3.3.23 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx\) [223]

Optimal. Leaf size=55 \[ \frac {2 x}{3 c^4 \sqrt {1-a^2 x^2}}+\frac {1}{3 a c^4 (1-a x) \sqrt {1-a^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {2 x}{3 c^4 \sqrt {1-a^2 x^2}}+\frac {1}{3 a c^4 (1-a x) \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/(3*c^4*Sqrt[1 - a^2*x^2]) + 1/(3*a*c^4*(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)^4} \, dx &=\frac {\int \frac {1}{(c-a c x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {1}{3 a c^4 (1-a x) \sqrt {1-a^2 x^2}}+\frac {2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^4}\\ &=\frac {2 x}{3 c^4 \sqrt {1-a^2 x^2}}+\frac {1}{3 a c^4 (1-a x) \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 45, normalized size = 0.82 \begin {gather*} \frac {-1-2 a x+2 a^2 x^2}{3 a c^4 (-1+a x) \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 - 2*a*x + 2*a^2*x^2)/(3*a*c^4*(-1 + a*x)*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 = 1.14, size = 1247, normalized size = 22.67

method	result	size
gosper	\(\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (2 a^{2} x^{2}-2 a x -1\right )}{3 \left (a x +1\right )^{2} c^{4} \left (a x -1\right )^{3} a}\)	\(49\)
trager	\(-\frac {\left (2 a^{2} x^{2}-2 a x -1\right ) \sqrt {-a^{2} x^{2}+1}}{3 c^{4} \left (a x -1\right )^{2} a \left (a x +1\right )}\)	\(49\)
default	\(\text {Expression too large to display}\)	\(1247\)

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)^4,x,method=_RETURNVERBOSE)

[Out]

1/c^4*(3/16/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^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))))))+5/32/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))))+1/16/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))))))+1/8/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)))))+1/8/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
)))))))-5/32/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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.40, size = 89, normalized size = 1.62 \begin {gather*} \frac {a^{3} x^{3} - a^{2} x^{2} - a x - {\left (2 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} + 1}{3 \, {\left (a^{4} c^{4} x^{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)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

1/3*(a^3*x^3 - a^2*x^2 - a*x - (2*a^2*x^2 - 2*a*x - 1)*sqrt(-a^2*x^2 + 1) + 1)/(a^4*c^4*x^3 - a^3*c^4*x^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 \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\right )\, dx}{c^{4}} \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)**4,x)

[Out]

(Integral(sqrt(-a**2*x**2 + 1)/(a**7*x**7 - a**6*x**6 - 3*a**5*x**5 + 3*a**4*x**4 + 3*a**3*x**3 - 3*a**2*x**2
- a*x + 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**7*x**7 - a**6*x**6 - 3*a**5*x**5 + 3*a**4*x**4 +
3*a**3*x**3 - 3*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(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.83, size = 48, normalized size = 0.87 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (-2\,a^2\,x^2+2\,a\,x+1\right )}{3\,a\,c^4\,{\left (a\,x-1\right )}^2\,\left (a\,x+1\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)^4*(a*x + 1)^3),x)

[Out]

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

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