Integrand size = 25, antiderivative size = 84 \[ \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\frac {(1+a x)^2}{a^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {c-a^2 c x^2}}{a^2 c}-\frac {2 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2 \sqrt {c}} \] Output:
(a*x+1)^2/a^2/(-a^2*c*x^2+c)^(1/2)+2*(-a^2*c*x^2+c)^(1/2)/a^2/c-2*arctan(a *c^(1/2)*x/(-a^2*c*x^2+c)^(1/2))/a^2/c^(1/2)
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93 \[ \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\frac {(-3+a x) \sqrt {c-a^2 c x^2}}{-1+a x}+2 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{a^2 c} \] Input:
Integrate[(E^(2*ArcTanh[a*x])*x)/Sqrt[c - a^2*c*x^2],x]
Output:
(((-3 + a*x)*Sqrt[c - a^2*c*x^2])/(-1 + a*x) + 2*Sqrt[c]*ArcTan[(a*x*Sqrt[ c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(a^2*c)
Time = 0.54 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6701, 527, 455, 224, 216}
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.
\(\displaystyle \int \frac {x e^{2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle c \int \frac {x (a x+1)^2}{\left (c-a^2 c x^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 527 |
\(\displaystyle c \left (\frac {2 (a x+1)}{a^2 c \sqrt {c-a^2 c x^2}}-\frac {\int \frac {a x+2}{\sqrt {c-a^2 c x^2}}dx}{a c}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c \left (\frac {2 (a x+1)}{a^2 c \sqrt {c-a^2 c x^2}}-\frac {2 \int \frac {1}{\sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{a c}}{a c}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle c \left (\frac {2 (a x+1)}{a^2 c \sqrt {c-a^2 c x^2}}-\frac {2 \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{a c}}{a c}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle c \left (\frac {2 (a x+1)}{a^2 c \sqrt {c-a^2 c x^2}}-\frac {\frac {2 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}-\frac {\sqrt {c-a^2 c x^2}}{a c}}{a c}\right )\) |
Input:
Int[(E^(2*ArcTanh[a*x])*x)/Sqrt[c - a^2*c*x^2],x]
Output:
c*((2*(1 + a*x))/(a^2*c*Sqrt[c - a^2*c*x^2]) - (-(Sqrt[c - a^2*c*x^2]/(a*c )) + (2*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a*Sqrt[c]))/(a*c))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_S ymbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b*x^2])), x] + Simp[1/(b*d^(m - 2)) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[( 2^(n - 1)*c^(m + n - 1) - d^m*x^m*(c + d*x)^(n - 1))/(c - d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^(n/2) Int[x^m*(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ [c, 0]) && IGtQ[n/2, 0]
Time = 0.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {\sqrt {-a^{2} c \,x^{2}+c}}{a^{2} c}-\frac {2 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a \sqrt {a^{2} c}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{a^{3} c \left (x -\frac {1}{a}\right )}\) | \(103\) |
risch | \(-\frac {a^{2} x^{2}-1}{a^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {2 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a \sqrt {a^{2} c}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{a^{3} c \left (x -\frac {1}{a}\right )}\) | \(111\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-a^2*c*x^2+c)^(1/2)/a^2/c-2/a/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2* c*x^2+c)^(1/2))-2/a^3/c/(x-1/a)*(-(x-1/a)^2*a^2*c-2*(x-1/a)*a*c)^(1/2)
Time = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.98 \[ \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\left [-\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (a x - 3\right )}}{a^{3} c x - a^{2} c}, \frac {2 \, {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (a x - 3\right )}}{a^{3} c x - a^{2} c}\right ] \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^(1/2),x, algorithm="fric as")
Output:
[-((a*x - 1)*sqrt(-c)*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)* x - c) - sqrt(-a^2*c*x^2 + c)*(a*x - 3))/(a^3*c*x - a^2*c), (2*(a*x - 1)*s qrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + sqrt(-a^ 2*c*x^2 + c)*(a*x - 3))/(a^3*c*x - a^2*c)]
\[ \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=- \int \frac {x}{a x \sqrt {- a^{2} c x^{2} + c} - \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {a x^{2}}{a x \sqrt {- a^{2} c x^{2} + c} - \sqrt {- a^{2} c x^{2} + c}}\, dx \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*x/(-a**2*c*x**2+c)**(1/2),x)
Output:
-Integral(x/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x) - In tegral(a*x**2/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x)
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.80 \[ \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=-a {\left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{4} c x - a^{3} c} + \frac {2 \, \arcsin \left (a x\right )}{a^{3} \sqrt {c}} - \frac {\sqrt {-a^{2} c x^{2} + c}}{a^{3} c}\right )} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^(1/2),x, algorithm="maxi ma")
Output:
-a*(2*sqrt(-a^2*c*x^2 + c)/(a^4*c*x - a^3*c) + 2*arcsin(a*x)/(a^3*sqrt(c)) - sqrt(-a^2*c*x^2 + c)/(a^3*c))
Exception generated. \[ \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^(1/2),x, algorithm="giac ")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\int -\frac {x\,{\left (a\,x+1\right )}^2}{\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )} \,d x \] Input:
int(-(x*(a*x + 1)^2)/((c - a^2*c*x^2)^(1/2)*(a^2*x^2 - 1)),x)
Output:
int(-(x*(a*x + 1)^2)/((c - a^2*c*x^2)^(1/2)*(a^2*x^2 - 1)), x)
Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.17 \[ \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-2 \mathit {asin} \left (a x \right ) a x +2 \mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a x -4 \sqrt {-a^{2} x^{2}+1}-a^{2} x^{2}+a x +4\right )}{a^{2} c \left (\sqrt {-a^{2} x^{2}+1}+a x -1\right )} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*x/(-a^2*c*x^2+c)^(1/2),x)
Output:
(sqrt(c)*( - 2*sqrt( - a**2*x**2 + 1)*asin(a*x) - 2*asin(a*x)*a*x + 2*asin (a*x) + sqrt( - a**2*x**2 + 1)*a*x - 4*sqrt( - a**2*x**2 + 1) - a**2*x**2 + a*x + 4))/(a**2*c*(sqrt( - a**2*x**2 + 1) + a*x - 1))