Integrand size = 22, antiderivative size = 121 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {1}{32 a c^4 (1-a x)^4}-\frac {1}{16 a c^4 (1-a x)^3}-\frac {3}{32 a c^4 (1-a x)^2}-\frac {5}{32 a c^4 (1-a x)}+\frac {1}{64 a c^4 (1+a x)^2}+\frac {5}{64 a c^4 (1+a x)}-\frac {15 \text {arctanh}(a x)}{64 a c^4} \] Output:
-1/32/a/c^4/(-a*x+1)^4-1/16/a/c^4/(-a*x+1)^3-3/32/a/c^4/(-a*x+1)^2-5/32/a/ c^4/(-a*x+1)+1/64/a/c^4/(a*x+1)^2+5/64/a/c^4/(a*x+1)-15/64*arctanh(a*x)/a/ c^4
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.68 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {16+17 a x-50 a^2 x^2+10 a^3 x^3+30 a^4 x^4-15 a^5 x^5+15 (-1+a x)^4 (1+a x)^2 \text {arctanh}(a x)}{64 a c^4 (-1+a x)^4 (1+a x)^2} \] Input:
Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^4,x]
Output:
-1/64*(16 + 17*a*x - 50*a^2*x^2 + 10*a^3*x^3 + 30*a^4*x^4 - 15*a^5*x^5 + 1 5*(-1 + a*x)^4*(1 + a*x)^2*ArcTanh[a*x])/(a*c^4*(-1 + a*x)^4*(1 + a*x)^2)
Time = 0.66 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6717, 27, 6690, 54, 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.
\(\displaystyle \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int \frac {e^{2 \text {arctanh}(a x)}}{c^4 \left (1-a^2 x^2\right )^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {e^{2 \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^4}dx}{c^4}\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle -\frac {\int \frac {1}{(1-a x)^5 (a x+1)^3}dx}{c^4}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle -\frac {\int \left (\frac {5}{32 (a x-1)^2}+\frac {5}{64 (a x+1)^2}-\frac {3}{16 (a x-1)^3}+\frac {1}{32 (a x+1)^3}+\frac {3}{16 (a x-1)^4}-\frac {1}{8 (a x-1)^5}-\frac {15}{64 \left (a^2 x^2-1\right )}\right )dx}{c^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {15 \text {arctanh}(a x)}{64 a}+\frac {5}{32 a (1-a x)}-\frac {5}{64 a (a x+1)}+\frac {3}{32 a (1-a x)^2}-\frac {1}{64 a (a x+1)^2}+\frac {1}{16 a (1-a x)^3}+\frac {1}{32 a (1-a x)^4}}{c^4}\) |
Input:
Int[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^4,x]
Output:
-((1/(32*a*(1 - a*x)^4) + 1/(16*a*(1 - a*x)^3) + 3/(32*a*(1 - a*x)^2) + 5/ (32*a*(1 - a*x)) - 1/(64*a*(1 + a*x)^2) - 5/(64*a*(1 + a*x)) + (15*ArcTanh [a*x])/(64*a))/c^4)
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_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a , c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
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 = 0.16 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {\frac {15 x^{5} a^{4}}{64}-\frac {15 a^{3} x^{4}}{32}-\frac {5 a^{2} x^{3}}{32}+\frac {25 a \,x^{2}}{32}-\frac {17 x}{64}-\frac {1}{4 a}}{\left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )^{2} c^{4}}-\frac {15 \ln \left (a x +1\right )}{128 a \,c^{4}}+\frac {15 \ln \left (-a x +1\right )}{128 a \,c^{4}}\) | \(92\) |
default | \(\frac {\frac {1}{64 a \left (a x +1\right )^{2}}+\frac {5}{64 a \left (a x +1\right )}-\frac {15 \ln \left (a x +1\right )}{128 a}-\frac {1}{32 a \left (a x -1\right )^{4}}+\frac {1}{16 a \left (a x -1\right )^{3}}-\frac {3}{32 a \left (a x -1\right )^{2}}+\frac {5}{32 \left (a x -1\right ) a}+\frac {15 \ln \left (a x -1\right )}{128 a}}{c^{4}}\) | \(100\) |
norman | \(\frac {-\frac {49 x}{64 c}-\frac {15 a \,x^{2}}{64 c}+\frac {11 a^{2} x^{3}}{8 c}+\frac {a^{3} x^{4}}{8 c}-\frac {63 a^{4} x^{5}}{64 c}-\frac {a^{5} x^{6}}{64 c}+\frac {a^{6} x^{7}}{4 c}}{c^{3} \left (a x +1\right )^{3} \left (a x -1\right )^{4}}+\frac {15 \ln \left (a x -1\right )}{128 a \,c^{4}}-\frac {15 \ln \left (a x +1\right )}{128 a \,c^{4}}\) | \(119\) |
parallelrisch | \(\frac {108 a^{3} x^{3}-98 a x -92 a^{4} x^{4}-34 a^{5} x^{5}+32 x^{6} a^{6}+30 \ln \left (a x +1\right ) x a -15 \ln \left (a x +1\right )+68 a^{2} x^{2}+15 \ln \left (a x -1\right )+15 \ln \left (a x +1\right ) x^{2} a^{2}-15 \ln \left (a x -1\right ) x^{4} a^{4}+15 \ln \left (a x +1\right ) x^{4} a^{4}-60 \ln \left (a x +1\right ) x^{3} a^{3}-30 a \ln \left (a x -1\right ) x +60 a^{3} \ln \left (a x -1\right ) x^{3}-30 \ln \left (a x -1\right ) x^{5} a^{5}+30 \ln \left (a x +1\right ) x^{5} a^{5}-15 a^{2} \ln \left (a x -1\right ) x^{2}+15 \ln \left (a x -1\right ) x^{6} a^{6}-15 \ln \left (a x +1\right ) x^{6} a^{6}}{128 c^{4} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )^{2} a}\) | \(248\) |
Input:
int(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
Output:
(15/64*x^5*a^4-15/32*a^3*x^4-5/32*a^2*x^3+25/32*a*x^2-17/64*x-1/4/a)/(a*x- 1)^2/(a^2*x^2-1)^2/c^4-15/128*ln(a*x+1)/a/c^4+15/128*ln(-a*x+1)/a/c^4
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (103) = 206\).
Time = 0.10 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.79 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {30 \, a^{5} x^{5} - 60 \, a^{4} x^{4} - 20 \, a^{3} x^{3} + 100 \, a^{2} x^{2} - 34 \, a x - 15 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 15 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 32}{128 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \] Input:
integrate(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^4,x, algorithm="fricas")
Output:
1/128*(30*a^5*x^5 - 60*a^4*x^4 - 20*a^3*x^3 + 100*a^2*x^2 - 34*a*x - 15*(a ^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)*log(a*x + 1) + 15*(a^6*x^6 - 2*a^5*x^5 - a^4*x^4 + 4*a^3*x^3 - a^2*x^2 - 2*a*x + 1)* log(a*x - 1) - 32)/(a^7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4* x^3 - a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4)
Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.17 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {15 a^{5} x^{5} - 30 a^{4} x^{4} - 10 a^{3} x^{3} + 50 a^{2} x^{2} - 17 a x - 16}{64 a^{7} c^{4} x^{6} - 128 a^{6} c^{4} x^{5} - 64 a^{5} c^{4} x^{4} + 256 a^{4} c^{4} x^{3} - 64 a^{3} c^{4} x^{2} - 128 a^{2} c^{4} x + 64 a c^{4}} + \frac {\frac {15 \log {\left (x - \frac {1}{a} \right )}}{128} - \frac {15 \log {\left (x + \frac {1}{a} \right )}}{128}}{a c^{4}} \] Input:
integrate(1/(a*x-1)*(a*x+1)/(-a**2*c*x**2+c)**4,x)
Output:
(15*a**5*x**5 - 30*a**4*x**4 - 10*a**3*x**3 + 50*a**2*x**2 - 17*a*x - 16)/ (64*a**7*c**4*x**6 - 128*a**6*c**4*x**5 - 64*a**5*c**4*x**4 + 256*a**4*c** 4*x**3 - 64*a**3*c**4*x**2 - 128*a**2*c**4*x + 64*a*c**4) + (15*log(x - 1/ a)/128 - 15*log(x + 1/a)/128)/(a*c**4)
Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {15 \, a^{5} x^{5} - 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 50 \, a^{2} x^{2} - 17 \, a x - 16}{64 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} - \frac {15 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac {15 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \] Input:
integrate(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^4,x, algorithm="maxima")
Output:
1/64*(15*a^5*x^5 - 30*a^4*x^4 - 10*a^3*x^3 + 50*a^2*x^2 - 17*a*x - 16)/(a^ 7*c^4*x^6 - 2*a^6*c^4*x^5 - a^5*c^4*x^4 + 4*a^4*c^4*x^3 - a^3*c^4*x^2 - 2* a^2*c^4*x + a*c^4) - 15/128*log(a*x + 1)/(a*c^4) + 15/128*log(a*x - 1)/(a* c^4)
Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.75 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=-\frac {15 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac {15 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} + \frac {15 \, a^{5} x^{5} - 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 50 \, a^{2} x^{2} - 17 \, a x - 16}{64 \, {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{4} a c^{4}} \] Input:
integrate(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^4,x, algorithm="giac")
Output:
-15/128*log(abs(a*x + 1))/(a*c^4) + 15/128*log(abs(a*x - 1))/(a*c^4) + 1/6 4*(15*a^5*x^5 - 30*a^4*x^4 - 10*a^3*x^3 + 50*a^2*x^2 - 17*a*x - 16)/((a*x + 1)^2*(a*x - 1)^4*a*c^4)
Time = 13.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\frac {17\,x}{64}-\frac {25\,a\,x^2}{32}+\frac {1}{4\,a}+\frac {5\,a^2\,x^3}{32}+\frac {15\,a^3\,x^4}{32}-\frac {15\,a^4\,x^5}{64}}{-a^6\,c^4\,x^6+2\,a^5\,c^4\,x^5+a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3+a^2\,c^4\,x^2+2\,a\,c^4\,x-c^4}-\frac {15\,\mathrm {atanh}\left (a\,x\right )}{64\,a\,c^4} \] Input:
int((a*x + 1)/((c - a^2*c*x^2)^4*(a*x - 1)),x)
Output:
((17*x)/64 - (25*a*x^2)/32 + 1/(4*a) + (5*a^2*x^3)/32 + (15*a^3*x^4)/32 - (15*a^4*x^5)/64)/(a^2*c^4*x^2 - c^4 - 4*a^3*c^4*x^3 + a^4*c^4*x^4 + 2*a^5* c^4*x^5 - a^6*c^4*x^6 + 2*a*c^4*x) - (15*atanh(a*x))/(64*a*c^4)
Time = 0.14 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.22 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {15 \,\mathrm {log}\left (a x -1\right ) a^{6} x^{6}-30 \,\mathrm {log}\left (a x -1\right ) a^{5} x^{5}-15 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}+60 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}-15 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}-30 \,\mathrm {log}\left (a x -1\right ) a x +15 \,\mathrm {log}\left (a x -1\right )-15 \,\mathrm {log}\left (a x +1\right ) a^{6} x^{6}+30 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}+15 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}-60 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+15 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}+30 \,\mathrm {log}\left (a x +1\right ) a x -15 \,\mathrm {log}\left (a x +1\right )+15 a^{6} x^{6}-75 a^{4} x^{4}+40 a^{3} x^{3}+85 a^{2} x^{2}-64 a x -17}{128 a \,c^{4} \left (a^{6} x^{6}-2 a^{5} x^{5}-a^{4} x^{4}+4 a^{3} x^{3}-a^{2} x^{2}-2 a x +1\right )} \] Input:
int(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^4,x)
Output:
(15*log(a*x - 1)*a**6*x**6 - 30*log(a*x - 1)*a**5*x**5 - 15*log(a*x - 1)*a **4*x**4 + 60*log(a*x - 1)*a**3*x**3 - 15*log(a*x - 1)*a**2*x**2 - 30*log( a*x - 1)*a*x + 15*log(a*x - 1) - 15*log(a*x + 1)*a**6*x**6 + 30*log(a*x + 1)*a**5*x**5 + 15*log(a*x + 1)*a**4*x**4 - 60*log(a*x + 1)*a**3*x**3 + 15* log(a*x + 1)*a**2*x**2 + 30*log(a*x + 1)*a*x - 15*log(a*x + 1) + 15*a**6*x **6 - 75*a**4*x**4 + 40*a**3*x**3 + 85*a**2*x**2 - 64*a*x - 17)/(128*a*c** 4*(a**6*x**6 - 2*a**5*x**5 - a**4*x**4 + 4*a**3*x**3 - a**2*x**2 - 2*a*x + 1))