Integrand size = 22, antiderivative size = 83 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2 (1+a x)^2}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}+\frac {1+a x}{15 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x}{15 c^2 \sqrt {1-a^2 x^2}} \] Output:
2/5*(a*x+1)^2/a/c^2/(-a^2*x^2+1)^(5/2)+1/15*(a*x+1)/a/c^2/(-a^2*x^2+1)^(3/ 2)+2/15*x/c^2/(-a^2*x^2+1)^(1/2)
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.52 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\sqrt {1+a x} \left (7-6 a x+2 a^2 x^2\right )}{15 a c^2 (1-a x)^{5/2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^2,x]
Output:
(Sqrt[1 + a*x]*(7 - 6*a*x + 2*a^2*x^2))/(15*a*c^2*(1 - a*x)^(5/2))
Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6688, 464, 461, 461, 460}
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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6688 |
\(\displaystyle \frac {\int \frac {(a x+1)^3}{\left (1-a^2 x^2\right )^{7/2}}dx}{c^2}\) |
\(\Big \downarrow \) 464 |
\(\displaystyle \frac {\int \frac {1}{(1-a x)^3 \sqrt {1-a^2 x^2}}dx}{c^2}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\frac {2}{5} \int \frac {1}{(1-a x)^2 \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{5 a (1-a x)^3}}{c^2}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\frac {2}{5} \left (\frac {1}{3} \int \frac {1}{(1-a x) \sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{3 a (1-a x)^2}\right )+\frac {\sqrt {1-a^2 x^2}}{5 a (1-a x)^3}}{c^2}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {\frac {2}{5} \left (\frac {\sqrt {1-a^2 x^2}}{3 a (1-a x)}+\frac {\sqrt {1-a^2 x^2}}{3 a (1-a x)^2}\right )+\frac {\sqrt {1-a^2 x^2}}{5 a (1-a x)^3}}{c^2}\) |
Input:
Int[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^2,x]
Output:
(Sqrt[1 - a^2*x^2]/(5*a*(1 - a*x)^3) + (2*(Sqrt[1 - a^2*x^2]/(3*a*(1 - a*x )^2) + Sqrt[1 - a^2*x^2]/(3*a*(1 - a*x))))/5)/c^2
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 2, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl ify[n + 2*p + 2]/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[( a + b*x^2)^(n + p)/(a/c + b*(x/d))^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b *c^2 + a*d^2, 0] && IntegerQ[n] && RationalQ[p] && (LtQ[0, -n, p] || LtQ[p, -n, 0]) && NeQ[n, 2] && NeQ[n, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && IGtQ[(n + 1)/2, 0] && !I ntegerQ[p - n/2]
Time = 0.43 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.51
method | result | size |
trager | \(-\frac {\left (2 a^{2} x^{2}-6 a x +7\right ) \sqrt {-a^{2} x^{2}+1}}{15 c^{2} \left (a x -1\right )^{3} a}\) | \(42\) |
gosper | \(-\frac {\left (2 a^{2} x^{2}-6 a x +7\right ) \left (a x +1\right )^{2}}{15 \left (a x -1\right ) c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\) | \(49\) |
orering | \(-\frac {\left (2 a^{2} x^{2}-6 a x +7\right ) \left (a x -1\right ) \left (a x +1\right )^{4}}{15 a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a^{2} c \,x^{2}+c \right )^{2}}\) | \(57\) |
default | \(\frac {\frac {\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a}+\frac {\frac {2}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {6 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{2}}}{c^{2}}\) | \(240\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
-1/15/c^2*(2*a^2*x^2-6*a*x+7)/(a*x-1)^3/a*(-a^2*x^2+1)^(1/2)
Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {7 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 21 \, a x - {\left (2 \, a^{2} x^{2} - 6 \, a x + 7\right )} \sqrt {-a^{2} x^{2} + 1} - 7}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^2,x, algorithm="fric as")
Output:
1/15*(7*a^3*x^3 - 21*a^2*x^2 + 21*a*x - (2*a^2*x^2 - 6*a*x + 7)*sqrt(-a^2* x^2 + 1) - 7)/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {3 a x}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**2,x)
Output:
(Integral(3*a*x/(-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(3*a**2*x**2/(-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**3*x**3/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*s qrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-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**2
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{2} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^2,x, algorithm="maxi ma")
Output:
integrate((a*x + 1)^3/((a^2*c*x^2 - c)^2*(-a^2*x^2 + 1)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (71) = 142\).
Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.75 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {2 \, {\left (\frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {40 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 7\right )}}{15 \, c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^2,x, algorithm="giac ")
Output:
-2/15*(20*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 40*(sqrt(-a^2*x^2 + 1) *abs(a) + a)^2/(a^4*x^2) + 30*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) - 7)/(c^2*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^5*abs(a))
Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.53 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a^3}{15\,c^2\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {a^3}{5\,c^2\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {2\,a^4}{15\,c^2\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^3\,\sqrt {-a^2}} \] Input:
int((a*x + 1)^3/((c - a^2*c*x^2)^2*(1 - a^2*x^2)^(3/2)),x)
Output:
((1 - a^2*x^2)^(1/2)*((2*a^3)/(15*c^2*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)) - a^3/(5*c^2*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^3) + (2*a^4)/(15*c^2*(x*(-a^ 2)^(1/2) - (-a^2)^(1/2)/a)^2*(-a^2)^(1/2))))/(a^3*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.16 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {-\frac {2 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{5}}{5}-\frac {4 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}}{3}+\frac {2 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )}{3}-\frac {8}{15}}{a \,c^{2} \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{5}-5 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{4}+10 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-10 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+5 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^2,x)
Output:
(2*( - 3*tan(asin(a*x)/2)**5 - 10*tan(asin(a*x)/2)**2 + 5*tan(asin(a*x)/2) - 4))/(15*a*c**2*(tan(asin(a*x)/2)**5 - 5*tan(asin(a*x)/2)**4 + 10*tan(as in(a*x)/2)**3 - 10*tan(asin(a*x)/2)**2 + 5*tan(asin(a*x)/2) - 1))