3.164 \(\int \frac {e^{\tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\)

Optimal. Leaf size=65 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{15 a c^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4} \]

[Out]

1/5*(-a^2*x^2+1)^(3/2)/a/c^3/(-a*x+1)^4+1/15*(-a^2*x^2+1)^(3/2)/a/c^3/(-a*x+1)^3

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Rubi [A]  time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6127, 659, 651} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{15 a c^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - a*c*x)^3,x]

[Out]

(1 - a^2*x^2)^(3/2)/(5*a*c^3*(1 - a*x)^4) + (1 - a^2*x^2)^(3/2)/(15*a*c^3*(1 - a*x)^3)

Rule 651

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*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 659

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 6127

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^{\tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4}+\frac {1}{5} \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{15 a c^3 (1-a x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 0.54 \[ \frac {(4-a x) (a x+1)^{3/2}}{15 a c^3 (1-a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((4 - a*x)*(1 + a*x)^(3/2))/(15*a*c^3*(1 - a*x)^(5/2))

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fricas [A]  time = 0.55, size = 89, normalized size = 1.37 \[ \frac {4 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 12 \, a x + {\left (a^{2} x^{2} - 3 \, a x - 4\right )} \sqrt {-a^{2} x^{2} + 1} - 4}{15 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(4*a^3*x^3 - 12*a^2*x^2 + 12*a*x + (a^2*x^2 - 3*a*x - 4)*sqrt(-a^2*x^2 + 1) - 4)/(a^4*c^3*x^3 - 3*a^3*c^3
*x^2 + 3*a^2*c^3*x - a*c^3)

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giac [B]  time = 0.36, size = 145, normalized size = 2.23 \[ -\frac {2 \, {\left (\frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {25 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 4\right )}}{15 \, c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(5*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 25*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 15*(sqrt(-
a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) - 4)/(c^3*((sqrt(-a^2*x^
2 + 1)*abs(a) + a)/(a^2*x) - 1)^5*abs(a))

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maple [A]  time = 0.03, size = 40, normalized size = 0.62 \[ -\frac {\left (a x -4\right ) \left (a x +1\right )^{2}}{15 \left (a x -1\right )^{2} c^{3} \sqrt {-a^{2} x^{2}+1}\, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(a*x-4)*(a*x+1)^2/(a*x-1)^2/c^3/(-a^2*x^2+1)^(1/2)/a

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maxima [B]  time = 0.44, size = 126, normalized size = 1.94 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{2} c^{3} x - a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*sqrt(-a^2*x^2 + 1)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3) - 1/15*sqrt(-a^2*x^2 + 1)/(a^3*c^3
*x^2 - 2*a^2*c^3*x + a*c^3) + 1/15*sqrt(-a^2*x^2 + 1)/(a^2*c^3*x - a*c^3)

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mupad [B]  time = 0.83, size = 183, normalized size = 2.82 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}-\frac {\sqrt {1-a^2\,x^2}}{15\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(3*c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^(1/2
) - 3*a*c^3*x^2*(-a^2)^(1/2))) - (1 - a^2*x^2)^(1/2)/(15*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))
/a)) - (a*(1 - a^2*x^2)^(1/2))/(15*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {a x}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(a*x/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1)
- sqrt(-a**2*x**2 + 1)), x) + Integral(1/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) +
3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3

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