Integrand size = 22, antiderivative size = 111 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {x}{c^3}-\frac {1}{8 a c^3 (1-a x)^4}+\frac {11}{12 a c^3 (1-a x)^3}-\frac {49}{16 a c^3 (1-a x)^2}+\frac {111}{16 a c^3 (1-a x)}+\frac {129 \log (1-a x)}{32 a c^3}-\frac {\log (1+a x)}{32 a c^3} \] Output:
x/c^3-1/8/a/c^3/(-a*x+1)^4+11/12/a/c^3/(-a*x+1)^3-49/16/a/c^3/(-a*x+1)^2+1 11/16/a/c^3/(-a*x+1)+129/32*ln(-a*x+1)/a/c^3-1/32*ln(a*x+1)/a/c^3
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {2 \left (224-701 a x+660 a^2 x^2-45 a^3 x^3-192 a^4 x^4+48 a^5 x^5\right )+387 (-1+a x)^4 \log (1-a x)-3 (-1+a x)^4 \log (1+a x)}{96 a c^3 (-1+a x)^4} \] Input:
Integrate[E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2))^3,x]
Output:
(2*(224 - 701*a*x + 660*a^2*x^2 - 45*a^3*x^3 - 192*a^4*x^4 + 48*a^5*x^5) + 387*(-1 + a*x)^4*Log[1 - a*x] - 3*(-1 + a*x)^4*Log[1 + a*x])/(96*a*c^3*(- 1 + a*x)^4)
Time = 0.86 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6717, 27, 6707, 6700, 99, 2009}
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.
\(\displaystyle \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle \int \frac {a^6 e^{4 \text {arctanh}(a x)}}{c^3 \left (a^2-\frac {1}{x^2}\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^6 \int \frac {e^{4 \text {arctanh}(a x)}}{\left (a^2-\frac {1}{x^2}\right )^3}dx}{c^3}\) |
\(\Big \downarrow \) 6707 |
\(\displaystyle -\frac {a^6 \int \frac {e^{4 \text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^3}dx}{c^3}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle -\frac {a^6 \int \frac {x^6}{(1-a x)^5 (a x+1)}dx}{c^3}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {a^6 \int \left (\frac {1}{32 a^6 (a x+1)}-\frac {1}{a^6}-\frac {129}{32 a^6 (a x-1)}-\frac {111}{16 a^6 (a x-1)^2}-\frac {49}{8 a^6 (a x-1)^3}-\frac {11}{4 a^6 (a x-1)^4}-\frac {1}{2 a^6 (a x-1)^5}\right )dx}{c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^6 \left (-\frac {111}{16 a^7 (1-a x)}+\frac {49}{16 a^7 (1-a x)^2}-\frac {11}{12 a^7 (1-a x)^3}+\frac {1}{8 a^7 (1-a x)^4}-\frac {129 \log (1-a x)}{32 a^7}+\frac {\log (a x+1)}{32 a^7}-\frac {x}{a^6}\right )}{c^3}\) |
Input:
Int[E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2))^3,x]
Output:
-((a^6*(-(x/a^6) + 1/(8*a^7*(1 - a*x)^4) - 11/(12*a^7*(1 - a*x)^3) + 49/(1 6*a^7*(1 - a*x)^2) - 111/(16*a^7*(1 - a*x)) - (129*Log[1 - a*x])/(32*a^7) + Log[1 + a*x]/(32*a^7)))/c^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x ] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {x}{c^{3}}+\frac {-\frac {111 a^{2} c^{3} x^{3}}{16}+\frac {71 a \,c^{3} x^{2}}{4}-\frac {749 c^{3} x}{48}+\frac {14 c^{3}}{3 a}}{c^{6} \left (a x -1\right )^{4}}+\frac {129 \ln \left (-a x +1\right )}{32 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{32 a \,c^{3}}\) | \(82\) |
default | \(\frac {a^{6} \left (\frac {x}{a^{6}}-\frac {\ln \left (a x +1\right )}{32 a^{7}}-\frac {1}{8 a^{7} \left (a x -1\right )^{4}}-\frac {11}{12 a^{7} \left (a x -1\right )^{3}}-\frac {49}{16 a^{7} \left (a x -1\right )^{2}}-\frac {111}{16 a^{7} \left (a x -1\right )}+\frac {129 \ln \left (a x -1\right )}{32 a^{7}}\right )}{c^{3}}\) | \(84\) |
norman | \(\frac {\frac {a^{6} x^{7}}{c}+\frac {65 x}{16 c}-\frac {49 a \,x^{2}}{8 c}-\frac {161 a^{2} x^{3}}{24 c}+\frac {301 a^{3} x^{4}}{24 c}+\frac {67 a^{4} x^{5}}{48 c}-\frac {20 a^{5} x^{6}}{3 c}}{c^{2} \left (a x +1\right )^{2} \left (a x -1\right )^{4}}+\frac {129 \ln \left (a x -1\right )}{32 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{32 a \,c^{3}}\) | \(118\) |
parallelrisch | \(\frac {1702 a^{3} x^{3}+390 a x -832 a^{4} x^{4}+96 a^{5} x^{5}+12 \ln \left (a x +1\right ) x a -3 \ln \left (a x +1\right )-1368 a^{2} x^{2}+387 \ln \left (a x -1\right )-18 \ln \left (a x +1\right ) x^{2} a^{2}+387 \ln \left (a x -1\right ) x^{4} a^{4}-3 \ln \left (a x +1\right ) x^{4} a^{4}+12 \ln \left (a x +1\right ) x^{3} a^{3}-1548 a \ln \left (a x -1\right ) x -1548 a^{3} \ln \left (a x -1\right ) x^{3}+2322 a^{2} \ln \left (a x -1\right ) x^{2}}{96 \left (a x -1\right )^{4} c^{3} a}\) | \(173\) |
Input:
int(1/(a*x-1)^2*(a*x+1)^2/(c-c/a^2/x^2)^3,x,method=_RETURNVERBOSE)
Output:
x/c^3+(-111/16*a^2*c^3*x^3+71/4*a*c^3*x^2-749/48*c^3*x+14/3*c^3/a)/c^6/(a* x-1)^4+129/32*ln(-a*x+1)/a/c^3-1/32*ln(a*x+1)/a/c^3
Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.47 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {96 \, a^{5} x^{5} - 384 \, a^{4} x^{4} - 90 \, a^{3} x^{3} + 1320 \, a^{2} x^{2} - 1402 \, a x - 3 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x + 1\right ) + 387 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 448}{96 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \] Input:
integrate(1/(a*x-1)^2*(a*x+1)^2/(c-c/a^2/x^2)^3,x, algorithm="fricas")
Output:
1/96*(96*a^5*x^5 - 384*a^4*x^4 - 90*a^3*x^3 + 1320*a^2*x^2 - 1402*a*x - 3* (a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x + 1) + 387*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(a*x - 1) + 448)/(a^5*c^3*x^4 - 4* a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)
Time = 0.36 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.03 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=a^{6} \left (\frac {- 333 a^{3} x^{3} + 852 a^{2} x^{2} - 749 a x + 224}{48 a^{11} c^{3} x^{4} - 192 a^{10} c^{3} x^{3} + 288 a^{9} c^{3} x^{2} - 192 a^{8} c^{3} x + 48 a^{7} c^{3}} + \frac {x}{a^{6} c^{3}} + \frac {\frac {129 \log {\left (x - \frac {1}{a} \right )}}{32} - \frac {\log {\left (x + \frac {1}{a} \right )}}{32}}{a^{7} c^{3}}\right ) \] Input:
integrate(1/(a*x-1)**2*(a*x+1)**2/(c-c/a**2/x**2)**3,x)
Output:
a**6*((-333*a**3*x**3 + 852*a**2*x**2 - 749*a*x + 224)/(48*a**11*c**3*x**4 - 192*a**10*c**3*x**3 + 288*a**9*c**3*x**2 - 192*a**8*c**3*x + 48*a**7*c* *3) + x/(a**6*c**3) + (129*log(x - 1/a)/32 - log(x + 1/a)/32)/(a**7*c**3))
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {333 \, a^{3} x^{3} - 852 \, a^{2} x^{2} + 749 \, a x - 224}{48 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac {x}{c^{3}} - \frac {\log \left (a x + 1\right )}{32 \, a c^{3}} + \frac {129 \, \log \left (a x - 1\right )}{32 \, a c^{3}} \] Input:
integrate(1/(a*x-1)^2*(a*x+1)^2/(c-c/a^2/x^2)^3,x, algorithm="maxima")
Output:
-1/48*(333*a^3*x^3 - 852*a^2*x^2 + 749*a*x - 224)/(a^5*c^3*x^4 - 4*a^4*c^3 *x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3) + x/c^3 - 1/32*log(a*x + 1)/(a *c^3) + 129/32*log(a*x - 1)/(a*c^3)
Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.17 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {a x - 1}{a c^{3}} - \frac {4 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{3}} - \frac {\log \left ({\left | -\frac {2}{a x - 1} - 1 \right |}\right )}{32 \, a c^{3}} - \frac {\frac {333 \, a^{11} c^{9}}{a x - 1} + \frac {147 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{2}} + \frac {44 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, a^{11} c^{9}}{{\left (a x - 1\right )}^{4}}}{48 \, a^{12} c^{12}} \] Input:
integrate(1/(a*x-1)^2*(a*x+1)^2/(c-c/a^2/x^2)^3,x, algorithm="giac")
Output:
(a*x - 1)/(a*c^3) - 4*log(abs(a*x - 1)/((a*x - 1)^2*abs(a)))/(a*c^3) - 1/3 2*log(abs(-2/(a*x - 1) - 1))/(a*c^3) - 1/48*(333*a^11*c^9/(a*x - 1) + 147* a^11*c^9/(a*x - 1)^2 + 44*a^11*c^9/(a*x - 1)^3 + 6*a^11*c^9/(a*x - 1)^4)/( a^12*c^12)
Time = 0.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {x}{c^3}-\frac {\frac {749\,x}{48}-\frac {71\,a\,x^2}{4}-\frac {14}{3\,a}+\frac {111\,a^2\,x^3}{16}}{a^4\,c^3\,x^4-4\,a^3\,c^3\,x^3+6\,a^2\,c^3\,x^2-4\,a\,c^3\,x+c^3}+\frac {129\,\ln \left (a\,x-1\right )}{32\,a\,c^3}-\frac {\ln \left (a\,x+1\right )}{32\,a\,c^3} \] Input:
int((a*x + 1)^2/((c - c/(a^2*x^2))^3*(a*x - 1)^2),x)
Output:
x/c^3 - ((749*x)/48 - (71*a*x^2)/4 - 14/(3*a) + (111*a^2*x^3)/16)/(c^3 + 6 *a^2*c^3*x^2 - 4*a^3*c^3*x^3 + a^4*c^3*x^4 - 4*a*c^3*x) + (129*log(a*x - 1 ))/(32*a*c^3) - log(a*x + 1)/(32*a*c^3)
Time = 0.15 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.70 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {774 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}-3096 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}+4644 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}-3096 \,\mathrm {log}\left (a x -1\right ) a x +774 \,\mathrm {log}\left (a x -1\right )-6 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}+24 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}-36 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}+24 \,\mathrm {log}\left (a x +1\right ) a x -6 \,\mathrm {log}\left (a x +1\right )+192 a^{5} x^{5}-813 a^{4} x^{4}+2370 a^{2} x^{2}-2624 a x +851}{192 a \,c^{3} \left (a^{4} x^{4}-4 a^{3} x^{3}+6 a^{2} x^{2}-4 a x +1\right )} \] Input:
int(1/(a*x-1)^2*(a*x+1)^2/(c-c/a^2/x^2)^3,x)
Output:
(774*log(a*x - 1)*a**4*x**4 - 3096*log(a*x - 1)*a**3*x**3 + 4644*log(a*x - 1)*a**2*x**2 - 3096*log(a*x - 1)*a*x + 774*log(a*x - 1) - 6*log(a*x + 1)* a**4*x**4 + 24*log(a*x + 1)*a**3*x**3 - 36*log(a*x + 1)*a**2*x**2 + 24*log (a*x + 1)*a*x - 6*log(a*x + 1) + 192*a**5*x**5 - 813*a**4*x**4 + 2370*a**2 *x**2 - 2624*a*x + 851)/(192*a*c**3*(a**4*x**4 - 4*a**3*x**3 + 6*a**2*x**2 - 4*a*x + 1))