3.11.76 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx\) [1076]

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

[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

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Rubi [A]
time = 0.06, 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 = {6275, 46, 213} \begin {gather*} \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} \end {gather*}

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 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)^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*}

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Mathematica [A]
time = 0.04, size = 82, normalized size = 0.68 \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)^4 (1+a x)^2 \tanh ^{-1}(a x)}{64 a c^4 (-1+a x)^4 (1+a x)^2} \end {gather*}

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)

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Maple [A]
time = 0.06, size = 100, normalized size = 0.83

method	result	size
risch	\(\frac {-\frac {15 a^{4} x^{5}}{64}+\frac {15 a^{3} x^{4}}{32}+\frac {5 a^{2} x^{3}}{32}-\frac {25 x^{2} a}{32}+\frac {17 x}{64}+\frac {1}{4 a}}{c^{4} \left (a x -1\right )^{2} \left (a^{2} x^{2}-1\right )^{2}}-\frac {15 \ln \left (a x -1\right )}{128 a \,c^{4}}+\frac {15 \ln \left (-a x -1\right )}{128 a \,c^{4}}\)	\(92\)
default	\(\frac {-\frac {1}{64 a \left (a x +1\right )^{2}}-\frac {5}{64 a \left (a x +1\right )}+\frac {15 \ln \left (a x +1\right )}{128 a}+\frac {1}{32 a \left (a x -1\right )^{4}}-\frac {1}{16 a \left (a x -1\right )^{3}}+\frac {3}{32 a \left (a x -1\right )^{2}}-\frac {5}{32 a \left (a x -1\right )}-\frac {15 \ln \left (a x -1\right )}{128 a}}{c^{4}}\)	\(100\)
norman	\(\frac {\frac {49 x}{64 c}-\frac {73 a^{2} x^{3}}{64 c}+\frac {55 a^{4} x^{5}}{64 c}-\frac {15 a^{6} x^{7}}{64 c}+\frac {a \,x^{2}}{c}-\frac {3 a^{3} x^{4}}{2 c}+\frac {a^{5} x^{6}}{c}-\frac {a^{7} x^{8}}{4 c}}{\left (a^{2} x^{2}-1\right )^{4} c^{3}}-\frac {15 \ln \left (a x -1\right )}{128 a \,c^{4}}+\frac {15 \ln \left (a x +1\right )}{128 a \,c^{4}}\)	\(125\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 140, normalized size = 1.16 \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((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 = 0.33, size = 217, normalized size = 1.79 \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((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 = 0.36, size = 143, 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 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((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 = 0.41, size = 91, normalized size = 0.75 \begin {gather*} \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}} \end {gather*}

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)

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Mupad [B]
time = 0.14, size = 122, normalized size = 1.01 \begin {gather*} \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} \end {gather*}

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)

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