3.166 \(\int \frac {e^{\tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx\)

Optimal. Leaf size=129 \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6} \]

[Out]

1/9*(-a^2*x^2+1)^(3/2)/a/c^5/(-a*x+1)^6+1/21*(-a^2*x^2+1)^(3/2)/a/c^5/(-a*x+1)^5+2/105*(-a^2*x^2+1)^(3/2)/a/c^
5/(-a*x+1)^4+2/315*(-a^2*x^2+1)^(3/2)/a/c^5/(-a*x+1)^3

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Rubi [A]  time = 0.09, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6127, 659, 651} \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(c - a*c*x)^5,x]

[Out]

(1 - a^2*x^2)^(3/2)/(9*a*c^5*(1 - a*x)^6) + (1 - a^2*x^2)^(3/2)/(21*a*c^5*(1 - a*x)^5) + (2*(1 - a^2*x^2)^(3/2
))/(105*a*c^5*(1 - a*x)^4) + (2*(1 - a^2*x^2)^(3/2))/(315*a*c^5*(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)^5} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^6} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {1}{3} \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx}{21 c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx}{105 c^2}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 51, normalized size = 0.40 \[ \frac {(a x+1)^{3/2} \left (-2 a^3 x^3+12 a^2 x^2-33 a x+58\right )}{315 a c^5 (1-a x)^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]/(c - a*c*x)^5,x]

[Out]

((1 + a*x)^(3/2)*(58 - 33*a*x + 12*a^2*x^2 - 2*a^3*x^3))/(315*a*c^5*(1 - a*x)^(9/2))

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fricas [A]  time = 0.76, size = 144, normalized size = 1.12 \[ \frac {58 \, a^{5} x^{5} - 290 \, a^{4} x^{4} + 580 \, a^{3} x^{3} - 580 \, a^{2} x^{2} + 290 \, a x + {\left (2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 25 \, a x - 58\right )} \sqrt {-a^{2} x^{2} + 1} - 58}{315 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

1/315*(58*a^5*x^5 - 290*a^4*x^4 + 580*a^3*x^3 - 580*a^2*x^2 + 290*a*x + (2*a^4*x^4 - 10*a^3*x^3 + 21*a^2*x^2 -
25*a*x - 58)*sqrt(-a^2*x^2 + 1) - 58)/(a^6*c^5*x^5 - 5*a^5*c^5*x^4 + 10*a^4*c^5*x^3 - 10*a^3*c^5*x^2 + 5*a^2*
c^5*x - a*c^5)

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giac [C]  time = 0.26, size = 321, normalized size = 2.49 \[ \frac {-\frac {16 i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}{c^{3}} - \frac {\frac {35 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 180 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 378 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 420 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{c^{3}} + \frac {9 \, {\left (5 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 21 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 35 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} - 35 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}}{2520 \, c^{2} {\left | a \right |}} \]

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)^5,x, algorithm="giac")

[Out]

1/2520*(-16*I*sgn(1/(a*c*x - c))*sgn(a)*sgn(c)/c^3 - ((35*(2*c/(a*c*x - c) + 1)^4*sqrt(-2*c/(a*c*x - c) - 1) -
180*(2*c/(a*c*x - c) + 1)^3*sqrt(-2*c/(a*c*x - c) - 1) + 378*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) -
1) + 420*(-2*c/(a*c*x - c) - 1)^(3/2) + 315*sqrt(-2*c/(a*c*x - c) - 1))/c^3 + 9*(5*(2*c/(a*c*x - c) + 1)^3*sqr
t(-2*c/(a*c*x - c) - 1) - 21*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) - 1) - 35*(-2*c/(a*c*x - c) - 1)^(3
/2) - 35*sqrt(-2*c/(a*c*x - c) - 1))/c^3)/(sgn(1/(a*c*x - c))*sgn(a)*sgn(c)))/(c^2*abs(a))

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maple [A]  time = 0.03, size = 57, normalized size = 0.44 \[ -\frac {\left (2 x^{3} a^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )^{2}}{315 \left (a x -1\right )^{4} c^{5} \sqrt {-a^{2} x^{2}+1}\, a} \]

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)^5,x)

[Out]

-1/315*(2*a^3*x^3-12*a^2*x^2+33*a*x-58)*(a*x+1)^2/(a*x-1)^4/c^5/(-a^2*x^2+1)^(1/2)/a

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maxima [B]  time = 0.43, size = 264, normalized size = 2.05 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{9 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{63 \, {\left (a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 6 \, a^{3} c^{5} x^{2} - 4 \, a^{2} c^{5} x + a c^{5}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{315 \, {\left (a^{3} c^{5} x^{2} - 2 \, a^{2} c^{5} x + a c^{5}\right )}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{315 \, {\left (a^{2} c^{5} x - a c^{5}\right )}} \]

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)^5,x, algorithm="maxima")

[Out]

-2/9*sqrt(-a^2*x^2 + 1)/(a^6*c^5*x^5 - 5*a^5*c^5*x^4 + 10*a^4*c^5*x^3 - 10*a^3*c^5*x^2 + 5*a^2*c^5*x - a*c^5)
- 1/63*sqrt(-a^2*x^2 + 1)/(a^5*c^5*x^4 - 4*a^4*c^5*x^3 + 6*a^3*c^5*x^2 - 4*a^2*c^5*x + a*c^5) + 1/105*sqrt(-a^
2*x^2 + 1)/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5) - 2/315*sqrt(-a^2*x^2 + 1)/(a^3*c^5*x^2 - 2*a^2
*c^5*x + a*c^5) + 2/315*sqrt(-a^2*x^2 + 1)/(a^2*c^5*x - a*c^5)

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mupad [B]  time = 0.87, size = 57, normalized size = 0.44 \[ -\frac {\sqrt {1-a^2\,x^2}\,\left (-2\,a^4\,x^4+10\,a^3\,x^3-21\,a^2\,x^2+25\,a\,x+58\right )}{315\,a\,c^5\,{\left (a\,x-1\right )}^5} \]

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)^5),x)

[Out]

-((1 - a^2*x^2)^(1/2)*(25*a*x - 21*a^2*x^2 + 10*a^3*x^3 - 2*a^4*x^4 + 58))/(315*a*c^5*(a*x - 1)^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {a x}{a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \]

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)**5,x)

[Out]

-(Integral(a*x/(a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 10*a**3*x**3*sqrt(-a**2*x*
*2 + 1) - 10*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integra
l(1/(a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 10*a**3*x**3*sqrt(-a**2*x**2 + 1) - 1
0*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**5

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