Integrand size = 10, antiderivative size = 88 \[ \int e^{3 \text {arctanh}(a x)} x \, dx=\frac {9 \sqrt {1-a^2 x^2}}{2 a^2}+\frac {3 \left (1-a^2 x^2\right )^{3/2}}{2 a^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{5/2}}{a^2 (1-a x)^3}-\frac {9 \arcsin (a x)}{2 a^2} \]
3/2*(-a^2*x^2+1)^(3/2)/a^2/(-a*x+1)+(-a^2*x^2+1)^(5/2)/a^2/(-a*x+1)^3-9/2* arcsin(a*x)/a^2+9/2*(-a^2*x^2+1)^(1/2)/a^2
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int e^{3 \text {arctanh}(a x)} x \, dx=\sqrt {1-a^2 x^2} \left (\frac {3}{a^2}+\frac {x}{2 a}-\frac {4}{a^2 (-1+a x)}\right )-\frac {9 \arcsin (a x)}{2 a^2} \]
Time = 0.61 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {6674, 2164, 25, 2027, 2164, 25, 27, 563, 25, 2346, 27, 455, 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.
\(\displaystyle \int x e^{3 \text {arctanh}(a x)} \, dx\) |
\(\Big \downarrow \) 6674 |
\(\displaystyle \int \frac {x (a x+1)^2}{(1-a x) \sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 2164 |
\(\displaystyle -a \int -\frac {\left (x^2+\frac {x}{a}\right ) \sqrt {1-a^2 x^2}}{(1-a x)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a \int \frac {\left (x^2+\frac {x}{a}\right ) \sqrt {1-a^2 x^2}}{(1-a x)^2}dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle a \int \frac {x \left (x+\frac {1}{a}\right ) \sqrt {1-a^2 x^2}}{(1-a x)^2}dx\) |
\(\Big \downarrow \) 2164 |
\(\displaystyle -a^2 \int -\frac {x \left (1-a^2 x^2\right )^{3/2}}{a^2 (1-a x)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a^2 \int \frac {x \left (1-a^2 x^2\right )^{3/2}}{a^2 (1-a x)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3}dx\) |
\(\Big \downarrow \) 563 |
\(\displaystyle \frac {\int -\frac {a^2 x^2+3 a x+4}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {4 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}-\frac {\int \frac {a^2 x^2+3 a x+4}{\sqrt {1-a^2 x^2}}dx}{a}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}-\frac {-\frac {\int -\frac {3 a^2 (2 a x+3)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}-\frac {\frac {3}{2} \int \frac {2 a x+3}{\sqrt {1-a^2 x^2}}dx-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}-\frac {\frac {3}{2} \left (3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{a}\right )-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {4 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}-\frac {\frac {3}{2} \left (\frac {3 \arcsin (a x)}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a}\right )-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a}\) |
(4*Sqrt[1 - a^2*x^2])/(a^2*(1 - a*x)) - (-1/2*(x*Sqrt[1 - a^2*x^2]) + (3*( (-2*Sqrt[1 - a^2*x^2])/a + (3*ArcSin[a*x])/a))/2)/a
3.1.20.3.1 Defintions of rubi rules used
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_))*((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)^(p_), x_Symbo l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1) 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] && EqQ[b*c^2 + a*d^2 , 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*e Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a*e + b*d*x, x]* (a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + b*d*x, x], 0 ]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x )^m*((1 + a*x)^((n + 1)/2)/((1 - a*x)^((n - 1)/2)*Sqrt[1 - a^2*x^2])), x] / ; FreeQ[{a, c, m}, x] && IntegerQ[(n - 1)/2]
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18
method | result | size |
risch | \(-\frac {\left (a x +6\right ) \left (a^{2} x^{2}-1\right )}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \sqrt {a^{2}}}-\frac {4 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a^{3} \left (x -\frac {1}{a}\right )}\) | \(104\) |
default | \(\frac {1}{a^{2} \sqrt {-a^{2} x^{2}+1}}+a^{3} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+3 a^{2} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+3 a \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )\) | \(190\) |
meijerg | \(-\frac {\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}}{a^{2} \sqrt {\pi }}+\frac {\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}}{a \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {-6 \sqrt {\pi }+\frac {3 \sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}}{a^{2} \sqrt {\pi }}-\frac {3 \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{a \sqrt {\pi }\, \sqrt {-a^{2}}}\) | \(206\) |
-1/2*(a*x+6)*(a^2*x^2-1)/a^2/(-a^2*x^2+1)^(1/2)-9/2/a/(a^2)^(1/2)*arctan(( a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-4/a^3/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a )^(1/2)
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86 \[ \int e^{3 \text {arctanh}(a x)} x \, dx=\frac {14 \, a x + 18 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a^{2} x^{2} + 5 \, a x - 14\right )} \sqrt {-a^{2} x^{2} + 1} - 14}{2 \, {\left (a^{3} x - a^{2}\right )}} \]
1/2*(14*a*x + 18*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a^2*x ^2 + 5*a*x - 14)*sqrt(-a^2*x^2 + 1) - 14)/(a^3*x - a^2)
\[ \int e^{3 \text {arctanh}(a x)} x \, dx=\int \frac {x \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91 \[ \int e^{3 \text {arctanh}(a x)} x \, dx=-\frac {a x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, x^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {9 \, x}{2 \, \sqrt {-a^{2} x^{2} + 1} a} - \frac {9 \, \arcsin \left (a x\right )}{2 \, a^{2}} + \frac {7}{\sqrt {-a^{2} x^{2} + 1} a^{2}} \]
-1/2*a*x^3/sqrt(-a^2*x^2 + 1) - 3*x^2/sqrt(-a^2*x^2 + 1) + 9/2*x/(sqrt(-a^ 2*x^2 + 1)*a) - 9/2*arcsin(a*x)/a^2 + 7/(sqrt(-a^2*x^2 + 1)*a^2)
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int e^{3 \text {arctanh}(a x)} x \, dx=\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {x}{a} + \frac {6}{a^{2}}\right )} - \frac {9 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, a {\left | a \right |}} + \frac {8}{a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
1/2*sqrt(-a^2*x^2 + 1)*(x/a + 6/a^2) - 9/2*arcsin(a*x)*sgn(a)/(a*abs(a)) + 8/(a*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a))
Time = 0.06 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.16 \[ \int e^{3 \text {arctanh}(a x)} x \, dx=-\frac {\left (\frac {3}{\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2\,a}\right )\,\sqrt {1-a^2\,x^2}+\frac {9\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a}-\frac {4\,\sqrt {1-a^2\,x^2}}{a\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}}{\sqrt {-a^2}} \]