Integrand size = 24, antiderivative size = 51 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=-\sqrt {c-\frac {c}{a x}} x-\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \] Output:
-(c-c/a/x)^(1/2)*x-3*c^(1/2)*arctanh((c-c/a/x)^(1/2)/c^(1/2))/a
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=-\sqrt {c-\frac {c}{a x}} x-\frac {3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \] Input:
Integrate[E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)],x]
Output:
-(Sqrt[c - c/(a*x)]*x) - (3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a
Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6683, 1035, 281, 899, 87, 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.
\(\displaystyle \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) |
\(\Big \downarrow \) 6683 |
\(\displaystyle \int \frac {(a x+1) \sqrt {c-\frac {c}{a x}}}{1-a x}dx\) |
\(\Big \downarrow \) 1035 |
\(\displaystyle \int \frac {\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}{\frac {1}{x}-a}dx\) |
\(\Big \downarrow \) 281 |
\(\displaystyle -\frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}}}dx}{a}\) |
\(\Big \downarrow \) 899 |
\(\displaystyle \frac {c \int \frac {\left (a+\frac {1}{x}\right ) x^2}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{a}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {c \left (\frac {3}{2} \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-\frac {a x \sqrt {c-\frac {c}{a x}}}{c}\right )}{a}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {c \left (-\frac {3 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {a x \sqrt {c-\frac {c}{a x}}}{c}\right )}{a}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {c \left (-\frac {3 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {a x \sqrt {c-\frac {c}{a x}}}{c}\right )}{a}\) |
Input:
Int[E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)],x]
Output:
(c*(-((a*Sqrt[c - c/(a*x)]*x)/c) - (3*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/ Sqrt[c]))/a
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_))*((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)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ mn, -n] && IntegerQ[p] && IntegerQ[r]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[u*(c + d/x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && !G tQ[c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(43)=86\).
Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.94
method | result | size |
risch | \(-x \sqrt {\frac {c \left (a x -1\right )}{a x}}-\frac {3 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(99\) |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {a \,x^{2}-x}\, \sqrt {a}-4 \sqrt {x \left (a x -1\right )}\, \sqrt {a}-\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )-2 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}}\) | \(120\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-x*(c*(a*x-1)/a/x)^(1/2)-3/2*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)*(c*(a*x-1)*a*x)^(1/2)*(c*(a*x-1)/a/x)^(1/2)/ (a*x-1)
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.63 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [-\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} - 3 \, \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a}, -\frac {a x \sqrt {\frac {a c x - c}{a x}} - 3 \, \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right )}{a}\right ] \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2),x, algorithm="fricas")
Output:
[-1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) - 3*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c )*x*sqrt((a*c*x - c)/(a*x)) + c))/a, -(a*x*sqrt((a*c*x - c)/(a*x)) - 3*sqr t(-c)*arctan(a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)))/a]
\[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=- \int \frac {\sqrt {c - \frac {c}{a x}}}{a x - 1}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x - 1}\, dx \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2),x)
Output:
-Integral(sqrt(c - c/(a*x))/(a*x - 1), x) - Integral(a*x*sqrt(c - c/(a*x)) /(a*x - 1), x)
\[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2} - 1} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2),x, algorithm="maxima")
Output:
-integrate((a*x + 1)^2*sqrt(c - c/(a*x))/(a^2*x^2 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (43) = 86\).
Time = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.90 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=-\frac {3 \, \sqrt {c} \log \left ({\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\left (x\right )}{2 \, a} + \frac {3 \, \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} \sqrt {c} {\left | a \right |} + a c \right |}\right )}{2 \, a \mathrm {sgn}\left (x\right )} - \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |}}{a^{2} \mathrm {sgn}\left (x\right )} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2),x, algorithm="giac")
Output:
-3/2*sqrt(c)*log(abs(a)*abs(c))*sgn(x)/a + 3/2*sqrt(c)*log(abs(-2*(sqrt(a^ 2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*sqrt(c)*abs(a) + a*c))/(a*sgn(x)) - sqrt (a^2*c*x^2 - a*c*x)*abs(a)/(a^2*sgn(x))
Timed out. \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int -\frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \] Input:
int(-((c - c/(a*x))^(1/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
int(-((c - c/(a*x))^(1/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.67 \[ \int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c}\, \left (-\sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-3 \,\mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )\right )}{a} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2),x)
Output:
(sqrt(c)*( - sqrt(x)*sqrt(a)*sqrt(a*x - 1) - 3*log(sqrt(a*x - 1) + sqrt(x) *sqrt(a))))/a