Integrand size = 23, antiderivative size = 112 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\frac {1+a x}{a^5 c \sqrt {1-a^2 x^2}}+\frac {2 \sqrt {1-a^2 x^2}}{a^5 c}+\frac {x \sqrt {1-a^2 x^2}}{2 a^4 c}-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^5 c}-\frac {3 \arcsin (a x)}{2 a^5 c} \] Output:
(a*x+1)/a^5/c/(-a^2*x^2+1)^(1/2)+2*(-a^2*x^2+1)^(1/2)/a^5/c+1/2*x*(-a^2*x^ 2+1)^(1/2)/a^4/c-1/3*(-a^2*x^2+1)^(3/2)/a^5/c-3/2*arcsin(a*x)/a^5/c
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.66 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=-\frac {-16-9 a x+8 a^2 x^2+3 a^3 x^3+2 a^4 x^4+9 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^5 c \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^4)/(c - a^2*c*x^2),x]
Output:
-1/6*(-16 - 9*a*x + 8*a^2*x^2 + 3*a^3*x^3 + 2*a^4*x^4 + 9*Sqrt[1 - a^2*x^2 ]*ArcSin[a*x])/(a^5*c*Sqrt[1 - a^2*x^2])
Time = 0.48 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6698, 527, 2346, 25, 2346, 25, 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.
\(\displaystyle \int \frac {x^4 e^{\text {arctanh}(a x)}}{c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {x^4 (a x+1)}{\left (1-a^2 x^2\right )^{3/2}}dx}{c}\) |
\(\Big \downarrow \) 527 |
\(\displaystyle \frac {\frac {a x+1}{a^5 \sqrt {1-a^2 x^2}}-\frac {\int \frac {a^3 x^3+a^2 x^2+a x+1}{\sqrt {1-a^2 x^2}}dx}{a^4}}{c}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {\frac {a x+1}{a^5 \sqrt {1-a^2 x^2}}-\frac {-\frac {\int -\frac {3 x^2 a^4+5 x a^3+3 a^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {a x+1}{a^5 \sqrt {1-a^2 x^2}}-\frac {\frac {\int \frac {3 x^2 a^4+5 x a^3+3 a^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {\frac {a x+1}{a^5 \sqrt {1-a^2 x^2}}-\frac {\frac {-\frac {\int -\frac {a^4 (10 a x+9)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {a x+1}{a^5 \sqrt {1-a^2 x^2}}-\frac {\frac {\frac {\int \frac {a^4 (10 a x+9)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {a x+1}{a^5 \sqrt {1-a^2 x^2}}-\frac {\frac {\frac {1}{2} a^2 \int \frac {10 a x+9}{\sqrt {1-a^2 x^2}}dx-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {\frac {a x+1}{a^5 \sqrt {1-a^2 x^2}}-\frac {\frac {\frac {1}{2} a^2 \left (9 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {10 \sqrt {1-a^2 x^2}}{a}\right )-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {a x+1}{a^5 \sqrt {1-a^2 x^2}}-\frac {\frac {\frac {1}{2} a^2 \left (\frac {9 \arcsin (a x)}{a}-\frac {10 \sqrt {1-a^2 x^2}}{a}\right )-\frac {3}{2} a^2 x \sqrt {1-a^2 x^2}}{3 a^2}-\frac {1}{3} a x^2 \sqrt {1-a^2 x^2}}{a^4}}{c}\) |
Input:
Int[(E^ArcTanh[a*x]*x^4)/(c - a^2*c*x^2),x]
Output:
((1 + a*x)/(a^5*Sqrt[1 - a^2*x^2]) - (-1/3*(a*x^2*Sqrt[1 - a^2*x^2]) + ((- 3*a^2*x*Sqrt[1 - a^2*x^2])/2 + (a^2*((-10*Sqrt[1 - a^2*x^2])/a + (9*ArcSin [a*x])/a))/2)/(3*a^2))/a^4)/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_))*((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[(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_.))*(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]
Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08
method | result | size |
risch | \(-\frac {\left (2 a^{2} x^{2}+3 a x +10\right ) \left (a^{2} x^{2}-1\right )}{6 a^{5} \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{4} \sqrt {a^{2}}}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}}{c}\) | \(121\) |
default | \(-\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}+\frac {-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}}{a}}{c}\) | \(188\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)
Output:
-1/6*(2*a^2*x^2+3*a*x+10)*(a^2*x^2-1)/a^5/(-a^2*x^2+1)^(1/2)/c-(3/2/a^4/(a ^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/a^6/(x-1/a)*(-(x-1/a) ^2*a^2-2*a*(x-1/a))^(1/2))/c
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\frac {16 \, a x + 18 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} x^{3} + a^{2} x^{2} + 7 \, a x - 16\right )} \sqrt {-a^{2} x^{2} + 1} - 16}{6 \, {\left (a^{6} c x - a^{5} c\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c),x, algorithm="fric as")
Output:
1/6*(16*a*x + 18*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (2*a^3 *x^3 + a^2*x^2 + 7*a*x - 16)*sqrt(-a^2*x^2 + 1) - 16)/(a^6*c*x - a^5*c)
\[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\frac {\int \frac {x^{4}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a**2*c*x**2+c),x)
Output:
(Integral(x**4/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x ) + Integral(a*x**5/(-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. 204 vs. \(2 (98) = 196\).
Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.82 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=-\frac {1}{6} \, a {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{7} c^{2} x + a^{6} c^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{7} c^{2} x - a^{6} c^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{a^{7} c x + a^{6} c} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{a^{7} c x - a^{6} c} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4} c} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{5} c} + \frac {9 \, \arcsin \left (a x\right )}{a^{6} c} - \frac {10 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6} c}\right )} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c),x, algorithm="maxi ma")
Output:
-1/6*a*(3*sqrt(-a^2*x^2 + 1)*c/(a^7*c^2*x + a^6*c^2) + 3*sqrt(-a^2*x^2 + 1 )*c/(a^7*c^2*x - a^6*c^2) - 3*sqrt(-a^2*x^2 + 1)/(a^7*c*x + a^6*c) + 3*sqr t(-a^2*x^2 + 1)/(a^7*c*x - a^6*c) - 2*sqrt(-a^2*x^2 + 1)*x^2/(a^4*c) - 3*s qrt(-a^2*x^2 + 1)*x/(a^5*c) + 9*arcsin(a*x)/(a^6*c) - 10*sqrt(-a^2*x^2 + 1 )/(a^6*c))
Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left (x {\left (\frac {2 \, x}{a^{3} c} + \frac {3}{a^{4} c}\right )} + \frac {10}{a^{5} c}\right )} - \frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, a^{4} c {\left | a \right |}} + \frac {2}{a^{4} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c),x, algorithm="giac ")
Output:
1/6*sqrt(-a^2*x^2 + 1)*(x*(2*x/(a^3*c) + 3/(a^4*c)) + 10/(a^5*c)) - 3/2*ar csin(a*x)*sgn(a)/(a^4*c*abs(a)) + 2/(a^4*c*((sqrt(-a^2*x^2 + 1)*abs(a) + a )/(a^2*x) - 1)*abs(a))
Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\frac {5\,\sqrt {1-a^2\,x^2}}{3\,a^5\,c}-\frac {\sqrt {1-a^2\,x^2}}{\sqrt {-a^2}\,\left (a^3\,c\,\sqrt {-a^2}-a^4\,c\,x\,\sqrt {-a^2}\right )}+\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^4\,c}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^4\,c\,\sqrt {-a^2}}+\frac {x^2\,\sqrt {1-a^2\,x^2}}{3\,a^3\,c} \] Input:
int((x^4*(a*x + 1))/((c - a^2*c*x^2)*(1 - a^2*x^2)^(1/2)),x)
Output:
(5*(1 - a^2*x^2)^(1/2))/(3*a^5*c) - (1 - a^2*x^2)^(1/2)/((-a^2)^(1/2)*(a^3 *c*(-a^2)^(1/2) - a^4*c*x*(-a^2)^(1/2))) + (x*(1 - a^2*x^2)^(1/2))/(2*a^4* c) - (3*asinh(x*(-a^2)^(1/2)))/(2*a^4*c*(-a^2)^(1/2)) + (x^2*(1 - a^2*x^2) ^(1/2))/(3*a^3*c)
Time = 0.15 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\frac {-9 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-9 \mathit {asin} \left (a x \right ) a x +9 \mathit {asin} \left (a x \right )+2 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+7 \sqrt {-a^{2} x^{2}+1}\, a x -18 \sqrt {-a^{2} x^{2}+1}-2 a^{4} x^{4}-3 a^{3} x^{3}-8 a^{2} x^{2}+7 a x +18}{6 a^{5} c \left (\sqrt {-a^{2} x^{2}+1}+a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a^2*c*x^2+c),x)
Output:
( - 9*sqrt( - a**2*x**2 + 1)*asin(a*x) - 9*asin(a*x)*a*x + 9*asin(a*x) + 2 *sqrt( - a**2*x**2 + 1)*a**3*x**3 + sqrt( - a**2*x**2 + 1)*a**2*x**2 + 7*s qrt( - a**2*x**2 + 1)*a*x - 18*sqrt( - a**2*x**2 + 1) - 2*a**4*x**4 - 3*a* *3*x**3 - 8*a**2*x**2 + 7*a*x + 18)/(6*a**5*c*(sqrt( - a**2*x**2 + 1) + a* x - 1))