3.12.99 \(\int \frac {e^{-\tanh ^{-1}(a x)}}{(c-a^2 c x^2)^4} \, dx\) [1199]

Optimal. Leaf size=97 \[ -\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{35 c^4 \sqrt {1-a^2 x^2}} \]

[Out]

1/7*(a*x-1)/a/c^4/(-a^2*x^2+1)^(7/2)+6/35*x/c^4/(-a^2*x^2+1)^(5/2)+8/35*x/c^4/(-a^2*x^2+1)^(3/2)+16/35*x/c^4/(
-a^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6274, 653, 198, 197} \begin {gather*} \frac {16 x}{35 c^4 \sqrt {1-a^2 x^2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*(1 - a*x)/(a*c^4*(1 - a^2*x^2)^(7/2)) + (6*x)/(35*c^4*(1 - a^2*x^2)^(5/2)) + (8*x)/(35*c^4*(1 - a^2*x^2)^
(3/2)) + (16*x)/(35*c^4*Sqrt[1 - a^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 6274

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a^2*x^2)^(p + n/
2)/(1 - a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] && IntegerQ[p] && ILtQ[(n - 1)/2, 0] &&
!IntegerQ[p - n/2]

Rubi steps

\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^4}\\ &=-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^4}\\ &=-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {24 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^4}\\ &=-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^4}\\ &=-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{35 c^4 \sqrt {1-a^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 75, normalized size = 0.77 \begin {gather*} \frac {-5+30 a x+30 a^2 x^2-40 a^3 x^3-40 a^4 x^4+16 a^5 x^5+16 a^6 x^6}{35 a c^4 (1-a x)^{5/2} (1+a x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5 + 30*a*x + 30*a^2*x^2 - 40*a^3*x^3 - 40*a^4*x^4 + 16*a^5*x^5 + 16*a^6*x^6)/(35*a*c^4*(1 - a*x)^(5/2)*(1 +
a*x)^(7/2))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.06, size = 720, normalized size = 7.42

method	result	size
gosper	\(\frac {16 x^{6} a^{6}+16 x^{5} a^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5}{35 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a x +1\right ) a \,c^{4}}\)	\(74\)
trager	\(-\frac {\left (16 x^{6} a^{6}+16 x^{5} a^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5\right ) \sqrt {-a^{2} x^{2}+1}}{35 c^{4} \left (a x +1\right )^{4} \left (a x -1\right )^{3} a}\)	\(81\)
default	\(\frac {\frac {\frac {35 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{256}+\frac {35 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{256 \sqrt {a^{2}}}}{a}+\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{5 a \left (x -\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{15 \left (x -\frac {1}{a}\right )^{3}}}{32 a^{4}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}}{8 a^{4}}+\frac {-\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{32 a \left (x +\frac {1}{a}\right )^{2}}-\frac {5 a \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{32}}{a^{2}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{7 a \left (x +\frac {1}{a}\right )^{5}}+\frac {2 a \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}\right )}{7}}{16 a^{5}}-\frac {35 \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}\right )}{256 a}-\frac {5 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{192 a^{4} \left (x -\frac {1}{a}\right )^{3}}-\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{96 a^{4} \left (x +\frac {1}{a}\right )^{3}}+\frac {\frac {15 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{128 a \left (x -\frac {1}{a}\right )^{2}}+\frac {15 a \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}\right )}{128}}{a^{2}}}{c^{4}}\)	\(720\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(35/256/a*((-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+
1/a))^(1/2)))+1/32/a^4*(1/5/a/(x-1/a)^4*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)-1/15/(x-1/a)^3*(-a^2*(x-1/a)^2-2*(x
-1/a)*a)^(3/2))+1/8/a^4*(-1/5/a/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-1/15/(x+1/a)^3*(-a^2*(x+1/a)^2+2*
a*(x+1/a))^(3/2))+5/32/a^2*(-1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-a*((-a^2*(x+1/a)^2+2*a*(x+1/a))^
(1/2)+a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))))+1/16/a^5*(-1/7/a/(x+1/a)^5*(-a^
2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+2/7*a*(-1/5/a/(x+1/a)^4*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)-1/15/(x+1/a)^3*(-a^2
*(x+1/a)^2+2*a*(x+1/a))^(3/2)))-35/256/a*((-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)-a/(a^2)^(1/2)*arctan((a^2)^(1/2)*
x/(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)))-5/192/a^4/(x-1/a)^3*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)-5/96/a^4/(x+1/a)
^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+15/128/a^2*(1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(3/2)+a*((-a^2*(x
-1/a)^2-2*(x-1/a)*a)^(1/2)-a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*(x-1/a)*a)^(1/2)))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (80) = 160\).
time = 0.44, size = 197, normalized size = 2.03 \begin {gather*} -\frac {5 \, a^{7} x^{7} + 5 \, a^{6} x^{6} - 15 \, a^{5} x^{5} - 15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 15 \, a^{2} x^{2} - 5 \, a x + {\left (16 \, a^{6} x^{6} + 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 30 \, a x - 5\right )} \sqrt {-a^{2} x^{2} + 1} - 5}{35 \, {\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{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)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^4,x, algorithm="fricas")

[Out]

-1/35*(5*a^7*x^7 + 5*a^6*x^6 - 15*a^5*x^5 - 15*a^4*x^4 + 15*a^3*x^3 + 15*a^2*x^2 - 5*a*x + (16*a^6*x^6 + 16*a^
5*x^5 - 40*a^4*x^4 - 40*a^3*x^3 + 30*a^2*x^2 + 30*a*x - 5)*sqrt(-a^2*x^2 + 1) - 5)/(a^8*c^4*x^7 + a^7*c^4*x^6
- 3*a^6*c^4*x^5 - 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 + 3*a^3*c^4*x^2 - a^2*c^4*x - a*c^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} - a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 1.18, size = 145, normalized size = 1.49 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {8\,x}{35\,c^4}+\frac {1}{56\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {17\,x}{70\,c^4}-\frac {1}{7\,a\,c^4}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {\sqrt {1-a^2\,x^2}}{56\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {16\,x\,\sqrt {1-a^2\,x^2}}{35\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1 - a^2*x^2)^(1/2)*((8*x)/(35*c^4) + 1/(56*a*c^4)))/((a*x - 1)^2*(a*x + 1)^2) - ((1 - a^2*x^2)^(1/2)*((17*x)
/(70*c^4) - 1/(7*a*c^4)))/((a*x - 1)^3*(a*x + 1)^3) - (1 - a^2*x^2)^(1/2)/(56*a*c^4*(a*x + 1)^4) - (16*x*(1 -
a^2*x^2)^(1/2))/(35*c^4*(a*x - 1)*(a*x + 1))

________________________________________________________________________________________