3.355 \(\int \frac{e^{\tanh ^{-1}(a x)} x^4}{(c-a c x)^4} \, dx\)

Optimal. Leaf size=168 \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^2}+\frac{184 \left (1-a^2 x^2\right )^{3/2}}{105 a^5 c^4 (1-a x)^3}-\frac{26 \left (1-a^2 x^2\right )^{3/2}}{35 a^5 c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^5 c^4 (1-a x)^5}-\frac{10 \sqrt{1-a^2 x^2}}{a^5 c^4 (1-a x)}+\frac{5 \sin ^{-1}(a x)}{a^5 c^4} \]

[Out]

(-10*Sqrt[1 - a^2*x^2])/(a^5*c^4*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(7*a^5*c^4*(1 - a*x)^5) - (26*(1 - a^2*x^2)^
(3/2))/(35*a^5*c^4*(1 - a*x)^4) + (184*(1 - a^2*x^2)^(3/2))/(105*a^5*c^4*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)/(a
^5*c^4*(1 - a*x)^2) + (5*ArcSin[a*x])/(a^5*c^4)

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Rubi [A]  time = 0.402663, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6128, 1639, 1637, 659, 651, 663, 216} \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^2}+\frac{184 \left (1-a^2 x^2\right )^{3/2}}{105 a^5 c^4 (1-a x)^3}-\frac{26 \left (1-a^2 x^2\right )^{3/2}}{35 a^5 c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^5 c^4 (1-a x)^5}-\frac{10 \sqrt{1-a^2 x^2}}{a^5 c^4 (1-a x)}+\frac{5 \sin ^{-1}(a x)}{a^5 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - a^2*x^2])/(a^5*c^4*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(7*a^5*c^4*(1 - a*x)^5) - (26*(1 - a^2*x^2)^
(3/2))/(35*a^5*c^4*(1 - a*x)^4) + (184*(1 - a^2*x^2)^(3/2))/(105*a^5*c^4*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)/(a
^5*c^4*(1 - a*x)^2) + (5*ArcSin[a*x])/(a^5*c^4)

Rule 6128

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

Rule 1639

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1637

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^2)^p,
(d + e*x)^m*Pq, x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[m + Expon[Pq
, x] + 2*p + 1, 0] && ILtQ[m, 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 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 663

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^4}{(c-a c x)^4} \, dx &=c \int \frac{x^4 \sqrt{1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^2}-\frac{\int \frac{\sqrt{1-a^2 x^2} \left (2 a^2 c^4-7 a^3 c^4 x+9 a^4 c^4 x^2-5 a^5 c^4 x^3\right )}{(c-a c x)^5} \, dx}{a^6 c^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^2}-\frac{\int \left (\frac{a^2 \sqrt{1-a^2 x^2}}{c (-1+a x)^5}+\frac{4 a^2 \sqrt{1-a^2 x^2}}{c (-1+a x)^4}+\frac{6 a^2 \sqrt{1-a^2 x^2}}{c (-1+a x)^3}+\frac{5 a^2 \sqrt{1-a^2 x^2}}{c (-1+a x)^2}\right ) \, dx}{a^6 c^3}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^2}-\frac{\int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^5} \, dx}{a^4 c^4}-\frac{4 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^4} \, dx}{a^4 c^4}-\frac{5 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^2} \, dx}{a^4 c^4}-\frac{6 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{a^4 c^4}\\ &=-\frac{10 \sqrt{1-a^2 x^2}}{a^5 c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^5 c^4 (1-a x)^5}-\frac{4 \left (1-a^2 x^2\right )^{3/2}}{5 a^5 c^4 (1-a x)^4}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^2}+\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^4} \, dx}{7 a^4 c^4}+\frac{4 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 a^4 c^4}+\frac{5 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^4 c^4}\\ &=-\frac{10 \sqrt{1-a^2 x^2}}{a^5 c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^5 c^4 (1-a x)^5}-\frac{26 \left (1-a^2 x^2\right )^{3/2}}{35 a^5 c^4 (1-a x)^4}+\frac{26 \left (1-a^2 x^2\right )^{3/2}}{15 a^5 c^4 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^2}+\frac{5 \sin ^{-1}(a x)}{a^5 c^4}-\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{35 a^4 c^4}\\ &=-\frac{10 \sqrt{1-a^2 x^2}}{a^5 c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^5 c^4 (1-a x)^5}-\frac{26 \left (1-a^2 x^2\right )^{3/2}}{35 a^5 c^4 (1-a x)^4}+\frac{184 \left (1-a^2 x^2\right )^{3/2}}{105 a^5 c^4 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^5 c^4 (1-a x)^2}+\frac{5 \sin ^{-1}(a x)}{a^5 c^4}\\ \end{align*}

Mathematica [C]  time = 0.0867586, size = 95, normalized size = 0.57 \[ -\frac{\sqrt{a x+1} \left (105 a^4 x^4-44 a^3 x^3-244 a^2 x^2+29 a x+124\right )-700 \sqrt{2} (a x-1)^2 \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},\frac{1}{2} (1-a x)\right )}{105 a^5 c^4 (1-a x)^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[1 + a*x]*(124 + 29*a*x - 244*a^2*x^2 - 44*a^3*x^3 + 105*a^4*x^4) - 700*Sqrt[2]*(-1 + a*x)^2*Hypergeomet
ric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(105*a^5*c^4*(1 - a*x)^(7/2))

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Maple [A]  time = 0.056, size = 231, normalized size = 1.4 \begin{align*} -{\frac{1}{{c}^{4}{a}^{5}}\sqrt{-{a}^{2}{x}^{2}+1}}+5\,{\frac{1}{{c}^{4}{a}^{4}\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{57}{35\,{c}^{4}{a}^{8}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}+{\frac{446}{105\,{c}^{4}{a}^{7}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{1024}{105\,{a}^{6}{c}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}+{\frac{2}{7\,{c}^{4}{a}^{9}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c^4/a^5*(-a^2*x^2+1)^(1/2)+5/c^4/a^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+57/35/c^4/a^8/(x-
1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+446/105/c^4/a^7/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1024/10
5/c^4/a^6/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+2/7/c^4/a^9/(x-1/a)^4*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59119, size = 423, normalized size = 2.52 \begin{align*} -\frac{824 \, a^{4} x^{4} - 3296 \, a^{3} x^{3} + 4944 \, a^{2} x^{2} - 3296 \, a x + 1050 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (105 \, a^{4} x^{4} - 1444 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 2771 \, a x + 824\right )} \sqrt{-a^{2} x^{2} + 1} + 824}{105 \,{\left (a^{9} c^{4} x^{4} - 4 \, a^{8} c^{4} x^{3} + 6 \, a^{7} c^{4} x^{2} - 4 \, a^{6} c^{4} x + a^{5} c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(824*a^4*x^4 - 3296*a^3*x^3 + 4944*a^2*x^2 - 3296*a*x + 1050*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x +
1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (105*a^4*x^4 - 1444*a^3*x^3 + 3256*a^2*x^2 - 2771*a*x + 824)*sqrt
(-a^2*x^2 + 1) + 824)/(a^9*c^4*x^4 - 4*a^8*c^4*x^3 + 6*a^7*c^4*x^2 - 4*a^6*c^4*x + a^5*c^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.199, size = 325, normalized size = 1.93 \begin{align*} \frac{5 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a^{4} c^{4}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{5} c^{4}} + \frac{2 \,{\left (\frac{4508 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{11529 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{15050 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{10115 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{3570 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{525 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 719\right )}}{105 \, a^{4} c^{4}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{7}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*arcsin(a*x)*sgn(a)/(a^4*c^4*abs(a)) - sqrt(-a^2*x^2 + 1)/(a^5*c^4) + 2/105*(4508*(sqrt(-a^2*x^2 + 1)*abs(a)
+ a)/(a^2*x) - 11529*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 15050*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^
6*x^3) - 10115*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 3570*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5)
- 525*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) - 719)/(a^4*c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) -
1)^7*abs(a))