3.345 \(\int \frac{e^{\tanh ^{-1}(a x)} x^4}{(c-a c x)^3} \, dx\)

Optimal. Leaf size=135 \[ \frac{(a x+1)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{19 (a x+1)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 (a x+1)^2}{a^5 c^3 \sqrt{1-a^2 x^2}}+\frac{(a x+20) \sqrt{1-a^2 x^2}}{2 a^5 c^3}-\frac{19 \sin ^{-1}(a x)}{2 a^5 c^3} \]

[Out]

(1 + a*x)^4/(5*a^5*c^3*(1 - a^2*x^2)^(5/2)) - (19*(1 + a*x)^3)/(15*a^5*c^3*(1 - a^2*x^2)^(3/2)) + (6*(1 + a*x)
^2)/(a^5*c^3*Sqrt[1 - a^2*x^2]) + ((20 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^5*c^3) - (19*ArcSin[a*x])/(2*a^5*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.415101, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6128, 852, 1635, 780, 216} \[ \frac{(a x+1)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{19 (a x+1)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 (a x+1)^2}{a^5 c^3 \sqrt{1-a^2 x^2}}+\frac{(a x+20) \sqrt{1-a^2 x^2}}{2 a^5 c^3}-\frac{19 \sin ^{-1}(a x)}{2 a^5 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)^4/(5*a^5*c^3*(1 - a^2*x^2)^(5/2)) - (19*(1 + a*x)^3)/(15*a^5*c^3*(1 - a^2*x^2)^(3/2)) + (6*(1 + a*x)
^2)/(a^5*c^3*Sqrt[1 - a^2*x^2]) + ((20 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^5*c^3) - (19*ArcSin[a*x])/(2*a^5*c^3)

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 852

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

Rule 1635

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

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

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)^3} \, dx &=c \int \frac{x^4 \sqrt{1-a^2 x^2}}{(c-a c x)^4} \, dx\\ &=\frac{\int \frac{x^4 (c+a c x)^4}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^7}\\ &=\frac{(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{\int \frac{(c+a c x)^3 \left (\frac{4}{a^4}+\frac{5 x}{a^3}+\frac{5 x^2}{a^2}+\frac{5 x^3}{a}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^6}\\ &=\frac{(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{19 (1+a x)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{\int \frac{(c+a c x)^2 \left (\frac{45}{a^4}+\frac{30 x}{a^3}+\frac{15 x^2}{a^2}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^5}\\ &=\frac{(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{19 (1+a x)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 (1+a x)^2}{a^5 c^3 \sqrt{1-a^2 x^2}}-\frac{\int \frac{\left (\frac{135}{a^4}+\frac{15 x}{a^3}\right ) (c+a c x)}{\sqrt{1-a^2 x^2}} \, dx}{15 c^4}\\ &=\frac{(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{19 (1+a x)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 (1+a x)^2}{a^5 c^3 \sqrt{1-a^2 x^2}}+\frac{(20+a x) \sqrt{1-a^2 x^2}}{2 a^5 c^3}-\frac{19 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a^4 c^3}\\ &=\frac{(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac{19 (1+a x)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 (1+a x)^2}{a^5 c^3 \sqrt{1-a^2 x^2}}+\frac{(20+a x) \sqrt{1-a^2 x^2}}{2 a^5 c^3}-\frac{19 \sin ^{-1}(a x)}{2 a^5 c^3}\\ \end{align*}

Mathematica [C]  time = 0.135883, size = 122, normalized size = 0.9 \[ \frac{140 \sqrt{2} (a x-1) \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},\frac{1}{2} (1-a x)\right )+\sqrt{a x+1} \left (-15 a^4 x^4-75 a^3 x^3+433 a^2 x^2-639 a x+308\right )+360 (1-a x)^{5/2} \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{30 a^5 c^3 (1-a x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 + a*x]*(308 - 639*a*x + 433*a^2*x^2 - 75*a^3*x^3 - 15*a^4*x^4) + 360*(1 - a*x)^(5/2)*ArcSin[Sqrt[1 - a
*x]/Sqrt[2]] + 140*Sqrt[2]*(-1 + a*x)*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(30*a^5*c^3*(1 - a*x)^
(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.052, size = 208, normalized size = 1.5 \begin{align*}{\frac{x}{2\,{a}^{4}{c}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{19}{2\,{a}^{4}{c}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+4\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{c}^{3}{a}^{5}}}-{\frac{2}{5\,{c}^{3}{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{41}{15\,{c}^{3}{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{199}{15\,{a}^{6}{c}^{3}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \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)^3,x)

[Out]

1/2/c^3/a^4*x*(-a^2*x^2+1)^(1/2)-19/2/c^3/a^4/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+4/c^3/a^5*(
-a^2*x^2+1)^(1/2)-2/5/c^3/a^8/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-41/15/c^3/a^7/(x-1/a)^2*(-a^2*(x-1/
a)^2-2*a*(x-1/a))^(1/2)-199/15/c^3/a^6/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)

________________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.69128, size = 355, normalized size = 2.63 \begin{align*} \frac{448 \, a^{3} x^{3} - 1344 \, a^{2} x^{2} + 1344 \, a x + 570 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (15 \, a^{4} x^{4} + 75 \, a^{3} x^{3} - 713 \, a^{2} x^{2} + 1059 \, a x - 448\right )} \sqrt{-a^{2} x^{2} + 1} - 448}{30 \,{\left (a^{8} c^{3} x^{3} - 3 \, a^{7} c^{3} x^{2} + 3 \, a^{6} c^{3} x - a^{5} c^{3}\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)^3,x, algorithm="fricas")

[Out]

1/30*(448*a^3*x^3 - 1344*a^2*x^2 + 1344*a*x + 570*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1)
- 1)/(a*x)) + (15*a^4*x^4 + 75*a^3*x^3 - 713*a^2*x^2 + 1059*a*x - 448)*sqrt(-a^2*x^2 + 1) - 448)/(a^8*c^3*x^3
- 3*a^7*c^3*x^2 + 3*a^6*c^3*x - a^5*c^3)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{x^{4}}{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{a x^{5}}{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}} \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)**3,x)

[Out]

-(Integral(x**4/(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(a*x**5/(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

________________________________________________________________________________________

Giac [A]  time = 1.24176, size = 267, normalized size = 1.98 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{x}{a^{4} c^{3}} + \frac{8}{a^{5} c^{3}}\right )} - \frac{19 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \, a^{4} c^{3}{\left | a \right |}} - \frac{2 \,{\left (\frac{685 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{1025 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{615 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{135 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 164\right )}}{15 \, a^{4} c^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{5}{\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)^3,x, algorithm="giac")

[Out]

1/2*sqrt(-a^2*x^2 + 1)*(x/(a^4*c^3) + 8/(a^5*c^3)) - 19/2*arcsin(a*x)*sgn(a)/(a^4*c^3*abs(a)) - 2/15*(685*(sqr
t(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1025*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 615*(sqrt(-a^2*x^2 +
1)*abs(a) + a)^3/(a^6*x^3) - 135*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) - 164)/(a^4*c^3*((sqrt(-a^2*x^2 +
1)*abs(a) + a)/(a^2*x) - 1)^5*abs(a))