (a x + 1\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^3,x, algorithm="giac")
Output:
-1/2*(3*c^3*arctan(sqrt(a^2*x^2 - 1)) - 2*sqrt(a^2*x^2 - 1)*c^3 - sqrt(a^2 *x^2 - 1)*c^3/(a^2*x^2))/(a*sgn(a*x + 1))
Time = 13.74 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.95 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {3\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+c^3\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a^2\,x}+\frac {c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a^3\,x^2} \] Input:
int((c - c/(a*x))^3/((a*x - 1)/(a*x + 1))^(3/2),x)
Output:
(c^3*((a*x - 1)/(a*x + 1))^(1/2))/a - (3*c^3*atan(((a*x - 1)/(a*x + 1))^(1 /2)))/a + c^3*x*((a*x - 1)/(a*x + 1))^(1/2) + (c^3*((a*x - 1)/(a*x + 1))^( 1/2))/(2*a^2*x) + (c^3*((a*x - 1)/(a*x + 1))^(1/2))/(2*a^3*x^2)
Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (-6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{2} x^{2}+6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{2} x^{2}+2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+\sqrt {a x +1}\, \sqrt {a x -1}\right )}{2 a^{3} x^{2}} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^3,x)
Output:
(c**3*( - 6*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**2*x**2 + 6*atan(sqr t(a*x - 1) + sqrt(a*x + 1) + 1)*a**2*x**2 + 2*sqrt(a*x + 1)*sqrt(a*x - 1)* a**2*x**2 + sqrt(a*x + 1)*sqrt(a*x - 1)))/(2*a**3*x**2)