Integrand size = 17, antiderivative size = 97 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^4} \, dx=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 c^4 (1-a x)^4}-\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a^2 c^4 (1-a x)^3} \]
1/7*(-a^2*x^2+1)^(3/2)/a^2/c^4/(-a*x+1)^5-1/7*(-a^2*x^2+1)^(3/2)/a^2/c^4/( -a*x+1)^4-1/21*(-a^2*x^2+1)^(3/2)/a^2/c^4/(-a*x+1)^3
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.43 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^4} \, dx=-\frac {(1+a x)^{3/2} \left (1-5 a x+a^2 x^2\right )}{21 a^2 c^4 (1-a x)^{7/2}} \]
Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6678, 27, 571, 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.
\(\displaystyle \int \frac {x e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {x \sqrt {1-a^2 x^2}}{c^5 (1-a x)^5}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {x \sqrt {1-a^2 x^2}}{(1-a x)^5}dx}{c^4}\) |
\(\Big \downarrow \) 571 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 (1-a x)^5}-\frac {5 \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^4}dx}{7 a}}{c^4}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 (1-a x)^5}-\frac {5 \left (\frac {1}{5} \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^3}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a (1-a x)^4}\right )}{7 a}}{c^4}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^2 (1-a x)^5}-\frac {5 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{15 a (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a (1-a x)^4}\right )}{7 a}}{c^4}\) |
((1 - a^2*x^2)^(3/2)/(7*a^2*(1 - a*x)^5) - (5*((1 - a^2*x^2)^(3/2)/(5*a*(1 - a*x)^4) + (1 - a^2*x^2)^(3/2)/(15*a*(1 - a*x)^3)))/(7*a))/c^4
3.4.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*(n + p + 1))), x] + Simp[n/(2*d* (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] && ((LtQ[n, -1] && !IGtQ[n + p + 1, 0]) || (LtQ[n, 0] && LtQ[p, -1]) || EqQ[n + 2*p + 2, 0]) && NeQ[n + p + 1, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}-5 a x +1\right ) \left (a x +1\right )^{2}}{21 \left (a x -1\right )^{3} c^{4} \sqrt {-a^{2} x^{2}+1}\, a^{2}}\) | \(48\) |
trager | \(-\frac {\left (a^{3} x^{3}-4 a^{2} x^{2}-4 a x +1\right ) \sqrt {-a^{2} x^{2}+1}}{21 c^{4} \left (a x -1\right )^{4} a^{2}}\) | \(49\) |
default | \(\frac {\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{a^{3}}+\frac {\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{4}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{5}}}{c^{4}}\) | \(395\) |
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^4} \, dx=-\frac {a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + {\left (a^{3} x^{3} - 4 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-a^{2} x^{2} + 1} + 1}{21 \, {\left (a^{6} c^{4} x^{4} - 4 \, a^{5} c^{4} x^{3} + 6 \, a^{4} c^{4} x^{2} - 4 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} \]
-1/21*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + (a^3*x^3 - 4*a^2*x^2 - 4* a*x + 1)*sqrt(-a^2*x^2 + 1) + 1)/(a^6*c^4*x^4 - 4*a^5*c^4*x^3 + 6*a^4*c^4* x^2 - 4*a^3*c^4*x + a^2*c^4)
\[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^4} \, dx=\frac {\int \frac {x}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
(Integral(x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqr t(-a**2*x**2 + 1)), x) + Integral(a*x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a *x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (82) = 164\).
Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^4} \, dx=\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{6} c^{4} x^{4} - 4 \, a^{5} c^{4} x^{3} + 6 \, a^{4} c^{4} x^{2} - 4 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{5} c^{4} x^{3} - 3 \, a^{4} c^{4} x^{2} + 3 \, a^{3} c^{4} x - a^{2} c^{4}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{21 \, {\left (a^{4} c^{4} x^{2} - 2 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{21 \, {\left (a^{3} c^{4} x - a^{2} c^{4}\right )}} \]
2/7*sqrt(-a^2*x^2 + 1)/(a^6*c^4*x^4 - 4*a^5*c^4*x^3 + 6*a^4*c^4*x^2 - 4*a^ 3*c^4*x + a^2*c^4) + 3/7*sqrt(-a^2*x^2 + 1)/(a^5*c^4*x^3 - 3*a^4*c^4*x^2 + 3*a^3*c^4*x - a^2*c^4) + 1/21*sqrt(-a^2*x^2 + 1)/(a^4*c^4*x^2 - 2*a^3*c^4 *x + a^2*c^4) - 1/21*sqrt(-a^2*x^2 + 1)/(a^3*c^4*x - a^2*c^4)
Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.53 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^4} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} + \frac {28 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {21 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - 1\right )}}{21 \, a c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \]
2/21*(7*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 28*(sqrt(-a^2*x^2 + 1)*a bs(a) + a)^3/(a^6*x^3) - 7*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 2 1*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 1)/(a*c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))
Time = 3.54 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.04 \[ \int \frac {e^{\text {arctanh}(a x)} x}{(c-a c x)^4} \, dx=\frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{21\,\left (c^4\,\sqrt {-a^2}-a\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {4\,a^2\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {3\,\sqrt {1-a^2\,x^2}}{7\,\sqrt {-a^2}\,\left (c^4\,\sqrt {-a^2}+3\,a^2\,c^4\,x^2\,\sqrt {-a^2}-a^3\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x\,\sqrt {-a^2}\right )} \]
(2*(1 - a^2*x^2)^(1/2))/(7*(a^2*c^4 - 4*a^3*c^4*x + 6*a^4*c^4*x^2 - 4*a^5* c^4*x^3 + a^6*c^4*x^4)) - (1 - a^2*x^2)^(1/2)/(15*(a^2*c^4 - 2*a^3*c^4*x + a^4*c^4*x^2)) - (1 - a^2*x^2)^(1/2)/(21*(c^4*(-a^2)^(1/2) - a*c^4*x*(-a^2 )^(1/2))*(-a^2)^(1/2)) + (4*a^2*(1 - a^2*x^2)^(1/2))/(35*(a^4*c^4 - 2*a^5* c^4*x + a^6*c^4*x^2)) + (3*(1 - a^2*x^2)^(1/2))/(7*(-a^2)^(1/2)*(c^4*(-a^2 )^(1/2) + 3*a^2*c^4*x^2*(-a^2)^(1/2) - a^3*c^4*x^3*(-a^2)^(1/2) - 3*a*c^4* x*(-a^2)^(1/2)))