Integrand size = 22, antiderivative size = 92 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {(1+a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {4 (1+a x)}{a c \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c}-\frac {3 \arcsin (a x)}{a c} \] Output:
-1/3*(a*x+1)^3/a/c/(-a^2*x^2+1)^(3/2)+4*(a*x+1)/a/c/(-a^2*x^2+1)^(1/2)+(-a ^2*x^2+1)^(1/2)/a/c-3*arcsin(a*x)/a/c
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.85 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {-14+5 a x+16 a^2 x^2-3 a^3 x^3-9 (-1+a x) \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a c (-1+a x) \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2)),x]
Output:
(-14 + 5*a*x + 16*a^2*x^2 - 3*a^3*x^3 - 9*(-1 + a*x)*Sqrt[1 - a^2*x^2]*Arc Sin[a*x])/(3*a*c*(-1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6707, 6698, 529, 27, 462, 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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6707 |
\(\displaystyle -\frac {a^2 \int \frac {e^{3 \text {arctanh}(a x)} x^2}{1-a^2 x^2}dx}{c}\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle -\frac {a^2 \int \frac {x^2 (a x+1)^3}{\left (1-a^2 x^2\right )^{5/2}}dx}{c}\) |
\(\Big \downarrow \) 529 |
\(\displaystyle -\frac {a^2 \left (\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {3 (a x+1)^3}{a^2 \left (1-a^2 x^2\right )^{3/2}}dx\right )}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^2 \left (\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(a x+1)^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{a^2}\right )}{c}\) |
\(\Big \downarrow \) 462 |
\(\displaystyle -\frac {a^2 \left (\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}-\int \frac {a x+3}{\sqrt {1-a^2 x^2}}dx}{a^2}\right )}{c}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -\frac {a^2 \left (\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {-3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a}}{a^2}\right )}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {a^2 \left (\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a}-\frac {3 \arcsin (a x)}{a}}{a^2}\right )}{c}\) |
Input:
Int[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2)),x]
Output:
-((a^2*((1 + a*x)^3/(3*a^3*(1 - a^2*x^2)^(3/2)) - ((4*(1 + a*x))/(a*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/a - (3*ArcSin[a*x])/a)/a^2))/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[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp [(-2^(n - 1))*d*c^(n - 2)*((c + d*x)/(b*Sqrt[a + b*x^2])), x] + Simp[d^2/b Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(n - 1)*c^(n - 1) - (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[n, 2]
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[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]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x ] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.65
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a c \sqrt {-a^{2} x^{2}+1}}-\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2}}{c}\) | \(152\) |
default | \(\frac {a^{2} \left (a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {7 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {4}{a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {4}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {4 \left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right )}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{3}}\right )}{c}\) | \(209\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x,method=_RETURNVERBOSE)
Output:
-1/a/c*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)-(3/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2 )*x/(-a^2*x^2+1)^(1/2))+2/3/a^5/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/ 2)+13/3/a^4/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))/c*a^2
Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x, algorithm="fricas" )
Output:
1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^3 *c*x^2 - 2*a^2*c*x + a*c)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {a^{2} \left (\int \frac {x^{2}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {2 a x^{3}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{2} x^{4}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2),x)
Output:
a**2*(Integral(x**2/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a** 2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integ ral(2*a*x**3/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a** 2*x**4/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x, algorithm="maxima" )
Output:
integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))), x)
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),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 = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.40 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}}-\frac {2\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^4\,x^2-2\,c\,a^3\,x+c\,a^2\right )} \] Input:
int((a*x + 1)^3/((c - c/(a^2*x^2))*(1 - a^2*x^2)^(3/2)),x)
Output:
(1 - a^2*x^2)^(1/2)/(a*c) - (13*(1 - a^2*x^2)^(1/2))/(3*((c*(-a^2)^(1/2))/ a - c*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (3*asinh(x*(-a^2)^(1/2)))/(c*(-a^2)^ (1/2)) - (2*a*(1 - a^2*x^2)^(1/2))/(3*(a^2*c + a^4*c*x^2 - 2*a^3*c*x))
Time = 0.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.97 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {-9 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x +9 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-9 \mathit {asin} \left (a x \right ) a^{2} x^{2}+18 \mathit {asin} \left (a x \right ) a x -9 \mathit {asin} \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-11 \sqrt {-a^{2} x^{2}+1}\, a x +6 \sqrt {-a^{2} x^{2}+1}-3 a^{3} x^{3}+24 a^{2} x^{2}-11 a x -6}{3 a c \left (\sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}-2 a x +1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x)
Output:
( - 9*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x + 9*sqrt( - a**2*x**2 + 1)*asin (a*x) - 9*asin(a*x)*a**2*x**2 + 18*asin(a*x)*a*x - 9*asin(a*x) + 3*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 11*sqrt( - a**2*x**2 + 1)*a*x + 6*sqrt( - a**2 *x**2 + 1) - 3*a**3*x**3 + 24*a**2*x**2 - 11*a*x - 6)/(3*a*c*(sqrt( - a**2 *x**2 + 1)*a*x - sqrt( - a**2*x**2 + 1) + a**2*x**2 - 2*a*x + 1))