Integrand size = 23, antiderivative size = 87 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{c-a^2 c x^2} \, dx=\frac {1+a x}{a^4 c \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a^4 c}+\frac {x \sqrt {1-a^2 x^2}}{2 a^3 c}-\frac {3 \arcsin (a x)}{2 a^4 c} \] Output:
(a*x+1)/a^4/c/(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)/a^4/c+1/2*x*(-a^2*x^2+ 1)^(1/2)/a^3/c-3/2*arcsin(a*x)/a^4/c
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{c-a^2 c x^2} \, dx=-\frac {-4-3 a x+2 a^2 x^2+a^3 x^3+3 \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^4 c \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^3)/(c - a^2*c*x^2),x]
Output:
-1/2*(-4 - 3*a*x + 2*a^2*x^2 + a^3*x^3 + 3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/ (a^4*c*Sqrt[1 - a^2*x^2])
Time = 0.38 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6698, 527, 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.
\(\displaystyle \int \frac {x^3 e^{\text {arctanh}(a x)}}{c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {x^3 (a x+1)}{\left (1-a^2 x^2\right )^{3/2}}dx}{c}\) |
\(\Big \downarrow \) 527 |
\(\displaystyle \frac {\frac {a x+1}{a^4 \sqrt {1-a^2 x^2}}-\frac {\int \frac {a^2 x^2+a x+1}{\sqrt {1-a^2 x^2}}dx}{a^3}}{c}\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {\frac {a x+1}{a^4 \sqrt {1-a^2 x^2}}-\frac {-\frac {\int -\frac {a^2 (2 a x+3)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a^3}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {a x+1}{a^4 \sqrt {1-a^2 x^2}}-\frac {\frac {\int \frac {a^2 (2 a x+3)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a^3}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {a x+1}{a^4 \sqrt {1-a^2 x^2}}-\frac {\frac {1}{2} \int \frac {2 a x+3}{\sqrt {1-a^2 x^2}}dx-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a^3}}{c}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {\frac {a x+1}{a^4 \sqrt {1-a^2 x^2}}-\frac {\frac {1}{2} \left (3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{a}\right )-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a^3}}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {a x+1}{a^4 \sqrt {1-a^2 x^2}}-\frac {\frac {1}{2} \left (\frac {3 \arcsin (a x)}{a}-\frac {2 \sqrt {1-a^2 x^2}}{a}\right )-\frac {1}{2} x \sqrt {1-a^2 x^2}}{a^3}}{c}\) |
Input:
Int[(E^ArcTanh[a*x]*x^3)/(c - a^2*c*x^2),x]
Output:
((1 + a*x)/(a^4*Sqrt[1 - a^2*x^2]) - (-1/2*(x*Sqrt[1 - a^2*x^2]) + ((-2*Sq rt[1 - a^2*x^2])/a + (3*ArcSin[a*x])/a)/2)/a^3)/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 = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.29
method | result | size |
risch | \(-\frac {\left (a x +2\right ) \left (a^{2} x^{2}-1\right )}{2 a^{4} \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{3} \sqrt {a^{2}}}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{5} \left (x -\frac {1}{a}\right )}}{c}\) | \(112\) |
default | \(-\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{4}}+\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{5} \left (x -\frac {1}{a}\right )}}{c}\) | \(146\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)
Output:
-1/2*(a*x+2)*(a^2*x^2-1)/a^4/(-a^2*x^2+1)^(1/2)/c-(3/2/a^3/(a^2)^(1/2)*arc tan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/a^5/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x -1/a))^(1/2))/c
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{c-a^2 c x^2} \, dx=\frac {4 \, a x + 6 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a^{2} x^{2} + a x - 4\right )} \sqrt {-a^{2} x^{2} + 1} - 4}{2 \, {\left (a^{5} c x - a^{4} c\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c),x, algorithm="fric as")
Output:
1/2*(4*a*x + 6*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a^2*x^2 + a*x - 4)*sqrt(-a^2*x^2 + 1) - 4)/(a^5*c*x - a^4*c)
\[ \int \frac {e^{\text {arctanh}(a x)} x^3}{c-a^2 c x^2} \, dx=\frac {\int \frac {x^{3}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{- a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a**2*c*x**2+c),x)
Output:
(Integral(x**3/(-a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x ) + Integral(a*x**4/(-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. 178 vs. \(2 (77) = 154\).
Time = 0.17 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.05 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{c-a^2 c x^2} \, dx=-\frac {1}{2} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{6} c^{2} x + a^{5} c^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a^{6} c^{2} x - a^{5} c^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{6} c x + a^{5} c} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{6} c x - a^{5} c} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{4} c} + \frac {3 \, \arcsin \left (a x\right )}{a^{5} c} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{5} c}\right )} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c),x, algorithm="maxi ma")
Output:
-1/2*a*(sqrt(-a^2*x^2 + 1)*c/(a^6*c^2*x + a^5*c^2) + sqrt(-a^2*x^2 + 1)*c/ (a^6*c^2*x - a^5*c^2) - sqrt(-a^2*x^2 + 1)/(a^6*c*x + a^5*c) + sqrt(-a^2*x ^2 + 1)/(a^6*c*x - a^5*c) - sqrt(-a^2*x^2 + 1)*x/(a^4*c) + 3*arcsin(a*x)/( a^5*c) - 2*sqrt(-a^2*x^2 + 1)/(a^5*c))
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c),x, algorithm="giac ")
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 = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.47 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{c-a^2 c x^2} \, dx=\frac {\sqrt {1-a^2\,x^2}}{a^3\,c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^3\,c\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {1}{a^2\,c\,\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2\,a^3\,c}\right )}{\sqrt {-a^2}} \] Input:
int((x^3*(a*x + 1))/((c - a^2*c*x^2)*(1 - a^2*x^2)^(1/2)),x)
Output:
(1 - a^2*x^2)^(1/2)/(a^3*c*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)*(-a^2)^(1/2)) - (3*asinh(x*(-a^2)^(1/2)))/(2*a^3*c*(-a^2)^(1/2)) - ((1 - a^2*x^2)^(1/2) *(1/(a^2*c*(-a^2)^(1/2)) - (x*(-a^2)^(1/2))/(2*a^3*c)))/(-a^2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{c-a^2 c x^2} \, dx=\frac {-3 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-3 \mathit {asin} \left (a x \right ) a x +3 \mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+\sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}-a^{3} x^{3}-2 a^{2} x^{2}+a x +6}{2 a^{4} c \left (\sqrt {-a^{2} x^{2}+1}+a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c),x)
Output:
( - 3*sqrt( - a**2*x**2 + 1)*asin(a*x) - 3*asin(a*x)*a*x + 3*asin(a*x) + s qrt( - a**2*x**2 + 1)*a**2*x**2 + sqrt( - a**2*x**2 + 1)*a*x - 6*sqrt( - a **2*x**2 + 1) - a**3*x**3 - 2*a**2*x**2 + a*x + 6)/(2*a**4*c*(sqrt( - a**2 *x**2 + 1) + a*x - 1))