Integrand size = 17, antiderivative size = 74 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^2} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{a^2 c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 c^2 (1-a x)^3}+\frac {\arcsin (a x)}{a^2 c^2} \] Output:
-2*(-a^2*x^2+1)^(1/2)/a^2/c^2/(-a*x+1)+1/3*(-a^2*x^2+1)^(3/2)/a^2/c^2/(-a* x+1)^3+arcsin(a*x)/a^2/c^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^2} \, dx=-\frac {(1+a x)^{3/2}-4 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1-a x)\right )}{3 a^2 c^2 (1-a x)^{3/2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x)/(c - a*c*x)^2,x]
Output:
-1/3*((1 + a*x)^(3/2) - 4*Sqrt[2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(a^2*c^2*(1 - a*x)^(3/2))
Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6678, 27, 571, 463, 25, 223}
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.
\(\displaystyle \int \frac {x e^{\text {arctanh}(a x)}}{(c-a c x)^2} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {x \sqrt {1-a^2 x^2}}{c^3 (1-a x)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {x \sqrt {1-a^2 x^2}}{(1-a x)^3}dx}{c^2}\) |
\(\Big \downarrow \) 571 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 (1-a x)^3}-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^2}dx}{a}}{c^2}\) |
\(\Big \downarrow \) 463 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 (1-a x)^3}-\frac {\int -\frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {2 \sqrt {1-a^2 x^2}}{a (1-a x)}}{a}}{c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 (1-a x)^3}-\frac {\frac {2 \sqrt {1-a^2 x^2}}{a (1-a x)}-\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}}{c^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 (1-a x)^3}-\frac {\frac {2 \sqrt {1-a^2 x^2}}{a (1-a x)}-\frac {\arcsin (a x)}{a}}{a}}{c^2}\) |
Input:
Int[(E^ArcTanh[a*x]*x)/(c - a*c*x)^2,x]
Output:
((1 - a^2*x^2)^(3/2)/(3*a^2*(1 - a*x)^3) - ((2*Sqrt[1 - a^2*x^2])/(a*(1 - a*x)) - ArcSin[a*x]/a)/a)/c^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(-c)^(-n - 2))*d^(2*n + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b^(n + 2)*(c + d*x ))), x] - Simp[d^(2*n + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[ (2^(-n - 1)*(-c)^(-n - 1) - (-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; F reeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*(n + p + 1))), x] + Simp[n/(2*d* (n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ((LtQ[n, -1] && !IGtQ[n + p + 1, 0]) || (LtQ[n, 0] && LtQ[p, -1]) || EqQ[n + 2*p + 2, 0]) && NeQ[n + p + 1, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(160\) vs. \(2(68)=136\).
Time = 0.21 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.18
method | result | size |
default | \(\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {a^{2}}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{3}}+\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{3} \left (x -\frac {1}{a}\right )}}{c^{2}}\) | \(161\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/c^2*(1/a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/a^3*(1/3 /a/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^ 2-2*a*(x-1/a))^(1/2))+3/a^3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))
Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^2} \, dx=-\frac {5 \, a^{2} x^{2} - 10 \, a x + 6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x - 5\right )} + 5}{3 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c)^2,x, algorithm="fricas")
Output:
-1/3*(5*a^2*x^2 - 10*a*x + 6*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1)*(7*a*x - 5) + 5)/(a^4*c^2*x^2 - 2*a^3 *c^2*x + a^2*c^2)
\[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^2} \, dx=\frac {\int \frac {x}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x/(-a*c*x+c)**2,x)
Output:
(Integral(x/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**2/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^2} \, dx=\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} + \frac {7 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{3} c^{2} x - a^{2} c^{2}\right )}} + \frac {\arcsin \left (a x\right )}{a^{2} c^{2}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c)^2,x, algorithm="maxima")
Output:
2/3*sqrt(-a^2*x^2 + 1)/(a^4*c^2*x^2 - 2*a^3*c^2*x + a^2*c^2) + 7/3*sqrt(-a ^2*x^2 + 1)/(a^3*c^2*x - a^2*c^2) + arcsin(a*x)/(a^2*c^2)
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^2} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c)^2,x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: 1/ab s(sageVARa)/sageVARc^2/sageVARa/sageVARc*(-(-6*sageVARc*atan(i)-(-7*i)*sag eVARc)/3*sign((sageVARa*sageVARc*sageVARx-sageVARc)^-1)*sign(sageVARa)*sig n(sageVARc)-2*sageV
Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.46 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^2} \, dx=\frac {4}{3\,a^2\,c^2\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {3}{a^2\,c^2\,\sqrt {1-a^2\,x^2}}-\frac {7\,x}{3\,a\,c^2\,\sqrt {1-a^2\,x^2}}+\frac {4\,x}{3\,a\,c^2\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{a^3\,c^2} \] Input:
int((x*(a*x + 1))/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^2),x)
Output:
4/(3*a^2*c^2*(1 - a^2*x^2)^(3/2)) - 3/(a^2*c^2*(1 - a^2*x^2)^(1/2)) - (7*x )/(3*a*c^2*(1 - a^2*x^2)^(1/2)) + (4*x)/(3*a*c^2*(1 - a^2*x^2)^(3/2)) - (a sinh(x*(-a^2)^(1/2))*(-a^2)^(1/2))/(a^3*c^2)
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.51 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^2} \, dx=\frac {3 \mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-9 \mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+9 \mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-3 \mathit {asin} \left (a x \right )+2 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-18 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+8}{3 a^{2} c^{2} \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x/(-a*c*x+c)^2,x)
Output:
(3*asin(a*x)*tan(asin(a*x)/2)**3 - 9*asin(a*x)*tan(asin(a*x)/2)**2 + 9*asi n(a*x)*tan(asin(a*x)/2) - 3*asin(a*x) + 2*tan(asin(a*x)/2)**3 - 18*tan(asi n(a*x)/2) + 8)/(3*a**2*c**2*(tan(asin(a*x)/2)**3 - 3*tan(asin(a*x)/2)**2 + 3*tan(asin(a*x)/2) - 1))