Integrand size = 20, antiderivative size = 119 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=-\frac {4 c \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a} \] Output:
-4*c*(-a^2*x^2+1)^(1/2)/a/(-a*c*x+c)^(1/2)-2/3*c^2*(-a^2*x^2+1)^(3/2)/a/(- a*c*x+c)^(3/2)+4*2^(1/2)*c^(1/2)*arctanh(1/2*c^(1/2)*(-a^2*x^2+1)^(1/2)*2^ (1/2)/(-a*c*x+c)^(1/2))/a
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.56 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=-\frac {2 \sqrt {c-a c x} \left (\sqrt {1+a x} (7+a x)-6 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{3 a \sqrt {1-a x}} \] Input:
Integrate[E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x],x]
Output:
(-2*Sqrt[c - a*c*x]*(Sqrt[1 + a*x]*(7 + a*x) - 6*Sqrt[2]*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/(3*a*Sqrt[1 - a*x])
Time = 0.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6677, 466, 466, 471, 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.
\(\displaystyle \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{5/2}}dx\) |
\(\Big \downarrow \) 466 |
\(\displaystyle c^3 \left (\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^{3/2}}dx}{c}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (c-a c x)^{3/2}}\right )\) |
\(\Big \downarrow \) 466 |
\(\displaystyle c^3 \left (\frac {2 \left (\frac {2 \int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}}dx}{c}-\frac {2 \sqrt {1-a^2 x^2}}{a c \sqrt {c-a c x}}\right )}{c}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (c-a c x)^{3/2}}\right )\) |
\(\Big \downarrow \) 471 |
\(\displaystyle c^3 \left (\frac {2 \left (-4 a \int \frac {1}{\frac {a^2 c^2 \left (1-a^2 x^2\right )}{c-a c x}-2 a^2 c}d\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {2 \sqrt {1-a^2 x^2}}{a c \sqrt {c-a c x}}\right )}{c}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (c-a c x)^{3/2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c^3 \left (\frac {2 \left (\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a c^{3/2}}-\frac {2 \sqrt {1-a^2 x^2}}{a c \sqrt {c-a c x}}\right )}{c}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (c-a c x)^{3/2}}\right )\) |
Input:
Int[E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x],x]
Output:
c^3*((-2*(1 - a^2*x^2)^(3/2))/(3*a*c*(c - a*c*x)^(3/2)) + (2*((-2*Sqrt[1 - a^2*x^2])/(a*c*Sqrt[c - a*c*x]) + (2*Sqrt[2]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^ 2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(a*c^(3/2))))/c)
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] - Simp[2*b*c*(p/(d^ 2*(n + 2*p + 1))) Int[(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; Fr eeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LeQ[-2, n, 0 ] || EqQ[n + p + 1, 0]) && NeQ[n + 2*p + 1, 0] && IntegerQ[2*p]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[2*d Subst[Int[1/(2*b*c + d^2*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^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[c^n Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (6 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-a x \sqrt {c \left (a x +1\right )}-7 \sqrt {c \left (a x +1\right )}\right )}{3 \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, a}\) | \(95\) |
risch | \(\frac {2 \left (a x +7\right ) \left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{3 a \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(153\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(6*c^(1/2)*2^(1/2)*arctanh(1/2* (c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))-a*x*(c*(a*x+1))^(1/2)-7*(c*(a*x+1))^(1/ 2))/(a*x-1)/(c*(a*x+1))^(1/2)/a
Time = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.82 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 7\right )}\right )}}{3 \, {\left (a^{2} x - a\right )}}, \frac {2 \, {\left (6 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{2 \, {\left (a c x - c\right )}}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 7\right )}\right )}}{3 \, {\left (a^{2} x - a\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2),x, algorithm="fric as")
Output:
[2/3*(3*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sq rt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 7))/(a^2*x - a), 2/3*(6*sqrt(2) *(a*x - 1)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c) *sqrt(-c)/(a*c*x - c)) + sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*(a*x + 7))/(a ^2*x - a)]
\[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2),x)
Output:
Integral(sqrt(-c*(a*x - 1))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)
\[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\int { \frac {\sqrt {-a c x + c} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2),x, algorithm="maxi ma")
Output:
integrate(sqrt(-a*c*x + c)*(a*x + 1)^3/(-a^2*x^2 + 1)^(3/2), x)
Exception generated. \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2),x, algorithm="giac ")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c - a*c*x)^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
int(((c - a*c*x)^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2), x)
Time = 0.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.43 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, \left (-\sqrt {a x +1}\, a x -7 \sqrt {a x +1}-6 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )+8 \sqrt {2}\right )}{3 a} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2),x)
Output:
(2*sqrt(c)*( - sqrt(a*x + 1)*a*x - 7*sqrt(a*x + 1) - 6*sqrt(2)*log(tan(asi n(sqrt( - a*x + 1)/sqrt(2))/2)) + 8*sqrt(2)))/(3*a)