3.3.25 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^6} \, dx\) [225]

Optimal. Leaf size=119 \[ \frac {8 x}{35 c^6 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x) \sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6262, 673, 197} \begin {gather*} \frac {8 x}{35 c^6 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x) \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^6),x]

[Out]

(8*x)/(35*c^6*Sqrt[1 - a^2*x^2]) + 1/(7*a*c^6*(1 - a*x)^3*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^6*(1 - a*x)^2*Sqrt[1
- a^2*x^2]) + 4/(35*a*c^6*(1 - a*x)*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 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*c*d*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rule 6262

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

Rubi steps

\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^6} \, dx &=\frac {\int \frac {1}{(c-a c x)^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4 \int \frac {1}{(c-a c x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{7 c^4}\\ &=\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {12 \int \frac {1}{(c-a c x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^5}\\ &=\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x) \sqrt {1-a^2 x^2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^6}\\ &=\frac {8 x}{35 c^6 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^6 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^6 (1-a x) \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 61, normalized size = 0.51 \begin {gather*} \frac {-13+4 a x+20 a^2 x^2-24 a^3 x^3+8 a^4 x^4}{35 a c^6 (-1+a x)^3 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^6),x]

[Out]

(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4)/(35*a*c^6*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 1.10, size = 1373, normalized size = 11.54

method	result	size
gosper	\(\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right )}{35 \left (a x -1\right )^{5} c^{6} a \left (a x +1\right )^{2}}\)	\(65\)
trager	\(-\frac {\left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \sqrt {-a^{2} x^{2}+1}}{35 c^{6} \left (a x -1\right )^{4} a \left (a x +1\right )}\)	\(65\)
default	\(\text {Expression too large to display}\)	\(1373\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^6,x,method=_RETURNVERBOSE)

[Out]

1/c^6*(-3/80/a^6/(x-1/a)^5*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+15/128/a^2*(-1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(
x-1/a))^(5/2)-3*a*(1/3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a
*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))-5/32/a^3*(1/a/(x-1
/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+2*a*(-1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)-3*a*(1/3*(-a^2
*(x-1/a)^2-2*a*(x-1/a))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1
/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))))))+21/256/a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3
/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(
-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))))+1/64/a^3*(-1/a/(x+1/a)^3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-2*a*(1/a/(x+1
/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2
*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(
1/2))))))+3/64/a^2*(1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/
2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-
a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))))+3/16/a^4*(1/3/a/(x-1/a)^4*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+1/3*a*(1/a/(
x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)+2*a*(-1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2)-3*a*(1/3*(-
a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)
^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))))+1/8/a^6*(1/7/a/(x-1/a)^6*(-a^2*(x-1/a)^2-
2*a*(x-1/a))^(5/2)-1/35/(x-1/a)^5*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(5/2))-21/256/a*(1/3*(-a^2*(x-1/a)^2-2*a*(x-1/a
))^(3/2)-a*(-1/4*(-2*a^2*(x-1/a)-2*a)/a^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1/2
)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))))

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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)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^6,x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)/((a*c*x - c)^6*(a*x + 1)^3), x)

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Fricas [A]
time = 0.38, size = 144, normalized size = 1.21 \begin {gather*} \frac {13 \, a^{5} x^{5} - 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 39 \, a x - {\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {-a^{2} x^{2} + 1} + 13}{35 \, {\left (a^{6} c^{6} x^{5} - 3 \, a^{5} c^{6} x^{4} + 2 \, a^{4} c^{6} x^{3} + 2 \, a^{3} c^{6} x^{2} - 3 \, a^{2} c^{6} x + a c^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^6,x, algorithm="fricas")

[Out]

1/35*(13*a^5*x^5 - 39*a^4*x^4 + 26*a^3*x^3 + 26*a^2*x^2 - 39*a*x - (8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*
x - 13)*sqrt(-a^2*x^2 + 1) + 13)/(a^6*c^6*x^5 - 3*a^5*c^6*x^4 + 2*a^4*c^6*x^3 + 2*a^3*c^6*x^2 - 3*a^2*c^6*x +
a*c^6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{9} x^{9} - 3 a^{8} x^{8} + 8 a^{6} x^{6} - 6 a^{5} x^{5} - 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x + 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{9} x^{9} - 3 a^{8} x^{8} + 8 a^{6} x^{6} - 6 a^{5} x^{5} - 6 a^{4} x^{4} + 8 a^{3} x^{3} - 3 a x + 1}\right )\, dx}{c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**6,x)

[Out]

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

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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(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^6,x, algorithm="giac")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)/((a*c*x - c)^6*(a*x + 1)^3), x)

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Mupad [B]
time = 0.82, size = 347, normalized size = 2.92 \begin {gather*} \frac {3\,a\,\sqrt {1-a^2\,x^2}}{40\,\left (a^4\,c^6\,x^2-2\,a^3\,c^6\,x+a^2\,c^6\right )}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^6\,x^2-2\,a^5\,c^6\,x+a^4\,c^6\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{14\,\left (a^6\,c^6\,x^4-4\,a^5\,c^6\,x^3+6\,a^4\,c^6\,x^2-4\,a^3\,c^6\,x+a^2\,c^6\right )}+\frac {\sqrt {1-a^2\,x^2}}{16\,\sqrt {-a^2}\,\left (c^6\,x\,\sqrt {-a^2}+\frac {c^6\,\sqrt {-a^2}}{a}\right )}+\frac {93\,\sqrt {1-a^2\,x^2}}{560\,\sqrt {-a^2}\,\left (c^6\,x\,\sqrt {-a^2}-\frac {c^6\,\sqrt {-a^2}}{a}\right )}+\frac {13\,\sqrt {1-a^2\,x^2}}{140\,\sqrt {-a^2}\,\left (3\,c^6\,x\,\sqrt {-a^2}-\frac {c^6\,\sqrt {-a^2}}{a}+a^2\,c^6\,x^3\,\sqrt {-a^2}-3\,a\,c^6\,x^2\,\sqrt {-a^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - a^2*x^2)^(3/2)/((c - a*c*x)^6*(a*x + 1)^3),x)

[Out]

(3*a*(1 - a^2*x^2)^(1/2))/(40*(a^2*c^6 - 2*a^3*c^6*x + a^4*c^6*x^2)) + (a^3*(1 - a^2*x^2)^(1/2))/(35*(a^4*c^6
- 2*a^5*c^6*x + a^6*c^6*x^2)) + (a*(1 - a^2*x^2)^(1/2))/(14*(a^2*c^6 - 4*a^3*c^6*x + 6*a^4*c^6*x^2 - 4*a^5*c^6
*x^3 + a^6*c^6*x^4)) + (1 - a^2*x^2)^(1/2)/(16*(-a^2)^(1/2)*(c^6*x*(-a^2)^(1/2) + (c^6*(-a^2)^(1/2))/a)) + (93
*(1 - a^2*x^2)^(1/2))/(560*(-a^2)^(1/2)*(c^6*x*(-a^2)^(1/2) - (c^6*(-a^2)^(1/2))/a)) + (13*(1 - a^2*x^2)^(1/2)
)/(140*(-a^2)^(1/2)*(3*c^6*x*(-a^2)^(1/2) - (c^6*(-a^2)^(1/2))/a + a^2*c^6*x^3*(-a^2)^(1/2) - 3*a*c^6*x^2*(-a^
2)^(1/2)))

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