Integrand size = 18, antiderivative size = 84 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\frac {(1-a x)^{-2-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^3 (4+n)}+\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^3 \left (8+6 n+n^2\right )} \] Output:
(-a*x+1)^(-2-1/2*n)*(a*x+1)^(1+1/2*n)/a/c^3/(4+n)+(-a*x+1)^(-1-1/2*n)*(a*x +1)^(1+1/2*n)/a/c^3/(n^2+6*n+8)
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.61 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\frac {(1-a x)^{-2-\frac {n}{2}} (3+n-a x) (1+a x)^{1+\frac {n}{2}}}{a c^3 (2+n) (4+n)} \] Input:
Integrate[E^(n*ArcTanh[a*x])/(c - a*c*x)^3,x]
Output:
((1 - a*x)^(-2 - n/2)*(3 + n - a*x)*(1 + a*x)^(1 + n/2))/(a*c^3*(2 + n)*(4 + n))
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6679, 55, 48}
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.
\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {\int (1-a x)^{-\frac {n}{2}-3} (a x+1)^{n/2}dx}{c^3}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {\frac {\int (1-a x)^{-\frac {n}{2}-2} (a x+1)^{n/2}dx}{n+4}+\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-2}}{a (n+4)}}{c^3}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-2}}{a (n+4)}+\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a (n+2) (n+4)}}{c^3}\) |
Input:
Int[E^(n*ArcTanh[a*x])/(c - a*c*x)^3,x]
Output:
(((1 - a*x)^(-2 - n/2)*(1 + a*x)^((2 + n)/2))/(a*(4 + n)) + ((1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a*(2 + n)*(4 + n)))/c^3
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
Time = 4.87 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \left (a x -n -3\right ) \left (a x +1\right )}{\left (a x -1\right )^{2} c^{3} \left (n^{2}+6 n +8\right ) a}\) | \(46\) |
orering | \(\frac {\left (a x -n -3\right ) \left (a x -1\right ) \left (a x +1\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (n^{2}+6 n +8\right ) a \left (-a c x +c \right )^{3}}\) | \(49\) |
parallelrisch | \(\frac {3 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}-x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2}+{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} n +x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a n +2 a \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x}{c^{3} \left (a x -1\right )^{2} \left (n^{2}+6 n +8\right ) a}\) | \(81\) |
Input:
int(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-exp(n*arctanh(a*x))*(a*x-n-3)*(a*x+1)/(a*x-1)^2/c^3/(n^2+6*n+8)/a
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.54 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=-\frac {{\left (a^{2} x^{2} - {\left (a n + 2 \, a\right )} x - n - 3\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{3} n^{2} + 6 \, a c^{3} n + 8 \, a c^{3} + {\left (a^{3} c^{3} n^{2} + 6 \, a^{3} c^{3} n + 8 \, a^{3} c^{3}\right )} x^{2} - 2 \, {\left (a^{2} c^{3} n^{2} + 6 \, a^{2} c^{3} n + 8 \, a^{2} c^{3}\right )} x} \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x, algorithm="fricas")
Output:
-(a^2*x^2 - (a*n + 2*a)*x - n - 3)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^3*n ^2 + 6*a*c^3*n + 8*a*c^3 + (a^3*c^3*n^2 + 6*a^3*c^3*n + 8*a^3*c^3)*x^2 - 2 *(a^2*c^3*n^2 + 6*a^2*c^3*n + 8*a^2*c^3)*x)
Leaf count of result is larger than twice the leaf count of optimal. 1103 vs. \(2 (65) = 130\).
Time = 43.86 (sec) , antiderivative size = 1103, normalized size of antiderivative = 13.13 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\text {Too large to display} \] Input:
integrate(exp(n*atanh(a*x))/(-a*c*x+c)**3,x)
Output:
Piecewise((x/c**3, Eq(a, 0)), (a**2*x**2*atanh(a*x)/(2*a**3*c**3*x**2*exp( 4*atanh(a*x)) - 4*a**2*c**3*x*exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x ))) + 2*a*x*atanh(a*x)/(2*a**3*c**3*x**2*exp(4*atanh(a*x)) - 4*a**2*c**3*x *exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x))) - a*x/(2*a**3*c**3*x**2*e xp(4*atanh(a*x)) - 4*a**2*c**3*x*exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh( a*x))) + atanh(a*x)/(2*a**3*c**3*x**2*exp(4*atanh(a*x)) - 4*a**2*c**3*x*ex p(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x))) - 1/(2*a**3*c**3*x**2*exp(4* atanh(a*x)) - 4*a**2*c**3*x*exp(4*atanh(a*x)) + 2*a*c**3*exp(4*atanh(a*x)) ), Eq(n, -4)), (-a**2*x**2*atanh(a*x)/(2*a**3*c**3*x**2*exp(2*atanh(a*x)) - 4*a**2*c**3*x*exp(2*atanh(a*x)) + 2*a*c**3*exp(2*atanh(a*x))) + a*x/(2*a **3*c**3*x**2*exp(2*atanh(a*x)) - 4*a**2*c**3*x*exp(2*atanh(a*x)) + 2*a*c* *3*exp(2*atanh(a*x))) + atanh(a*x)/(2*a**3*c**3*x**2*exp(2*atanh(a*x)) - 4 *a**2*c**3*x*exp(2*atanh(a*x)) + 2*a*c**3*exp(2*atanh(a*x))) + 1/(2*a**3*c **3*x**2*exp(2*atanh(a*x)) - 4*a**2*c**3*x*exp(2*atanh(a*x)) + 2*a*c**3*ex p(2*atanh(a*x))), Eq(n, -2)), (-a**2*x**2*exp(n*atanh(a*x))/(a**3*c**3*n** 2*x**2 + 6*a**3*c**3*n*x**2 + 8*a**3*c**3*x**2 - 2*a**2*c**3*n**2*x - 12*a **2*c**3*n*x - 16*a**2*c**3*x + a*c**3*n**2 + 6*a*c**3*n + 8*a*c**3) + a*n *x*exp(n*atanh(a*x))/(a**3*c**3*n**2*x**2 + 6*a**3*c**3*n*x**2 + 8*a**3*c* *3*x**2 - 2*a**2*c**3*n**2*x - 12*a**2*c**3*n*x - 16*a**2*c**3*x + a*c**3* n**2 + 6*a*c**3*n + 8*a*c**3) + 2*a*x*exp(n*atanh(a*x))/(a**3*c**3*n**2...
\[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\int { -\frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{3}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x, algorithm="maxima")
Output:
-integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c)^3, x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\int { -\frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{3}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^3,x, algorithm="giac")
Output:
integrate(-(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c)^3, x)
Time = 14.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.19 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\frac {{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}\,\left (\frac {n+3}{a^3\,c^3\,\left (n^2+6\,n+8\right )}-\frac {x^2}{a\,c^3\,\left (n^2+6\,n+8\right )}+\frac {x\,\left (n+2\right )}{a^2\,c^3\,\left (n^2+6\,n+8\right )}\right )}{\frac {1}{a^2}-\frac {2\,x}{a}+x^2} \] Input:
int(exp(n*atanh(a*x))/(c - a*c*x)^3,x)
Output:
(exp((n*log(a*x + 1))/2 - (n*log(1 - a*x))/2)*((n + 3)/(a^3*c^3*(6*n + n^2 + 8)) - x^2/(a*c^3*(6*n + n^2 + 8)) + (x*(n + 2))/(a^2*c^3*(6*n + n^2 + 8 ))))/(1/a^2 - (2*x)/a + x^2)
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.04 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\frac {e^{\mathit {atanh} \left (a x \right ) n} \left (-a^{2} x^{2}+a n x +2 a x +n +3\right )}{a \,c^{3} \left (a^{2} n^{2} x^{2}+6 a^{2} n \,x^{2}+8 a^{2} x^{2}-2 a \,n^{2} x -12 a n x -16 a x +n^{2}+6 n +8\right )} \] Input:
int(exp(n*atanh(a*x))/(-a*c*x+c)^3,x)
Output:
(e**(atanh(a*x)*n)*( - a**2*x**2 + a*n*x + 2*a*x + n + 3))/(a*c**3*(a**2*n **2*x**2 + 6*a**2*n*x**2 + 8*a**2*x**2 - 2*a*n**2*x - 12*a*n*x - 16*a*x + n**2 + 6*n + 8))