3.1076 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx\)

Optimal. Leaf size=121 \[ \frac {5}{32 a c^4 (1-a x)}-\frac {5}{64 a c^4 (a x+1)}+\frac {3}{32 a c^4 (1-a x)^2}-\frac {1}{64 a c^4 (a x+1)^2}+\frac {1}{16 a c^4 (1-a x)^3}+\frac {1}{32 a c^4 (1-a x)^4}+\frac {15 \tanh ^{-1}(a x)}{64 a c^4} \]

[Out]

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)^2-5/6
4/a/c^4/(a*x+1)+15/64*arctanh(a*x)/a/c^4

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6140, 44, 207} \[ \frac {5}{32 a c^4 (1-a x)}-\frac {5}{64 a c^4 (a x+1)}+\frac {3}{32 a c^4 (1-a x)^2}-\frac {1}{64 a c^4 (a x+1)^2}+\frac {1}{16 a c^4 (1-a x)^3}+\frac {1}{32 a c^4 (1-a x)^4}+\frac {15 \tanh ^{-1}(a x)}{64 a c^4} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^4,x]

[Out]

1/(32*a*c^4*(1 - a*x)^4) + 1/(16*a*c^4*(1 - a*x)^3) + 3/(32*a*c^4*(1 - a*x)^2) + 5/(32*a*c^4*(1 - a*x)) - 1/(6
4*a*c^4*(1 + a*x)^2) - 5/(64*a*c^4*(1 + a*x)) + (15*ArcTanh[a*x])/(64*a*c^4)

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[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])

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac {\int \frac {1}{(1-a x)^5 (1+a x)^3} \, dx}{c^4}\\ &=\frac {\int \left (-\frac {1}{8 (-1+a x)^5}+\frac {3}{16 (-1+a x)^4}-\frac {3}{16 (-1+a x)^3}+\frac {5}{32 (-1+a x)^2}+\frac {1}{32 (1+a x)^3}+\frac {5}{64 (1+a x)^2}-\frac {15}{64 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^4}\\ &=\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 \int \frac {1}{-1+a^2 x^2} \, dx}{64 c^4}\\ &=\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 \tanh ^{-1}(a x)}{64 a c^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 82, normalized size = 0.68 \[ \frac {-15 a^5 x^5+30 a^4 x^4+10 a^3 x^3-50 a^2 x^2+17 a x+15 (a x-1)^4 (a x+1)^2 \tanh ^{-1}(a x)+16}{64 a c^4 (a x-1)^4 (a x+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^4,x]

[Out]

(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*ArcTanh[a*x])/(
64*a*c^4*(-1 + a*x)^4*(1 + a*x)^2)

________________________________________________________________________________________

fricas [B]  time = 0.59, size = 217, normalized size = 1.79 \[ -\frac {30 \, a^{5} x^{5} - 60 \, a^{4} x^{4} - 20 \, a^{3} x^{3} + 100 \, a^{2} x^{2} - 34 \, a x - 15 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 15 \, {\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 32}{128 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

-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)

________________________________________________________________________________________

giac [A]  time = 1.12, size = 91, normalized size = 0.75 \[ \frac {15 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} - \frac {15 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} - \frac {15 \, a^{5} x^{5} - 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 50 \, a^{2} x^{2} - 17 \, a x - 16}{64 \, {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{4} a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

15/128*log(abs(a*x + 1))/(a*c^4) - 15/128*log(abs(a*x - 1))/(a*c^4) - 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*x + 1)^2*(a*x - 1)^4*a*c^4)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 120, normalized size = 0.99 \[ \frac {1}{32 c^{4} a \left (a x -1\right )^{4}}-\frac {1}{16 c^{4} a \left (a x -1\right )^{3}}+\frac {3}{32 c^{4} a \left (a x -1\right )^{2}}-\frac {5}{32 c^{4} a \left (a x -1\right )}-\frac {15 \ln \left (a x -1\right )}{128 c^{4} a}-\frac {1}{64 a \,c^{4} \left (a x +1\right )^{2}}-\frac {5}{64 a \,c^{4} \left (a x +1\right )}+\frac {15 \ln \left (a x +1\right )}{128 a \,c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^4,x)

[Out]

1/32/c^4/a/(a*x-1)^4-1/16/c^4/a/(a*x-1)^3+3/32/c^4/a/(a*x-1)^2-5/32/c^4/a/(a*x-1)-15/128/c^4/a*ln(a*x-1)-1/64/
a/c^4/(a*x+1)^2-5/64/a/c^4/(a*x+1)+15/128*ln(a*x+1)/a/c^4

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 140, normalized size = 1.16 \[ -\frac {15 \, a^{5} x^{5} - 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 50 \, a^{2} x^{2} - 17 \, a x - 16}{64 \, {\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {15 \, \log \left (a x + 1\right )}{128 \, a c^{4}} - \frac {15 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

-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)

________________________________________________________________________________________

mupad [B]  time = 0.14, size = 122, normalized size = 1.01 \[ \frac {15\,\mathrm {atanh}\left (a\,x\right )}{64\,a\,c^4}-\frac {\frac {17\,x}{64}-\frac {25\,a\,x^2}{32}+\frac {1}{4\,a}+\frac {5\,a^2\,x^3}{32}+\frac {15\,a^3\,x^4}{32}-\frac {15\,a^4\,x^5}{64}}{-a^6\,c^4\,x^6+2\,a^5\,c^4\,x^5+a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3+a^2\,c^4\,x^2+2\,a\,c^4\,x-c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*x + 1)^2/((c - a^2*c*x^2)^4*(a^2*x^2 - 1)),x)

[Out]

(15*atanh(a*x))/(64*a*c^4) - ((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)

________________________________________________________________________________________

sympy [A]  time = 0.61, size = 143, normalized size = 1.18 \[ - \frac {15 a^{5} x^{5} - 30 a^{4} x^{4} - 10 a^{3} x^{3} + 50 a^{2} x^{2} - 17 a x - 16}{64 a^{7} c^{4} x^{6} - 128 a^{6} c^{4} x^{5} - 64 a^{5} c^{4} x^{4} + 256 a^{4} c^{4} x^{3} - 64 a^{3} c^{4} x^{2} - 128 a^{2} c^{4} x + 64 a c^{4}} - \frac {\frac {15 \log {\left (x - \frac {1}{a} \right )}}{128} - \frac {15 \log {\left (x + \frac {1}{a} \right )}}{128}}{a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)/(-a**2*c*x**2+c)**4,x)

[Out]

-(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)

________________________________________________________________________________________