3.354 \(\int \frac{e^{\tanh ^{-1}(a x)} x^5}{(c-a c x)^4} \, dx\)

Optimal. Leaf size=166 \[ \frac{(a x+1)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{33 (a x+1)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{317 (a x+1)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{10 (a x+1)^2}{a^6 c^4 \sqrt{1-a^2 x^2}}-\frac{(a x+30) \sqrt{1-a^2 x^2}}{2 a^6 c^4}+\frac{29 \sin ^{-1}(a x)}{2 a^6 c^4} \]

[Out]

(1 + a*x)^5/(7*a^6*c^4*(1 - a^2*x^2)^(7/2)) - (33*(1 + a*x)^4)/(35*a^6*c^4*(1 - a^2*x^2)^(5/2)) + (317*(1 + a*
x)^3)/(105*a^6*c^4*(1 - a^2*x^2)^(3/2)) - (10*(1 + a*x)^2)/(a^6*c^4*Sqrt[1 - a^2*x^2]) - ((30 + a*x)*Sqrt[1 -
a^2*x^2])/(2*a^6*c^4) + (29*ArcSin[a*x])/(2*a^6*c^4)

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Rubi [A]  time = 0.533396, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{(a x+1)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{33 (a x+1)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{317 (a x+1)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{10 (a x+1)^2}{a^6 c^4 \sqrt{1-a^2 x^2}}-\frac{(a x+30) \sqrt{1-a^2 x^2}}{2 a^6 c^4}+\frac{29 \sin ^{-1}(a x)}{2 a^6 c^4} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^5)/(c - a*c*x)^4,x]

[Out]

(1 + a*x)^5/(7*a^6*c^4*(1 - a^2*x^2)^(7/2)) - (33*(1 + a*x)^4)/(35*a^6*c^4*(1 - a^2*x^2)^(5/2)) + (317*(1 + a*
x)^3)/(105*a^6*c^4*(1 - a^2*x^2)^(3/2)) - (10*(1 + a*x)^2)/(a^6*c^4*Sqrt[1 - a^2*x^2]) - ((30 + a*x)*Sqrt[1 -
a^2*x^2])/(2*a^6*c^4) + (29*ArcSin[a*x])/(2*a^6*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 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^5}{(c-a c x)^4} \, dx &=c \int \frac{x^5 \sqrt{1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac{\int \frac{x^5 (c+a c x)^5}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^9}\\ &=\frac{(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{\int \frac{(c+a c x)^4 \left (\frac{5}{a^5}+\frac{7 x}{a^4}+\frac{7 x^2}{a^3}+\frac{7 x^3}{a^2}+\frac{7 x^4}{a}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^8}\\ &=\frac{(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{\int \frac{(c+a c x)^3 \left (\frac{107}{a^5}+\frac{105 x}{a^4}+\frac{70 x^2}{a^3}+\frac{35 x^3}{a^2}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^7}\\ &=\frac{(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{317 (1+a x)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{(c+a c x)^2 \left (\frac{630}{a^5}+\frac{315 x}{a^4}+\frac{105 x^2}{a^3}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^6}\\ &=\frac{(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{317 (1+a x)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{10 (1+a x)^2}{a^6 c^4 \sqrt{1-a^2 x^2}}+\frac{\int \frac{\left (\frac{1470}{a^5}+\frac{105 x}{a^4}\right ) (c+a c x)}{\sqrt{1-a^2 x^2}} \, dx}{105 c^5}\\ &=\frac{(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{317 (1+a x)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{10 (1+a x)^2}{a^6 c^4 \sqrt{1-a^2 x^2}}-\frac{(30+a x) \sqrt{1-a^2 x^2}}{2 a^6 c^4}+\frac{29 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a^5 c^4}\\ &=\frac{(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{317 (1+a x)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{10 (1+a x)^2}{a^6 c^4 \sqrt{1-a^2 x^2}}-\frac{(30+a x) \sqrt{1-a^2 x^2}}{2 a^6 c^4}+\frac{29 \sin ^{-1}(a x)}{2 a^6 c^4}\\ \end{align*}

Mathematica [A]  time = 0.323606, size = 126, normalized size = 0.76 \[ -\frac{(a x+1) \left (\sqrt{1-a^2 x^2} \left (105 a^5 x^5+630 a^4 x^4-8404 a^3 x^3+18916 a^2 x^2-16091 a x+4784\right )-945 (a x-1)^4 \sin ^{-1}(a x)+4200 (a x-1)^4 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{210 a^6 c^4 (a x-1)^3 \left (a^2 x^2-1\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^ArcTanh[a*x]*x^5)/(c - a*c*x)^4,x]

[Out]

-((1 + a*x)*(Sqrt[1 - a^2*x^2]*(4784 - 16091*a*x + 18916*a^2*x^2 - 8404*a^3*x^3 + 630*a^4*x^4 + 105*a^5*x^5) -
945*(-1 + a*x)^4*ArcSin[a*x] + 4200*(-1 + a*x)^4*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(210*a^6*c^4*(-1 + a*x)^3*(-
1 + a^2*x^2))

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Maple [A]  time = 0.057, size = 252, normalized size = 1.5 \begin{align*} -{\frac{x}{2\,{c}^{4}{a}^{5}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{29}{2\,{c}^{4}{a}^{5}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-5\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{{a}^{6}{c}^{4}}}+{\frac{71}{35\,{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 ) ^{-3}}+{\frac{733}{105\,{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 ) ^{-2}}+{\frac{2417}{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 ) ^{-1}}+{\frac{2}{7\,{c}^{4}{a}^{10}}\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^5/(-a*c*x+c)^4,x)

[Out]

-1/2/c^4/a^5*x*(-a^2*x^2+1)^(1/2)+29/2/c^4/a^5/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-5/c^4/a^6*
(-a^2*x^2+1)^(1/2)+71/35/c^4/a^9/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+733/105/c^4/a^8/(x-1/a)^2*(-a^2*
(x-1/a)^2-2*a*(x-1/a))^(1/2)+2417/105/c^4/a^7/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+2/7/c^4/a^10/(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^5/(-a*c*x+c)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73425, size = 454, normalized size = 2.73 \begin{align*} -\frac{4784 \, a^{4} x^{4} - 19136 \, a^{3} x^{3} + 28704 \, a^{2} x^{2} - 19136 \, a x + 6090 \,{\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^{5} x^{5} + 630 \, a^{4} x^{4} - 8404 \, a^{3} x^{3} + 18916 \, a^{2} x^{2} - 16091 \, a x + 4784\right )} \sqrt{-a^{2} x^{2} + 1} + 4784}{210 \,{\left (a^{10} c^{4} x^{4} - 4 \, a^{9} c^{4} x^{3} + 6 \, a^{8} c^{4} x^{2} - 4 \, a^{7} c^{4} x + a^{6} 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^5/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

-1/210*(4784*a^4*x^4 - 19136*a^3*x^3 + 28704*a^2*x^2 - 19136*a*x + 6090*(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^5*x^5 + 630*a^4*x^4 - 8404*a^3*x^3 + 18916*a^2*x^2 - 1
6091*a*x + 4784)*sqrt(-a^2*x^2 + 1) + 4784)/(a^10*c^4*x^4 - 4*a^9*c^4*x^3 + 6*a^8*c^4*x^2 - 4*a^7*c^4*x + a^6*
c^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{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 + \int \frac{a x^{6}}{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**5/(-a*c*x+c)**4,x)

[Out]

(Integral(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) + Integral(a*x**6/(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 = 1.27729, size = 340, normalized size = 2.05 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{x}{a^{5} c^{4}} + \frac{10}{a^{6} c^{4}}\right )} + \frac{29 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \, a^{5} c^{4}{\left | a \right |}} + \frac{2 \,{\left (\frac{11599 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{29442 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{38500 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{26845 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{9765 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{1470 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 1867\right )}}{105 \, a^{5} 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^5/(-a*c*x+c)^4,x, algorithm="giac")

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*(x/(a^5*c^4) + 10/(a^6*c^4)) + 29/2*arcsin(a*x)*sgn(a)/(a^5*c^4*abs(a)) + 2/105*(11599
*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 29442*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 38500*(sqrt(-a^
2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 26845*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 9765*(sqrt(-a^2*x^2 +
1)*abs(a) + a)^5/(a^10*x^5) - 1470*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) - 1867)/(a^5*c^4*((sqrt(-a^2*
x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))