3.7.13 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx\) [613]

Optimal. Leaf size=127 \[ \frac {16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4}+\frac {e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac {10 e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}-\frac {8 e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{21 a c^4 \left (1-a^2 x^2\right )} \]

[Out]

16/63/a/c^4*((a*x-1)/(a*x+1))^(3/2)+1/9*(2*a*x+1)/a/c^4*((a*x-1)/(a*x+1))^(3/2)/(-a^2*x^2+1)^3+10/63*(4*a*x+3)
/a/c^4*((a*x-1)/(a*x+1))^(3/2)/(-a^2*x^2+1)^2-8/21*(2*a*x+3)/a/c^4*((a*x-1)/(a*x+1))^(3/2)/(-a^2*x^2+1)

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Rubi [A]
time = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \begin {gather*} \frac {(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac {8 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}+\frac {10 (4 a x+3) e^{-3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac {16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^4),x]

[Out]

16/(63*a*c^4*E^(3*ArcCoth[a*x])) + (1 + 2*a*x)/(9*a*c^4*E^(3*ArcCoth[a*x])*(1 - a^2*x^2)^3) + (10*(3 + 4*a*x))
/(63*a*c^4*E^(3*ArcCoth[a*x])*(1 - a^2*x^2)^2) - (8*(3 + 2*a*x))/(21*a*c^4*E^(3*ArcCoth[a*x])*(1 - a^2*x^2))

Rule 6318

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rule 6320

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(n + 2*a*(p + 1)*x)*(c + d*x^2
)^(p + 1)*(E^(n*ArcCoth[a*x])/(a*c*(n^2 - 4*(p + 1)^2))), x] - Dist[2*(p + 1)*((2*p + 3)/(c*(n^2 - 4*(p + 1)^2
))), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !In
tegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] ||  !IntegerQ[n])

Rubi steps

\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac {e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac {10 \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{9 c}\\ &=\frac {e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac {10 e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac {40 \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{21 c^2}\\ &=\frac {e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac {10 e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}-\frac {8 e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}-\frac {16 \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{21 c^3}\\ &=\frac {16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4}+\frac {e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac {10 e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}-\frac {8 e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 82, normalized size = 0.65 \begin {gather*} \frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (19-6 a x-66 a^2 x^2-56 a^3 x^3+24 a^4 x^4+48 a^5 x^5+16 a^6 x^6\right )}{63 c^4 (-1+a x)^2 (1+a x)^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^4),x]

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*(19 - 6*a*x - 66*a^2*x^2 - 56*a^3*x^3 + 24*a^4*x^4 + 48*a^5*x^5 + 16*a^6*x^6))/(63*c^
4*(-1 + a*x)^2*(1 + a*x)^5)

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Maple [A]
time = 0.20, size = 84, normalized size = 0.66

method	result	size
gosper	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right )}{63 \left (a^{2} x^{2}-1\right )^{3} c^{4} a}\)	\(81\)
default	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right )}{63 \left (a x -1\right )^{3} c^{4} \left (a x +1\right )^{3} a}\)	\(84\)
trager	\(\frac {\left (16 a^{6} x^{6}+48 a^{5} x^{5}+24 a^{4} x^{4}-56 a^{3} x^{3}-66 a^{2} x^{2}-6 a x +19\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{63 a \,c^{4} \left (a x -1\right )^{2} \left (a x +1\right )^{4}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)

[Out]

1/63*((a*x-1)/(a*x+1))^(3/2)*(16*a^6*x^6+48*a^5*x^5+24*a^4*x^4-56*a^3*x^3-66*a^2*x^2-6*a*x+19)/(a*x-1)^3/c^4/(
a*x+1)^3/a

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Maxima [A]
time = 0.27, size = 136, normalized size = 1.07 \begin {gather*} \frac {1}{4032} \, a {\left (\frac {7 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 54 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 189 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 420 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 945 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} + \frac {21 \, {\left (\frac {18 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm="maxima")

[Out]

1/4032*a*((7*((a*x - 1)/(a*x + 1))^(9/2) - 54*((a*x - 1)/(a*x + 1))^(7/2) + 189*((a*x - 1)/(a*x + 1))^(5/2) -
420*((a*x - 1)/(a*x + 1))^(3/2) + 945*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) + 21*(18*(a*x - 1)/(a*x + 1) - 1)/(
a^2*c^4*((a*x - 1)/(a*x + 1))^(3/2)))

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Fricas [A]
time = 0.35, size = 134, normalized size = 1.06 \begin {gather*} \frac {{\left (16 \, a^{6} x^{6} + 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} - 6 \, a x + 19\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{63 \, {\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)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

1/63*(16*a^6*x^6 + 48*a^5*x^5 + 24*a^4*x^4 - 56*a^3*x^3 - 66*a^2*x^2 - 6*a*x + 19)*sqrt((a*x - 1)/(a*x + 1))/(
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\, dx}{c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))**(3/2)/(-a**2*c*x**2+c)**4,x)

[Out]

(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**9*x**9 + a**8*x**8 - 4*a**7*x**7 - 4*a**6*x**6 + 6*a**5*x**5
+ 6*a**4*x**4 - 4*a**3*x**3 - 4*a**2*x**2 + a*x + 1), x) + Integral(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**
9*x**9 + a**8*x**8 - 4*a**7*x**7 - 4*a**6*x**6 + 6*a**5*x**5 + 6*a**4*x**4 - 4*a**3*x**3 - 4*a**2*x**2 + a*x +
1), x))/c**4

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm="giac")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(3/2)/(a^2*c*x^2 - c)^4, x)

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Mupad [B]
time = 0.04, size = 155, normalized size = 1.22 \begin {gather*} \frac {15\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{48\,a\,c^4}+\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{64\,a\,c^4}-\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{576\,a\,c^4}+\frac {\frac {6\,\left (a\,x-1\right )}{a\,x+1}-\frac {1}{3}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x - 1)/(a*x + 1))^(3/2)/(c - a^2*c*x^2)^4,x)

[Out]

(15*((a*x - 1)/(a*x + 1))^(1/2))/(64*a*c^4) - (5*((a*x - 1)/(a*x + 1))^(3/2))/(48*a*c^4) + (3*((a*x - 1)/(a*x
+ 1))^(5/2))/(64*a*c^4) - (3*((a*x - 1)/(a*x + 1))^(7/2))/(224*a*c^4) + ((a*x - 1)/(a*x + 1))^(9/2)/(576*a*c^4
) + ((6*(a*x - 1))/(a*x + 1) - 1/3)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(3/2))

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