Integrand size = 20, antiderivative size = 76 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a}+\frac {2 (c-a c x)^{3/2}}{3 a c}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \] Output:
4*(-a*c*x+c)^(1/2)/a+2/3*(-a*c*x+c)^(3/2)/a/c-4*2^(1/2)*c^(1/2)*arctanh(1/ 2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))/a
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=-\frac {2 (-7+a x) \sqrt {c-a c x}+12 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{3 a} \] Input:
Integrate[Sqrt[c - a*c*x]/E^(2*ArcTanh[a*x]),x]
Output:
-1/3*(2*(-7 + a*x)*Sqrt[c - a*c*x] + 12*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a *c*x]/(Sqrt[2]*Sqrt[c])])/a
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6680, 35, 60, 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.
\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(1-a x) \sqrt {c-a c x}}{a x+1}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {\int \frac {(c-a c x)^{3/2}}{a x+1}dx}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 c \int \frac {\sqrt {c-a c x}}{a x+1}dx+\frac {2 (c-a c x)^{3/2}}{3 a}}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 c \left (2 c \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx+\frac {2 \sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {4 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}}{c}\) |
Input:
Int[Sqrt[c - a*c*x]/E^(2*ArcTanh[a*x]),x]
Output:
((2*(c - a*c*x)^(3/2))/(3*a) + 2*c*((2*Sqrt[c - a*c*x])/a - (2*Sqrt[2]*Sqr t[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 = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {2 \left (a x -7\right ) \left (a x -1\right ) c}{3 a \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a}\) | \(57\) |
derivativedivides | \(\frac {\frac {2 \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 \sqrt {-a c x +c}\, c -4 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\) | \(59\) |
default | \(-\frac {2 \left (-\frac {\left (-a c x +c \right )^{\frac {3}{2}}}{3}-2 \sqrt {-a c x +c}\, c +2 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\) | \(59\) |
pseudoelliptic | \(\frac {-4 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-\frac {2 a x \sqrt {-c \left (a x -1\right )}}{3}+\frac {14 \sqrt {-c \left (a x -1\right )}}{3}}{a}\) | \(60\) |
Input:
int((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)
Output:
2/3*(a*x-7)*(a*x-1)/a/(-c*(a*x-1))^(1/2)*c-4*2^(1/2)*c^(1/2)*arctanh(1/2*( -a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))/a
Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.64 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}, -\frac {2 \, {\left (6 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}\right ] \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")
Output:
[2/3*(3*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - sqrt(-a*c*x + c)*(a*x - 7))/a, -2/3*(6*sqrt(2)*sqrt(-c)* arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + sqrt(-a*c*x + c)*( a*x - 7))/a]
Time = 5.66 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.25 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \left (- \frac {2 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - 2 c \sqrt {- a c x + c} - \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (- x + 2 \left (\begin {cases} x & \text {for}\: a = 0 \\\frac {\log {\left (a x + 1 \right )}}{a} & \text {otherwise} \end {cases}\right )\right ) & \text {otherwise} \end {cases} \] Input:
integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)
Output:
Piecewise((-2*(-2*sqrt(2)*c**2*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c))) /sqrt(-c) - 2*c*sqrt(-a*c*x + c) - (-a*c*x + c)**(3/2)/3)/(a*c), Ne(a*c, 0 )), (sqrt(c)*(-x + 2*Piecewise((x, Eq(a, 0)), (log(a*x + 1)/a, True))), Tr ue))
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \, {\left (3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + {\left (-a c x + c\right )}^{\frac {3}{2}} + 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")
Output:
2/3*(3*sqrt(2)*c^(3/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)* sqrt(c) + sqrt(-a*c*x + c))) + (-a*c*x + c)^(3/2) + 6*sqrt(-a*c*x + c)*c)/ (a*c)
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c^{2} + 6 \, \sqrt {-a c x + c} a^{2} c^{3}\right )}}{3 \, a^{3} c^{3}} \] Input:
integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
Output:
4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)) + 2 /3*((-a*c*x + c)^(3/2)*a^2*c^2 + 6*sqrt(-a*c*x + c)*a^2*c^3)/(a^3*c^3)
Time = 14.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {c-a\,c\,x}}{a}+\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c}-\frac {4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a} \] Input:
int(-((a^2*x^2 - 1)*(c - a*c*x)^(1/2))/(a*x + 1)^2,x)
Output:
(4*(c - a*c*x)^(1/2))/a + (2*(c - a*c*x)^(3/2))/(3*a*c) - (4*2^(1/2)*c^(1/ 2)*atanh((2^(1/2)*(c - a*c*x)^(1/2))/(2*c^(1/2))))/a
Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, \left (-\sqrt {-a x +1}\, a x +7 \sqrt {-a x +1}+3 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right )-3 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right )\right )}{3 a} \] Input:
int((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x)
Output:
(2*sqrt(c)*( - sqrt( - a*x + 1)*a*x + 7*sqrt( - a*x + 1) + 3*sqrt(2)*log(s qrt( - a*x + 1) - sqrt(2)) - 3*sqrt(2)*log(sqrt( - a*x + 1) + sqrt(2))))/( 3*a)