Integrand size = 23, antiderivative size = 137 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1+a x}{5 a^6 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2}{3 a^6 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {11 x}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {23 x}{15 a^5 c^3 \sqrt {1-a^2 x^2}}-\frac {\arcsin (a x)}{a^6 c^3} \] Output:
1/5*(a*x+1)/a^6/c^3/(-a^2*x^2+1)^(5/2)-2/3/a^6/c^3/(-a^2*x^2+1)^(3/2)-11/1 5*x/a^5/c^3/(-a^2*x^2+1)^(3/2)+1/a^6/c^3/(-a^2*x^2+1)^(1/2)+23/15*x/a^5/c^ 3/(-a^2*x^2+1)^(1/2)-arcsin(a*x)/a^6/c^3
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {8+7 a x-27 a^2 x^2-8 a^3 x^3+23 a^4 x^4-15 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2} \arcsin (a x)}{15 a^6 c^3 (-1+a x)^2 (1+a x) \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^3,x]
Output:
(8 + 7*a*x - 27*a^2*x^2 - 8*a^3*x^3 + 23*a^4*x^4 - 15*(-1 + a*x)^2*(1 + a* x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(15*a^6*c^3*(-1 + a*x)^2*(1 + a*x)*Sqrt[ 1 - a^2*x^2])
Time = 0.47 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6698, 529, 2345, 2345, 27, 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.
\(\displaystyle \int \frac {x^5 e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {x^5 (a x+1)}{\left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\) |
\(\Big \downarrow \) 529 |
\(\displaystyle \frac {\frac {a x+1}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int \frac {\frac {5 x^4}{a}+\frac {5 x^3}{a^2}+\frac {5 x^2}{a^3}+\frac {5 x}{a^4}+\frac {1}{a^5}}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^3}\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\frac {15 x^2}{a^3}+\frac {15 x}{a^4}+\frac {8}{a^5}}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {11 a x+10}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {23 a x+15}{a^6 \sqrt {1-a^2 x^2}}-\int \frac {15}{a^5 \sqrt {1-a^2 x^2}}dx\right )-\frac {11 a x+10}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {23 a x+15}{a^6 \sqrt {1-a^2 x^2}}-\frac {15 \int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a^5}\right )-\frac {11 a x+10}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {23 a x+15}{a^6 \sqrt {1-a^2 x^2}}-\frac {15 \arcsin (a x)}{a^6}\right )-\frac {11 a x+10}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
Input:
Int[(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^3,x]
Output:
((1 + a*x)/(5*a^6*(1 - a^2*x^2)^(5/2)) + (-1/3*(10 + 11*a*x)/(a^6*(1 - a^2 *x^2)^(3/2)) + ((15 + 23*a*x)/(a^6*Sqrt[1 - a^2*x^2]) - (15*ArcSin[a*x])/a ^6)/3)/5)/c^3
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[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(398\) vs. \(2(119)=238\).
Time = 0.28 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.91
method | result | size |
default | \(-\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{5} \sqrt {a^{2}}}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{7}}+\frac {23 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a^{7} \left (x -\frac {1}{a}\right )}+\frac {7 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a^{7} \left (x +\frac {1}{a}\right )}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{8}}+\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a^{7}}}{c^{3}}\) | \(399\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOS E)
Output:
-1/c^3*(1/a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/a^7*( 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))+23/16/a^7/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1 /2)+7/16/a^7/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/4/a^8*(1/5/a/(x- 1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-2/5*a*(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))) +1/8/a^7*(-1/3/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)-1/3/(x+1/a)* (-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))
Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {8 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 8 \, a x + 30 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (23 \, a^{4} x^{4} - 8 \, a^{3} x^{3} - 27 \, a^{2} x^{2} + 7 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} - 8}{15 \, {\left (a^{11} c^{3} x^{5} - a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{8} c^{3} x^{2} + a^{7} c^{3} x - a^{6} c^{3}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^3,x, algorithm="fr icas")
Output:
1/15*(8*a^5*x^5 - 8*a^4*x^4 - 16*a^3*x^3 + 16*a^2*x^2 + 8*a*x + 30*(a^5*x^ 5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (23*a^4*x^4 - 8*a^3*x^3 - 27*a^2*x^2 + 7*a*x + 8)*sqrt(-a^2* x^2 + 1) - 8)/(a^11*c^3*x^5 - a^10*c^3*x^4 - 2*a^9*c^3*x^3 + 2*a^8*c^3*x^2 + a^7*c^3*x - a^6*c^3)
\[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{5}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{6}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a**2*c*x**2+c)**3,x)
Output:
(Integral(x**5/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x **2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**6/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2* x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/ c**3
\[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {{\left (a x + 1\right )} x^{5}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^3,x, algorithm="ma xima")
Output:
-a*integrate(x^6/((a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*sqrt (a*x + 1)*sqrt(-a*x + 1)), x) + 1/15*(10*a^2*x^2 + 15*(a^2*x^2 - 1)^2 - 7) /((-a^2*x^2 + 1)^(5/2)*a^6*c^3)
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^3,x, algorithm="gi ac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 14.38 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.28 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^8\,c^3\,x^2+2\,a^7\,c^3\,x+a^6\,c^3\right )}-\frac {3\,\sqrt {1-a^2\,x^2}}{10\,\left (a^8\,c^3\,x^2-2\,a^7\,c^3\,x+a^6\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^4\,c^3\,\sqrt {-a^2}+3\,a^6\,c^3\,x^2\,\sqrt {-a^2}-a^7\,c^3\,x^3\,\sqrt {-a^2}-3\,a^5\,c^3\,x\,\sqrt {-a^2}\right )}+\frac {19\,\sqrt {1-a^2\,x^2}}{48\,\left (a^4\,c^3\,\sqrt {-a^2}+a^5\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {91\,\sqrt {1-a^2\,x^2}}{80\,\left (a^4\,c^3\,\sqrt {-a^2}-a^5\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^5\,c^3\,\sqrt {-a^2}} \] Input:
int((x^5*(a*x + 1))/((c - a^2*c*x^2)^3*(1 - a^2*x^2)^(1/2)),x)
Output:
(1 - a^2*x^2)^(1/2)/(24*(a^6*c^3 + 2*a^7*c^3*x + a^8*c^3*x^2)) - (3*(1 - a ^2*x^2)^(1/2))/(10*(a^6*c^3 - 2*a^7*c^3*x + a^8*c^3*x^2)) - (1 - a^2*x^2)^ (1/2)/(20*(-a^2)^(1/2)*(a^4*c^3*(-a^2)^(1/2) + 3*a^6*c^3*x^2*(-a^2)^(1/2) - a^7*c^3*x^3*(-a^2)^(1/2) - 3*a^5*c^3*x*(-a^2)^(1/2))) + (19*(1 - a^2*x^2 )^(1/2))/(48*(a^4*c^3*(-a^2)^(1/2) + a^5*c^3*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (91*(1 - a^2*x^2)^(1/2))/(80*(a^4*c^3*(-a^2)^(1/2) - a^5*c^3*x*(-a^2)^( 1/2))*(-a^2)^(1/2)) - asinh(x*(-a^2)^(1/2))/(a^5*c^3*(-a^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{3} x^{3}+15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}+15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -15 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-7 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+7 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+7 \sqrt {-a^{2} x^{2}+1}\, a x -7 \sqrt {-a^{2} x^{2}+1}+23 a^{4} x^{4}-8 a^{3} x^{3}-27 a^{2} x^{2}+7 a x +8}{15 \sqrt {-a^{2} x^{2}+1}\, a^{6} c^{3} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^3,x)
Output:
( - 15*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**3*x**3 + 15*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 + 15*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 15*sqr t( - a**2*x**2 + 1)*asin(a*x) - 7*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 7*sqr t( - a**2*x**2 + 1)*a**2*x**2 + 7*sqrt( - a**2*x**2 + 1)*a*x - 7*sqrt( - a **2*x**2 + 1) + 23*a**4*x**4 - 8*a**3*x**3 - 27*a**2*x**2 + 7*a*x + 8)/(15 *sqrt( - a**2*x**2 + 1)*a**6*c**3*(a**3*x**3 - a**2*x**2 - a*x + 1))