Integrand size = 18, antiderivative size = 135 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {(1-a x)^{-3-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^4 (6+n)}+\frac {2 (1-a x)^{-2-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^4 (4+n) (6+n)}+\frac {2 (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^4 (6+n) \left (8+6 n+n^2\right )} \] Output:
(-a*x+1)^(-3-1/2*n)*(a*x+1)^(1+1/2*n)/a/c^4/(6+n)+2*(-a*x+1)^(-2-1/2*n)*(a *x+1)^(1+1/2*n)/a/c^4/(4+n)/(6+n)+2*(-a*x+1)^(-1-1/2*n)*(a*x+1)^(1+1/2*n)/ a/c^4/(6+n)/(n^2+6*n+8)
Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.55 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {(1-a x)^{-3-\frac {n}{2}} (1+a x)^{1+\frac {n}{2}} \left (14+8 n+n^2-8 a x-2 a n x+2 a^2 x^2\right )}{a c^4 (2+n) (4+n) (6+n)} \] Input:
Integrate[E^(n*ArcTanh[a*x])/(c - a*c*x)^4,x]
Output:
((1 - a*x)^(-3 - n/2)*(1 + a*x)^(1 + n/2)*(14 + 8*n + n^2 - 8*a*x - 2*a*n* x + 2*a^2*x^2))/(a*c^4*(2 + n)*(4 + n)*(6 + n))
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6679, 55, 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.
\(\displaystyle \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {\int (1-a x)^{-\frac {n}{2}-4} (a x+1)^{n/2}dx}{c^4}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {\frac {2 \int (1-a x)^{-\frac {n}{2}-3} (a x+1)^{n/2}dx}{n+6}+\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-3}}{a (n+6)}}{c^4}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {\frac {2 \left (\frac {\int (1-a x)^{-\frac {n}{2}-2} (a x+1)^{n/2}dx}{n+4}+\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-2}}{a (n+4)}\right )}{n+6}+\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-3}}{a (n+6)}}{c^4}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-3}}{a (n+6)}+\frac {2 \left (\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-2}}{a (n+4)}+\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{-\frac {n}{2}-1}}{a (n+2) (n+4)}\right )}{n+6}}{c^4}\) |
Input:
Int[E^(n*ArcTanh[a*x])/(c - a*c*x)^4,x]
Output:
(((1 - a*x)^(-3 - n/2)*(1 + a*x)^((2 + n)/2))/(a*(6 + n)) + (2*(((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))))/(6 + n))/c^4
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 = 13.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {\left (a x +1\right ) \left (2 a^{2} x^{2}-2 n a x -8 a x +n^{2}+8 n +14\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (a x -1\right )^{3} c^{4} a \left (n^{2}+8 n +12\right ) \left (4+n \right )}\) | \(68\) |
orering | \(-\frac {\left (2 a^{2} x^{2}-2 n a x -8 a x +n^{2}+8 n +14\right ) \left (a x -1\right ) \left (a x +1\right ) {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{a \left (n^{3}+12 n^{2}+44 n +48\right ) \left (-a c x +c \right )^{4}}\) | \(72\) |
parallelrisch | \(\frac {-14 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}+2 x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2} n -x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a \,n^{2}-2 a^{3} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{3}+6 x^{2} {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a^{2}-{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} n^{2}-8 \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} n -6 x \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} a n -6 a \,{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x}{c^{4} \left (a x -1\right )^{3} a \left (n^{3}+12 n^{2}+44 n +48\right )}\) | \(145\) |
Input:
int(exp(n*arctanh(a*x))/(-a*c*x+c)^4,x,method=_RETURNVERBOSE)
Output:
-(a*x+1)*(2*a^2*x^2-2*a*n*x-8*a*x+n^2+8*n+14)*exp(n*arctanh(a*x))/(a*x-1)^ 3/c^4/a/(n^2+8*n+12)/(4+n)
Time = 0.10 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.70 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {{\left (2 \, a^{3} x^{3} - 2 \, {\left (a^{2} n + 3 \, a^{2}\right )} x^{2} + n^{2} + {\left (a n^{2} + 6 \, a n + 6 \, a\right )} x + 8 \, n + 14\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{4} n^{3} + 12 \, a c^{4} n^{2} + 44 \, a c^{4} n + 48 \, a c^{4} - {\left (a^{4} c^{4} n^{3} + 12 \, a^{4} c^{4} n^{2} + 44 \, a^{4} c^{4} n + 48 \, a^{4} c^{4}\right )} x^{3} + 3 \, {\left (a^{3} c^{4} n^{3} + 12 \, a^{3} c^{4} n^{2} + 44 \, a^{3} c^{4} n + 48 \, a^{3} c^{4}\right )} x^{2} - 3 \, {\left (a^{2} c^{4} n^{3} + 12 \, a^{2} c^{4} n^{2} + 44 \, a^{2} c^{4} n + 48 \, a^{2} c^{4}\right )} x} \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^4,x, algorithm="fricas")
Output:
(2*a^3*x^3 - 2*(a^2*n + 3*a^2)*x^2 + n^2 + (a*n^2 + 6*a*n + 6*a)*x + 8*n + 14)*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c^4*n^3 + 12*a*c^4*n^2 + 44*a*c^4*n + 48*a*c^4 - (a^4*c^4*n^3 + 12*a^4*c^4*n^2 + 44*a^4*c^4*n + 48*a^4*c^4)*x ^3 + 3*(a^3*c^4*n^3 + 12*a^3*c^4*n^2 + 44*a^3*c^4*n + 48*a^3*c^4)*x^2 - 3* (a^2*c^4*n^3 + 12*a^2*c^4*n^2 + 44*a^2*c^4*n + 48*a^2*c^4)*x)
Leaf count of result is larger than twice the leaf count of optimal. 3526 vs. \(2 (105) = 210\).
Time = 148.10 (sec) , antiderivative size = 3526, normalized size of antiderivative = 26.12 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\text {Too large to display} \] Input:
integrate(exp(n*atanh(a*x))/(-a*c*x+c)**4,x)
Output:
Piecewise((x/c**4, Eq(a, 0)), (-a**3*x**3*atanh(a*x)/(4*a**4*c**4*x**3*exp (6*atanh(a*x)) - 12*a**3*c**4*x**2*exp(6*atanh(a*x)) + 12*a**2*c**4*x*exp( 6*atanh(a*x)) - 4*a*c**4*exp(6*atanh(a*x))) - 3*a**2*x**2*atanh(a*x)/(4*a* *4*c**4*x**3*exp(6*atanh(a*x)) - 12*a**3*c**4*x**2*exp(6*atanh(a*x)) + 12* a**2*c**4*x*exp(6*atanh(a*x)) - 4*a*c**4*exp(6*atanh(a*x))) + a**2*x**2/(4 *a**4*c**4*x**3*exp(6*atanh(a*x)) - 12*a**3*c**4*x**2*exp(6*atanh(a*x)) + 12*a**2*c**4*x*exp(6*atanh(a*x)) - 4*a*c**4*exp(6*atanh(a*x))) - 3*a*x*ata nh(a*x)/(4*a**4*c**4*x**3*exp(6*atanh(a*x)) - 12*a**3*c**4*x**2*exp(6*atan h(a*x)) + 12*a**2*c**4*x*exp(6*atanh(a*x)) - 4*a*c**4*exp(6*atanh(a*x))) + 3*a*x/(4*a**4*c**4*x**3*exp(6*atanh(a*x)) - 12*a**3*c**4*x**2*exp(6*atanh (a*x)) + 12*a**2*c**4*x*exp(6*atanh(a*x)) - 4*a*c**4*exp(6*atanh(a*x))) - atanh(a*x)/(4*a**4*c**4*x**3*exp(6*atanh(a*x)) - 12*a**3*c**4*x**2*exp(6*a tanh(a*x)) + 12*a**2*c**4*x*exp(6*atanh(a*x)) - 4*a*c**4*exp(6*atanh(a*x)) ) + 2/(4*a**4*c**4*x**3*exp(6*atanh(a*x)) - 12*a**3*c**4*x**2*exp(6*atanh( a*x)) + 12*a**2*c**4*x*exp(6*atanh(a*x)) - 4*a*c**4*exp(6*atanh(a*x))), Eq (n, -6)), (a**3*x**3*atanh(a*x)/(2*a**4*c**4*x**3*exp(4*atanh(a*x)) - 6*a* *3*c**4*x**2*exp(4*atanh(a*x)) + 6*a**2*c**4*x*exp(4*atanh(a*x)) - 2*a*c** 4*exp(4*atanh(a*x))) + a**2*x**2*atanh(a*x)/(2*a**4*c**4*x**3*exp(4*atanh( a*x)) - 6*a**3*c**4*x**2*exp(4*atanh(a*x)) + 6*a**2*c**4*x*exp(4*atanh(a*x )) - 2*a*c**4*exp(4*atanh(a*x))) - a**2*x**2/(2*a**4*c**4*x**3*exp(4*at...
\[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{4}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^4,x, algorithm="maxima")
Output:
integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c)^4, x)
\[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{4}} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))/(-a*c*x+c)^4,x, algorithm="giac")
Output:
integrate((-(a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c)^4, x)
Time = 14.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.24 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=-\frac {{\left (a\,x+1\right )}^{n/2}\,\left (\frac {2\,x^3}{a\,c^4\,\left (n^3+12\,n^2+44\,n+48\right )}+\frac {n^2+8\,n+14}{a^4\,c^4\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {x^2\,\left (2\,n+6\right )}{a^2\,c^4\,\left (n^3+12\,n^2+44\,n+48\right )}+\frac {x\,\left (n^2+6\,n+6\right )}{a^3\,c^4\,\left (n^3+12\,n^2+44\,n+48\right )}\right )}{{\left (1-a\,x\right )}^{n/2}\,\left (\frac {3\,x}{a^2}-\frac {1}{a^3}+x^3-\frac {3\,x^2}{a}\right )} \] Input:
int(exp(n*atanh(a*x))/(c - a*c*x)^4,x)
Output:
-((a*x + 1)^(n/2)*((2*x^3)/(a*c^4*(44*n + 12*n^2 + n^3 + 48)) + (8*n + n^2 + 14)/(a^4*c^4*(44*n + 12*n^2 + n^3 + 48)) - (x^2*(2*n + 6))/(a^2*c^4*(44 *n + 12*n^2 + n^3 + 48)) + (x*(6*n + n^2 + 6))/(a^3*c^4*(44*n + 12*n^2 + n ^3 + 48))))/((1 - a*x)^(n/2)*((3*x)/a^2 - 1/a^3 + x^3 - (3*x^2)/a))
Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.36 \[ \int \frac {e^{n \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {e^{\mathit {atanh} \left (a x \right ) n} \left (-2 a^{3} x^{3}+2 a^{2} n \,x^{2}+6 a^{2} x^{2}-a \,n^{2} x -6 a n x -6 a x -n^{2}-8 n -14\right )}{a \,c^{4} \left (a^{3} n^{3} x^{3}+12 a^{3} n^{2} x^{3}+44 a^{3} n \,x^{3}-3 a^{2} n^{3} x^{2}+48 a^{3} x^{3}-36 a^{2} n^{2} x^{2}-132 a^{2} n \,x^{2}+3 a \,n^{3} x -144 a^{2} x^{2}+36 a \,n^{2} x +132 a n x -n^{3}+144 a x -12 n^{2}-44 n -48\right )} \] Input:
int(exp(n*atanh(a*x))/(-a*c*x+c)^4,x)
Output:
(e**(atanh(a*x)*n)*( - 2*a**3*x**3 + 2*a**2*n*x**2 + 6*a**2*x**2 - a*n**2* x - 6*a*n*x - 6*a*x - n**2 - 8*n - 14))/(a*c**4*(a**3*n**3*x**3 + 12*a**3* n**2*x**3 + 44*a**3*n*x**3 + 48*a**3*x**3 - 3*a**2*n**3*x**2 - 36*a**2*n** 2*x**2 - 132*a**2*n*x**2 - 144*a**2*x**2 + 3*a*n**3*x + 36*a*n**2*x + 132* a*n*x + 144*a*x - n**3 - 12*n**2 - 44*n - 48))