Integrand size = 20, antiderivative size = 58 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \sqrt {c-a c x}}{a c}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}} \] Output:
2*(-a*c*x+c)^(1/2)/a/c-2*2^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1 /2))/a/c^(1/2)
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \sqrt {c-a c x}}{a c}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a \sqrt {c}} \] Input:
Integrate[1/(E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x]),x]
Output:
(2*Sqrt[c - a*c*x])/(a*c) - (2*Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sq rt[c])])/(a*Sqrt[c])
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6680, 35, 60, 73, 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.
\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {1-a x}{(a x+1) \sqrt {c-a c x}}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {\int \frac {\sqrt {c-a c x}}{a x+1}dx}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 c \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx+\frac {2 \sqrt {c-a c x}}{a}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 \sqrt {c-a c x}}{a}-\frac {4 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{a}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {2 \sqrt {c-a c x}}{a}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}}{c}\) |
Input:
Int[1/(E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x]),x]
Output:
((2*Sqrt[c - a*c*x])/a - (2*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[ 2]*Sqrt[c])])/a)/c
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_)^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[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, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {2 \sqrt {-a c x +c}-2 \sqrt {2}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\) | \(45\) |
default | \(-\frac {2 \left (-\sqrt {-a c x +c}+\sqrt {2}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\) | \(46\) |
pseudoelliptic | \(\frac {2 \sqrt {-c \left (a x -1\right )}-2 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\) | \(48\) |
risch | \(-\frac {2 \left (a x -1\right )}{a \sqrt {-c \left (a x -1\right )}}-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a \sqrt {c}}\) | \(51\) |
Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/c/a*((-a*c*x+c)^(1/2)-2^(1/2)*c^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/ 2)/c^(1/2)))
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.90 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\left [\frac {\sqrt {2} \sqrt {c} \log \left (\frac {a x + \frac {2 \, \sqrt {2} \sqrt {-a c x + c}}{\sqrt {c}} - 3}{a x + 1}\right ) + 2 \, \sqrt {-a c x + c}}{a c}, \frac {2 \, {\left (\sqrt {2} c \sqrt {-\frac {1}{c}} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-a c x + c} \sqrt {-\frac {1}{c}}\right ) + \sqrt {-a c x + c}\right )}}{a c}\right ] \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(1/2),x, algorithm="fricas")
Output:
[(sqrt(2)*sqrt(c)*log((a*x + 2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(c) - 3)/(a*x + 1)) + 2*sqrt(-a*c*x + c))/(a*c), 2*(sqrt(2)*c*sqrt(-1/c)*arctan(1/2*sqrt (2)*sqrt(-a*c*x + c)*sqrt(-1/c)) + sqrt(-a*c*x + c))/(a*c)]
Time = 5.66 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.34 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\begin {cases} - \frac {2 \left (- \frac {\sqrt {2} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - \sqrt {- a c x + c}\right )}{a c} & \text {for}\: a c \neq 0 \\\frac {- x + 2 \left (\begin {cases} x & \text {for}\: a = 0 \\\frac {\log {\left (a x + 1 \right )}}{a} & \text {otherwise} \end {cases}\right )}{\sqrt {c}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a*c*x+c)**(1/2),x)
Output:
Piecewise((-2*(-sqrt(2)*c*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt (-c) - sqrt(-a*c*x + c))/(a*c), Ne(a*c, 0)), ((-x + 2*Piecewise((x, Eq(a, 0)), (log(a*x + 1)/a, True)))/sqrt(c), True))
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {\sqrt {2} \sqrt {c} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 2 \, \sqrt {-a c x + c}}{a c} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(1/2),x, algorithm="maxima")
Output:
(sqrt(2)*sqrt(c)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c ) + sqrt(-a*c*x + c))) + 2*sqrt(-a*c*x + c))/(a*c)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {2 \, \sqrt {-a c x + c}}{a c} \] Input:
integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(1/2),x, algorithm="giac")
Output:
2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)) + 2*s qrt(-a*c*x + c)/(a*c)
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2\,\sqrt {c-a\,c\,x}}{a\,c}-\frac {2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a\,\sqrt {c}} \] Input:
int(-(a^2*x^2 - 1)/((c - a*c*x)^(1/2)*(a*x + 1)^2),x)
Output:
(2*(c - a*c*x)^(1/2))/(a*c) - (2*2^(1/2)*atanh((2^(1/2)*(c - a*c*x)^(1/2)) /(2*c^(1/2))))/(a*c^(1/2))
Time = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {\sqrt {c}\, \left (2 \sqrt {-a x +1}+\sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right )-\sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right )\right )}{a c} \] Input:
int(1/(a*x+1)^2*(-a^2*x^2+1)/(-a*c*x+c)^(1/2),x)
Output:
(sqrt(c)*(2*sqrt( - a*x + 1) + sqrt(2)*log(sqrt( - a*x + 1) - sqrt(2)) - s qrt(2)*log(sqrt( - a*x + 1) + sqrt(2))))/(a*c)