3.13.35 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx\) [1235]

Optimal. Leaf size=119 \[ \frac {1}{64 a c^4 (1-a x)^2}+\frac {5}{64 a c^4 (1-a x)}-\frac {1}{32 a c^4 (1+a x)^4}-\frac {1}{16 a c^4 (1+a x)^3}-\frac {3}{32 a c^4 (1+a x)^2}-\frac {5}{32 a c^4 (1+a x)}+\frac {15 \tanh ^{-1}(a x)}{64 a c^4} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 119, 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 = {6275, 46, 213} \begin {gather*} \frac {5}{64 a c^4 (1-a x)}-\frac {5}{32 a c^4 (a x+1)}+\frac {1}{64 a c^4 (1-a x)^2}-\frac {3}{32 a c^4 (a x+1)^2}-\frac {1}{16 a c^4 (a x+1)^3}-\frac {1}{32 a c^4 (a x+1)^4}+\frac {15 \tanh ^{-1}(a x)}{64 a c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^4),x]

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 213

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

Rule 6275

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

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Mathematica [A]
time = 0.04, size = 80, normalized size = 0.67 \begin {gather*} \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)^2 (1+a x)^4 \tanh ^{-1}(a x)}{64 a (-1+a x)^2 (c+a c x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [A]
time = 0.07, size = 100, normalized size = 0.84

method	result	size
risch	\(\frac {-\frac {15 a^{4} x^{5}}{64}-\frac {15 a^{3} x^{4}}{32}+\frac {5 a^{2} x^{3}}{32}+\frac {25 x^{2} a}{32}+\frac {17 x}{64}-\frac {1}{4 a}}{\left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right )^{2} c^{4}}-\frac {15 \ln \left (a x -1\right )}{128 a \,c^{4}}+\frac {15 \ln \left (-a x -1\right )}{128 a \,c^{4}}\)	\(92\)
default	\(\frac {-\frac {1}{32 a \left (a x +1\right )^{4}}-\frac {1}{16 a \left (a x +1\right )^{3}}-\frac {3}{32 a \left (a x +1\right )^{2}}-\frac {5}{32 a \left (a x +1\right )}+\frac {15 \ln \left (a x +1\right )}{128 a}+\frac {1}{64 a \left (a x -1\right )^{2}}-\frac {5}{64 a \left (a x -1\right )}-\frac {15 \ln \left (a x -1\right )}{128 a}}{c^{4}}\)	\(100\)
norman	\(\frac {\frac {5 a^{3} x^{4}}{4 c}-\frac {49 x}{64 c}-\frac {17 a \,x^{2}}{32 c}+\frac {103 a^{2} x^{3}}{64 c}-\frac {71 a^{4} x^{5}}{64 c}-\frac {31 a^{5} x^{6}}{32 c}+\frac {17 a^{6} x^{7}}{64 c}+\frac {a^{7} x^{8}}{4 c}}{\left (a x +1\right )^{5} c^{3} \left (a x -1\right )^{3}}-\frac {15 \ln \left (a x -1\right )}{128 a \,c^{4}}+\frac {15 \ln \left (a x +1\right )}{128 a \,c^{4}}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 140, normalized size = 1.18 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (103) = 206\).
time = 0.34, size = 217, normalized size = 1.82 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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Sympy [A]
time = 0.35, size = 143, normalized size = 1.20 \begin {gather*} - \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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Giac [A]
time = 0.42, size = 122, normalized size = 1.03 \begin {gather*} -\frac {15 \, \log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{128 \, a c^{4}} + \frac {\frac {24}{a x + 1} - 11}{256 \, a c^{4} {\left (\frac {2}{a x + 1} - 1\right )}^{2}} - \frac {\frac {5 \, a^{11} c^{12}}{a x + 1} + \frac {3 \, a^{11} c^{12}}{{\left (a x + 1\right )}^{2}} + \frac {2 \, a^{11} c^{12}}{{\left (a x + 1\right )}^{3}} + \frac {a^{11} c^{12}}{{\left (a x + 1\right )}^{4}}}{32 \, a^{12} c^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/128*log(abs(-2/(a*x + 1) + 1))/(a*c^4) + 1/256*(24/(a*x + 1) - 11)/(a*c^4*(2/(a*x + 1) - 1)^2) - 1/32*(5*a
^11*c^12/(a*x + 1) + 3*a^11*c^12/(a*x + 1)^2 + 2*a^11*c^12/(a*x + 1)^3 + a^11*c^12/(a*x + 1)^4)/(a^12*c^16)

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Mupad [B]
time = 0.99, size = 120, normalized size = 1.01 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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)/(c^4 - a^2*c^4*x^2
- 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)

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