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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.40, antiderivative size = 168, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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 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]

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 = 0.09, size = 95, normalized size = 0.57 \[ -\frac {\sqrt {a x+1} \left (105 a^4 x^4-44 a^3 x^3-244 a^2 x^2+29 a x+124\right )-700 \sqrt {2} (a x-1)^2 \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {1}{2} (1-a x)\right )}{105 a^5 c^4 (1-a x)^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/105*(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*Hyper
geometric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(a^5*c^4*(1 - a*x)^(7/2))

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 179, normalized size = 1.07 \[ -\frac {824 \, a^{4} x^{4} - 3296 \, a^{3} x^{3} + 4944 \, a^{2} x^{2} - 3296 \, a x + 1050 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (105 \, a^{4} x^{4} - 1444 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 2771 \, a x + 824\right )} \sqrt {-a^{2} x^{2} + 1} + 824}{105 \, {\left (a^{9} c^{4} x^{4} - 4 \, a^{8} c^{4} x^{3} + 6 \, a^{7} c^{4} x^{2} - 4 \, a^{6} c^{4} x + a^{5} c^{4}\right )}} \]

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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.05, size = 231, normalized size = 1.38 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{c^{4} a^{5}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{4} a^{4} \sqrt {a^{2}}}+\frac {446 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 c^{4} a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {1024 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 c^{4} a^{6} \left (x -\frac {1}{a}\right )}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 c^{4} a^{9} \left (x -\frac {1}{a}\right )^{4}}+\frac {57 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{35 c^{4} a^{8} \left (x -\frac {1}{a}\right )^{3}} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 229, normalized size = 1.36 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{9} c^{4} x^{4} - 4 \, a^{8} c^{4} x^{3} + 6 \, a^{7} c^{4} x^{2} - 4 \, a^{6} c^{4} x + a^{5} c^{4}\right )}} + \frac {57 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{8} c^{4} x^{3} - 3 \, a^{7} c^{4} x^{2} + 3 \, a^{6} c^{4} x - a^{5} c^{4}\right )}} + \frac {446 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{7} c^{4} x^{2} - 2 \, a^{6} c^{4} x + a^{5} c^{4}\right )}} + \frac {1024 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{6} c^{4} x - a^{5} c^{4}\right )}} + \frac {5 \, \arcsin \left (a x\right )}{a^{5} c^{4}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{5} c^{4}} \]

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]

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

________________________________________________________________________________________

mupad [B]  time = 0.84, size = 350, normalized size = 2.08 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^9\,c^4\,x^4-4\,a^8\,c^4\,x^3+6\,a^7\,c^4\,x^2-4\,a^6\,c^4\,x+a^5\,c^4\right )}+\frac {572\,\sqrt {1-a^2\,x^2}}{105\,\left (a^7\,c^4\,x^2-2\,a^6\,c^4\,x+a^5\,c^4\right )}+\frac {57\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a^3\,c^4\,\sqrt {-a^2}+3\,a^5\,c^4\,x^2\,\sqrt {-a^2}-a^6\,c^4\,x^3\,\sqrt {-a^2}-3\,a^4\,c^4\,x\,\sqrt {-a^2}\right )}-\frac {6\,a^4\,\sqrt {1-a^2\,x^2}}{5\,\left (a^{11}\,c^4\,x^2-2\,a^{10}\,c^4\,x+a^9\,c^4\right )}+\frac {1024\,\sqrt {1-a^2\,x^2}}{105\,\left (a^3\,c^4\,\sqrt {-a^2}-a^4\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^5\,c^4}+\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^4\,c^4\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(1 - a^2*x^2)^(1/2))/(7*(a^5*c^4 - 4*a^6*c^4*x + 6*a^7*c^4*x^2 - 4*a^8*c^4*x^3 + a^9*c^4*x^4)) + (572*(1 -
a^2*x^2)^(1/2))/(105*(a^5*c^4 - 2*a^6*c^4*x + a^7*c^4*x^2)) + (57*(1 - a^2*x^2)^(1/2))/(35*(-a^2)^(1/2)*(a^3*c
^4*(-a^2)^(1/2) + 3*a^5*c^4*x^2*(-a^2)^(1/2) - a^6*c^4*x^3*(-a^2)^(1/2) - 3*a^4*c^4*x*(-a^2)^(1/2))) - (6*a^4*
(1 - a^2*x^2)^(1/2))/(5*(a^9*c^4 - 2*a^10*c^4*x + a^11*c^4*x^2)) + (1024*(1 - a^2*x^2)^(1/2))/(105*(a^3*c^4*(-
a^2)^(1/2) - a^4*c^4*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a^5*c^4) + (5*asinh(x*(-a^2)^(1/2)))
/(a^4*c^4*(-a^2)^(1/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]

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

________________________________________________________________________________________