Integrand size = 22, antiderivative size = 78 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \] Output:
c*(1-1/a^2/x^2)^(1/2)*x/(c-c/a/x)^(1/2)+c^(1/2)*arctanh(c^(1/2)*(1-1/a^2/x ^2)^(1/2)/(c-c/a/x)^(1/2))/a
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c-\frac {c}{a x}} \left (1+a x+\sqrt {1+\frac {1}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}} \] Input:
Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a*x)],x]
Output:
(Sqrt[c - c/(a*x)]*(1 + a*x + Sqrt[1 + 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]] ))/(a*Sqrt[1 - 1/(a^2*x^2)])
Time = 0.56 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6731, 575, 573, 219}
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.
\(\displaystyle \int \sqrt {c-\frac {c}{a x}} e^{\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6731 |
\(\displaystyle -c \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 575 |
\(\displaystyle -c \left (\frac {\int \frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{2 a c}-\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )\) |
\(\Big \downarrow \) 573 |
\(\displaystyle -c \left (-\frac {\int \frac {1}{1-\frac {c}{x^2}}d\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}}{a}-\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -c \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}}-\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )\) |
Input:
Int[E^ArcCoth[a*x]*Sqrt[c - c/(a*x)],x]
Output:
-(c*(-((Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a*x)]) - ArcTanh[(Sqrt[c]*Sqr t[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]]/(a*Sqrt[c])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2*c Subst[Int[1/(a - c*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*x)^(m + 1)*(c + d*x)^n*((a + b*x^2)^p/(e*(m + 1))), x] + Simp[b*(n/(d*e*(m + 1))) Int[(e*x)^(m + 1)*(c + d*x)^(n + 1)*(a + b*x^ 2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0] && GtQ[p, 0] && LtQ[m, -1] && !(IntegerQ[m + p] && LeQ[m + p + 2, 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 = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+\ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {x \left (a x +1\right )}\, \sqrt {a}}\) | \(87\) |
risch | \(\frac {x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{\sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(127\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/((a*x-1)/(a*x+1))^(1/2)*(c*(a*x-1)/a/x)^(1/2)*x*(2*(x*(a*x+1))^(1/2)*a ^(1/2)+ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2)))/(x*(a*x+1))^ (1/2)/a^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (66) = 132\).
Time = 0.12 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.78 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2),x, algorithm="fricas")
Output:
[1/4*((a*x - 1)*sqrt(c)*log(-(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)) + 4*(a^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c) /(a*x)))/(a^2*x - a), -1/2*((a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sq rt(-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)) - 2*(a^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/( a*x)))/(a^2*x - a)]
\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(c-c/a/x)**(1/2),x)
Output:
Integral(sqrt(-c*(-1 + 1/(a*x)))/sqrt((a*x - 1)/(a*x + 1)), x)
\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(c - c/(a*x))/sqrt((a*x - 1)/(a*x + 1)), x)
\[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { \frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(c - c/(a*x))/sqrt((a*x - 1)/(a*x + 1)), x)
Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int((c - c/(a*x))^(1/2)/((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
int((c - c/(a*x))^(1/2)/((a*x - 1)/(a*x + 1))^(1/2), x)
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.40 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c}\, \left (\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )\right )}{a} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(1/2),x)
Output:
(sqrt(c)*(sqrt(x)*sqrt(a)*sqrt(a*x + 1) + log(sqrt(a*x + 1) + sqrt(x)*sqrt (a))))/a