Integrand size = 23, antiderivative size = 70 \[ \int \frac {e^{3 \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=\frac {(1+a x)^3}{3 a^2 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)}{a^2 c \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)}{a^2 c} \] Output:
1/3*(a*x+1)^3/a^2/c/(-a^2*x^2+1)^(3/2)-2*(a*x+1)/a^2/c/(-a^2*x^2+1)^(1/2)+ arcsin(a*x)/a^2/c
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {e^{3 \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=\frac {5-2 a x-7 a^2 x^2+3 (-1+a x) \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2 c (-1+a x) \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^(3*ArcTanh[a*x])*x)/(c - a^2*c*x^2),x]
Output:
(5 - 2*a*x - 7*a^2*x^2 + 3*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^ 2*c*(-1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.50 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6698, 531, 27, 457, 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.
\(\displaystyle \int \frac {x e^{3 \text {arctanh}(a x)}}{c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {x (a x+1)^3}{\left (1-a^2 x^2\right )^{5/2}}dx}{c}\) |
\(\Big \downarrow \) 531 |
\(\displaystyle \frac {\frac {\int -\frac {3 a (a x+1)^2}{\left (1-a^2 x^2\right )^{3/2}}dx}{3 a^2}+\frac {(a x+1)^3}{3 a^2 \left (1-a^2 x^2\right )^{3/2}}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(a x+1)^2}{\left (1-a^2 x^2\right )^{3/2}}dx}{a}}{c}\) |
\(\Big \downarrow \) 457 |
\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\frac {2 (a x+1)}{a \sqrt {1-a^2 x^2}}-\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\frac {2 (a x+1)}{a \sqrt {1-a^2 x^2}}-\frac {\arcsin (a x)}{a}}{a}}{c}\) |
Input:
Int[(E^(3*ArcTanh[a*x])*x)/(c - a^2*c*x^2),x]
Output:
((1 + a*x)^3/(3*a^2*(1 - a^2*x^2)^(3/2)) - ((2*(1 + a*x))/(a*Sqrt[1 - a^2* x^2]) - ArcSin[a*x]/a)/a)/c
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_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ b*c^2 + a*d^2, 0] && LtQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomi alRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a + b*x^2, x], x, 1]}, Simp[(c + d*x)^n*(a*f - b*e*x)*((a + b*x^2)^(p + 1)/( 2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b *x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(c + d*x)*Qx - a*d*f*n + b*c*e*(2*p + 3) + b*d*e*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGt Q[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && GtQ[n, 1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*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 + 1)/2, 0] && !IntegerQ[p - n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(186\) vs. \(2(64)=128\).
Time = 0.18 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.67
method | result | size |
default | \(-\frac {a \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )+\frac {4 x}{a \sqrt {-a^{2} x^{2}+1}}+\frac {3}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {4}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {4 \left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right )}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{2}}}{c}\) | \(187\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)
Output:
-1/c*(a*(x/a^2/(-a^2*x^2+1)^(1/2)-1/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/( -a^2*x^2+1)^(1/2)))+4*x/a/(-a^2*x^2+1)^(1/2)+3/a^2/(-a^2*x^2+1)^(1/2)+4/a^ 2*(1/3/a/(x-1/a)/(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+1/3/a*(-2*(x-1/a)*a^2- 2*a)/(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)))
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.37 \[ \int \frac {e^{3 \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=-\frac {5 \, a^{2} x^{2} - 10 \, a x + 6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x - 5\right )} + 5}{3 \, {\left (a^{4} c x^{2} - 2 \, a^{3} c x + a^{2} c\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x/(-a^2*c*x^2+c),x, algorithm="fric as")
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*x^2 - 2*a^3*c *x + a^2*c)
\[ \int \frac {e^{3 \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=\frac {\int \frac {x}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x/(-a**2*c*x**2+c),x)
Output:
(Integral(x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a*x**2/(a**4*x**4*sqrt(-a**2 *x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt( -a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**4/(a**4*x** 4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x** 2 + 1)), x))/c
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (64) = 128\).
Time = 0.18 (sec) , antiderivative size = 241, normalized size of antiderivative = 3.44 \[ \int \frac {e^{3 \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=\frac {1}{3} \, a {\left (\frac {2 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac {2 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac {2 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{4} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac {2 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{4} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}} - \frac {7 \, x}{\sqrt {-a^{2} x^{2} + 1} a^{2} c} + \frac {3 \, \arcsin \left (a x\right )}{a^{3} c} - \frac {9}{\sqrt {-a^{2} x^{2} + 1} a^{3} c}\right )} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x/(-a^2*c*x^2+c),x, algorithm="maxi ma")
Output:
1/3*a*(2*a*c/(sqrt(-a^2*x^2 + 1)*a^5*c^2*x + sqrt(-a^2*x^2 + 1)*a^4*c^2) - 2*a*c/(sqrt(-a^2*x^2 + 1)*a^5*c^2*x - sqrt(-a^2*x^2 + 1)*a^4*c^2) - 2*c/( sqrt(-a^2*x^2 + 1)*a^4*c^2*x + sqrt(-a^2*x^2 + 1)*a^3*c^2) - 2*c/(sqrt(-a^ 2*x^2 + 1)*a^4*c^2*x - sqrt(-a^2*x^2 + 1)*a^3*c^2) - 7*x/(sqrt(-a^2*x^2 + 1)*a^2*c) + 3*arcsin(a*x)/(a^3*c) - 9/(sqrt(-a^2*x^2 + 1)*a^3*c))
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.60 \[ \int \frac {e^{3 \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{a c {\left | a \right |}} + \frac {2 \, {\left (\frac {12 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 5\right )}}{3 \, a c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\left | a \right |}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x/(-a^2*c*x^2+c),x, algorithm="giac ")
Output:
arcsin(a*x)*sgn(a)/(a*c*abs(a)) + 2/3*(12*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/ (a^2*x) - 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) - 5)/(a*c*((sqrt(- a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^3*abs(a))
Time = 23.81 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.54 \[ \int \frac {e^{3 \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=\frac {4}{3\,a^2\,c\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {3}{a^2\,c\,\sqrt {1-a^2\,x^2}}-\frac {7\,x}{3\,a\,c\,\sqrt {1-a^2\,x^2}}+\frac {4\,x}{3\,a\,c\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{a^3\,c} \] Input:
int((x*(a*x + 1)^3)/((c - a^2*c*x^2)*(1 - a^2*x^2)^(3/2)),x)
Output:
4/(3*a^2*c*(1 - a^2*x^2)^(3/2)) - 3/(a^2*c*(1 - a^2*x^2)^(1/2)) - (7*x)/(3 *a*c*(1 - a^2*x^2)^(1/2)) + (4*x)/(3*a*c*(1 - a^2*x^2)^(3/2)) - (asinh(x*( -a^2)^(1/2))*(-a^2)^(1/2))/(a^3*c)
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.60 \[ \int \frac {e^{3 \text {arctanh}(a x)} x}{c-a^2 c x^2} \, dx=\frac {3 \mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-9 \mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+9 \mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-3 \mathit {asin} \left (a x \right )+2 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-18 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+8}{3 a^{2} c \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x/(-a^2*c*x^2+c),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*(tan(asin(a*x)/2)**3 - 3*tan(asin(a*x)/2)**2 + 3* tan(asin(a*x)/2) - 1))