Integrand size = 18, antiderivative size = 104 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {\left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^3 (4+n)}-\frac {(3+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^3 (2+n) (4+n)} \] Output:
(1-1/a/x)^(-2-1/2*n)*(1+1/a/x)^(1+1/2*n)/a/c^3/(4+n)-(3+n)*(1-1/a/x)^(-1-1 /2*n)*(1+1/a/x)^(1+1/2*n)/a/c^3/(2+n)/(4+n)
Time = 0.37 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.62 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {e^{n \coth ^{-1}(a x)} (3+n-a x) \left (\cosh \left (3 \coth ^{-1}(a x)\right )+\sinh \left (3 \coth ^{-1}(a x)\right )\right )}{a^2 c^3 (2+n) (4+n) \sqrt {1-\frac {1}{a^2 x^2}} x} \] Input:
Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^3,x]
Output:
(E^(n*ArcCoth[a*x])*(3 + n - a*x)*(Cosh[3*ArcCoth[a*x]] + Sinh[3*ArcCoth[a *x]]))/(a^2*c^3*(2 + n)*(4 + n)*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.46 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6725, 88, 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.
\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx\) |
\(\Big \downarrow \) 6725 |
\(\displaystyle \frac {\int \frac {\left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-3} \left (1+\frac {1}{a x}\right )^{n/2}}{x}d\frac {1}{x}}{a^3 c^3}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle \frac {\frac {a^2 \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+4}-\frac {a (n+3) \int \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (1+\frac {1}{a x}\right )^{n/2}d\frac {1}{x}}{n+4}}{a^3 c^3}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\frac {a^2 \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+4}-\frac {a^2 (n+3) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{(n+2) (n+4)}}{a^3 c^3}\) |
Input:
Int[E^(n*ArcCoth[a*x])/(c - a*c*x)^3,x]
Output:
((a^2*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(4 + n) - (a^2*( 3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/((2 + n)*(4 + n )))/(a^3*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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[-d^p Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a )^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && IntegerQ[p]
Time = 5.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (a x -n -3\right ) \left (a x +1\right )}{\left (a x -1\right )^{2} c^{3} \left (n^{2}+6 n +8\right ) a}\) | \(46\) |
orering | \(\frac {\left (a x -n -3\right ) \left (a x -1\right ) \left (a x +1\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (n^{2}+6 n +8\right ) a \left (-a c x +c \right )^{3}}\) | \(49\) |
parallelrisch | \(\frac {3 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}+{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n +2 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a +x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a n -x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2}}{c^{3} \left (a x -1\right )^{2} \left (n^{2}+6 n +8\right ) a}\) | \(81\) |
Input:
int(exp(n*arccoth(a*x))/(-a*c*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-exp(n*arccoth(a*x))*(a*x-n-3)*(a*x+1)/(a*x-1)^2/c^3/(n^2+6*n+8)/a
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.23 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=-\frac {{\left (a^{2} x^{2} - {\left (a n + 2 \, a\right )} x - n - 3\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{3} n^{2} + 6 \, a c^{3} n + 8 \, a c^{3} + {\left (a^{3} c^{3} n^{2} + 6 \, a^{3} c^{3} n + 8 \, a^{3} c^{3}\right )} x^{2} - 2 \, {\left (a^{2} c^{3} n^{2} + 6 \, a^{2} c^{3} n + 8 \, a^{2} c^{3}\right )} x} \] Input:
integrate(exp(n*arccoth(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)
Result contains complex when optimal does not.
Time = 55.94 (sec) , antiderivative size = 1112, normalized size of antiderivative = 10.69 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\text {Too large to display} \] Input:
integrate(exp(n*acoth(a*x))/(-a*c*x+c)**3,x)
Output:
Piecewise((x*exp(I*pi*n/2)/c**3, Eq(a, 0)), (a**2*x**2*acoth(a*x)/(2*a**3* c**3*x**2*exp(4*acoth(a*x)) - 4*a**2*c**3*x*exp(4*acoth(a*x)) + 2*a*c**3*e xp(4*acoth(a*x))) + 2*a*x*acoth(a*x)/(2*a**3*c**3*x**2*exp(4*acoth(a*x)) - 4*a**2*c**3*x*exp(4*acoth(a*x)) + 2*a*c**3*exp(4*acoth(a*x))) - a*x/(2*a* *3*c**3*x**2*exp(4*acoth(a*x)) - 4*a**2*c**3*x*exp(4*acoth(a*x)) + 2*a*c** 3*exp(4*acoth(a*x))) + acoth(a*x)/(2*a**3*c**3*x**2*exp(4*acoth(a*x)) - 4* a**2*c**3*x*exp(4*acoth(a*x)) + 2*a*c**3*exp(4*acoth(a*x))) - 1/(2*a**3*c* *3*x**2*exp(4*acoth(a*x)) - 4*a**2*c**3*x*exp(4*acoth(a*x)) + 2*a*c**3*exp (4*acoth(a*x))), Eq(n, -4)), (-a**2*x**2*acoth(a*x)/(2*a**3*c**3*x**2*exp( 2*acoth(a*x)) - 4*a**2*c**3*x*exp(2*acoth(a*x)) + 2*a*c**3*exp(2*acoth(a*x ))) + a*x/(2*a**3*c**3*x**2*exp(2*acoth(a*x)) - 4*a**2*c**3*x*exp(2*acoth( a*x)) + 2*a*c**3*exp(2*acoth(a*x))) + acoth(a*x)/(2*a**3*c**3*x**2*exp(2*a coth(a*x)) - 4*a**2*c**3*x*exp(2*acoth(a*x)) + 2*a*c**3*exp(2*acoth(a*x))) + 1/(2*a**3*c**3*x**2*exp(2*acoth(a*x)) - 4*a**2*c**3*x*exp(2*acoth(a*x)) + 2*a*c**3*exp(2*acoth(a*x))), Eq(n, -2)), (-a**2*x**2*exp(n*acoth(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*acoth(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*acoth(a*x))/(...
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\int { -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{3}} \,d x } \] Input:
integrate(exp(n*arccoth(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)
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\int { -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{3}} \,d x } \] Input:
integrate(exp(n*arccoth(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 = 13.93 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.09 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {n+3}{a^3\,c^3\,\left (n^2+6\,n+8\right )}-\frac {x^2}{a\,c^3\,\left (n^2+6\,n+8\right )}+\frac {x\,\left (n+2\right )}{a^2\,c^3\,\left (n^2+6\,n+8\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^2}-\frac {2\,x}{a}+x^2\right )} \] Input:
int(exp(n*acoth(a*x))/(c - a*c*x)^3,x)
Output:
(((a*x + 1)/(a*x))^(n/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))))/(((a*x - 1)/(a*x ))^(n/2)*(1/a^2 - (2*x)/a + x^2))
Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx=\frac {e^{\mathit {acoth} \left (a x \right ) n} \left (-a^{2} x^{2}-a n x +2 a x -n +3\right )}{a \,c^{3} \left (a^{2} n^{2} x^{2}-6 a^{2} n \,x^{2}+8 a^{2} x^{2}-2 a \,n^{2} x +12 a n x -16 a x +n^{2}-6 n +8\right )} \] Input:
int(exp(n*acoth(a*x))/(-a*c*x+c)^3,x)
Output:
(e**(acoth(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))