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

Optimal. Leaf size=171 \[ -\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{525 a+\frac{719}{x}}{105 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{5 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4} \]

[Out]

(-16*(a + x^(-1)))/(7*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (4*(7*a + 17/x))/(35*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2))
- (175*a + 307/x)/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (525*a + 719/x)/(105*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)])
+ (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (5*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^4)

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Rubi [A]  time = 0.500726, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{525 a+\frac{719}{x}}{105 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{5 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]/(c - c/(a*x))^4,x]

[Out]

(-16*(a + x^(-1)))/(7*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (4*(7*a + 17/x))/(35*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2))
- (175*a + 307/x)/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (525*a + 719/x)/(105*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)])
+ (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (5*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^4)

Rule 6177

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> -Dist[c^n, Subst[Int[((c + d*x)^(p -
n)*(1 - x^2/a^2)^(n/2))/x^2, x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)
/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1]) && 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 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^4} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x^2 \left (c-\frac{c x}{a}\right )^5} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^5}{x^2 \left (1-\frac{x^2}{a^2}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{c^9}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{\operatorname{Subst}\left (\int \frac{-7 c^5-\frac{35 c^5 x}{a}-\frac{61 c^5 x^2}{a^2}+\frac{7 c^5 x^3}{a^3}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 c^9}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{35 c^5+\frac{175 c^5 x}{a}+\frac{272 c^5 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{35 c^9}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-105 c^5-\frac{525 c^5 x}{a}-\frac{614 c^5 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{105 c^9}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{525 a+\frac{719}{x}}{105 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{105 c^5+\frac{525 c^5 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{105 c^9}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{525 a+\frac{719}{x}}{105 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^4}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{525 a+\frac{719}{x}}{105 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a c^4}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{525 a+\frac{719}{x}}{105 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}+\frac{(5 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c^4}\\ &=-\frac{16 \left (a+\frac{1}{x}\right )}{7 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{4 \left (7 a+\frac{17}{x}\right )}{35 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{525 a+\frac{719}{x}}{105 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}+\frac{5 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.0852049, size = 112, normalized size = 0.65 \[ \frac{105 a^5 x^5-1339 a^4 x^4+1812 a^3 x^3+485 a^2 x^2+525 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-1947 a x+824}{105 a^2 c^4 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(824 - 1947*a*x + 485*a^2*x^2 + 1812*a^3*x^3 - 1339*a^4*x^4 + 105*a^5*x^5 + 525*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1
+ a*x)^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(105*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3)

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Maple [B]  time = 0.138, size = 523, normalized size = 3.1 \begin{align*}{\frac{1}{105\,a \left ( ax-1 \right ) ^{4}{c}^{4}} \left ( 525\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{5}{a}^{6}+525\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{5}{a}^{5}-2625\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}-420\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{3}{a}^{3}-2625\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}+5250\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}+1076\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+5250\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-5250\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-970\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-5250\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+2625\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}+299\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+2625\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-525\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -525\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(525*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^5*a^6+525*(a^2)^(1/2)*((a*x-1)*(a*x+1
))^(1/2)*x^5*a^5-2625*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5-420*(a^2)^(1/2)*((a*
x-1)*(a*x+1))^(3/2)*x^3*a^3-2625*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+5250*ln((a^2*x+(a^2)^(1/2)*((a*x-
1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4+1076*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+5250*(a^2)^(1/2)*((a*
x-1)*(a*x+1))^(1/2)*x^3*a^3-5250*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3-970*(a^2)
^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a-5250*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2+2625*ln((a^2*x+(a^2)^(1/2)
*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2+299*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+2625*(a^2)^(1/2)*((a*x-1)
*(a*x+1))^(1/2)*x*a-525*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))-525*(a^2)^(1/2)*((a*x-1)
*(a*x+1))^(1/2))/a/(a^2)^(1/2)/(a*x-1)^4/c^4/((a*x-1)*(a*x+1))^(1/2)/((a*x-1)/(a*x+1))^(1/2)

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Maxima [A]  time = 1.01998, size = 228, normalized size = 1.33 \begin{align*} \frac{1}{420} \, a{\left (\frac{\frac{111 \,{\left (a x - 1\right )}}{a x + 1} + \frac{469 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{2765 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{4200 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 15}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}} + \frac{2100 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{2100 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*a*((111*(a*x - 1)/(a*x + 1) + 469*(a*x - 1)^2/(a*x + 1)^2 + 2765*(a*x - 1)^3/(a*x + 1)^3 - 4200*(a*x - 1
)^4/(a*x + 1)^4 + 15)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(9/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(7/2)) + 2100*log(s
qrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 2100*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))

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Fricas [A]  time = 1.63672, size = 475, normalized size = 2.78 \begin{align*} \frac{525 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 525 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (105 \, a^{5} x^{5} - 1339 \, a^{4} x^{4} + 1812 \, a^{3} x^{3} + 485 \, a^{2} x^{2} - 1947 \, a x + 824\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{105 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(525*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 525*(a^4*x^4 - 4
*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (105*a^5*x^5 - 1339*a^4*x^4 + 1812*a^3*
x^3 + 485*a^2*x^2 - 1947*a*x + 824)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 -
4*a^2*c^4*x + a*c^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15662, size = 246, normalized size = 1.44 \begin{align*} \frac{1}{420} \, a{\left (\frac{2100 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{2100 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac{{\left (a x + 1\right )}^{3}{\left (\frac{126 \,{\left (a x - 1\right )}}{a x + 1} + \frac{595 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{3360 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 15\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{840 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*a*(2100*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 2100*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2
*c^4) - (a*x + 1)^3*(126*(a*x - 1)/(a*x + 1) + 595*(a*x - 1)^2/(a*x + 1)^2 + 3360*(a*x - 1)^3/(a*x + 1)^3 + 15
)/((a*x - 1)^3*a^2*c^4*sqrt((a*x - 1)/(a*x + 1))) - 840*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^4*((a*x - 1)/(a*x + 1
) - 1)))