3.402 \(\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-\frac{c}{a x})^3} \, dx\)

Optimal. Leaf size=165 \[ -\frac{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{2 \left (7 a+\frac{13}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{42 a+\frac{59}{x}}{7 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}-\frac{16}{7 a^2 c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{6 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3} \]

[Out]

(-32*(a + x^(-1)))/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(7/2)) - (2*(7*a + 13/x))/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2))
- (42*a + 59/x)/(7*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) - 16/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2)*x) + (Sqrt[1 - 1/(a^
2*x^2)]*x)/c^3 + (6*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.521118, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{2 \left (7 a+\frac{13}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{42 a+\frac{59}{x}}{7 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}-\frac{16}{7 a^2 c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{6 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*(a + x^(-1)))/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(7/2)) - (2*(7*a + 13/x))/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2))
- (42*a + 59/x)/(7*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) - 16/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2)*x) + (Sqrt[1 - 1/(a^
2*x^2)]*x)/c^3 + (6*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^3)

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^{3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac{c x}{a}\right )^6} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^6}{x^2 \left (1-\frac{x^2}{a^2}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{c^9}\\ &=-\frac{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{\operatorname{Subst}\left (\int \frac{-7 c^6-\frac{42 c^6 x}{a}-\frac{80 c^6 x^2}{a^2}+\frac{42 c^6 x^3}{a^3}+\frac{7 c^6 x^4}{a^4}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 c^9}\\ &=-\frac{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{16}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}-\frac{\operatorname{Subst}\left (\int \frac{35 c^6+\frac{210 c^6 x}{a}+\frac{355 c^6 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{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{2 \left (7 a+\frac{13}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{16}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}+\frac{\operatorname{Subst}\left (\int \frac{-105 c^6-\frac{630 c^6 x}{a}-\frac{780 c^6 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{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{2 \left (7 a+\frac{13}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{42 a+\frac{59}{x}}{7 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}-\frac{\operatorname{Subst}\left (\int \frac{105 c^6+\frac{630 c^6 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{105 c^9}\\ &=-\frac{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{2 \left (7 a+\frac{13}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{42 a+\frac{59}{x}}{7 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}-\frac{6 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^3}\\ &=-\frac{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{2 \left (7 a+\frac{13}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{42 a+\frac{59}{x}}{7 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a c^3}\\ &=-\frac{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{2 \left (7 a+\frac{13}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{42 a+\frac{59}{x}}{7 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}+\frac{(6 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^3}\\ &=-\frac{32 \left (a+\frac{1}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{2 \left (7 a+\frac{13}{x}\right )}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{42 a+\frac{59}{x}}{7 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16}{7 a^2 c^3 \left (1-\frac{1}{a^2 x^2}\right )^{5/2} x}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^3}+\frac{6 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^3}\\ \end{align*}

Mathematica [A]  time = 0.0827369, size = 112, normalized size = 0.68 \[ \frac{7 a^5 x^5-109 a^4 x^4+145 a^3 x^3+39 a^2 x^2+42 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-156 a x+66}{7 a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(66 - 156*a*x + 39*a^2*x^2 + 145*a^3*x^3 - 109*a^4*x^4 + 7*a^5*x^5 + 42*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3
*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(7*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3)

________________________________________________________________________________________

Maple [B]  time = 0.174, size = 530, normalized size = 3.2 \begin{align*} -{\frac{1}{7\,a \left ( ax-1 \right ) ^{3}{c}^{3} \left ( ax+1 \right ) } \left ( -42\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{5}{a}^{5}-42\,\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}+35\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{3}{a}^{3}+210\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}+210\,\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}-87\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}-420\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-420\,\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}+78\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa+420\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+420\,\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}-24\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-210\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-210\,\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}+42\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+42\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x)

[Out]

-1/7*(-42*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5-42*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^
(1/2))*x^5*a^6+35*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+210*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+
210*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5-87*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)
*x^2*a^2-420*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3-420*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a
^2)^(1/2))*x^3*a^4+78*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a+420*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2+
420*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3-24*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)
-210*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a-210*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x
*a^2+42*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)+42*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2)))/
a/(a^2)^(1/2)/(a*x-1)^3/c^3/((a*x-1)*(a*x+1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.07271, size = 228, normalized size = 1.38 \begin{align*} \frac{1}{14} \, a{\left (\frac{\frac{6 \,{\left (a x - 1\right )}}{a x + 1} + \frac{21 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{112 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{168 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}} + \frac{84 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{84 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="maxima")

[Out]

1/14*a*((6*(a*x - 1)/(a*x + 1) + 21*(a*x - 1)^2/(a*x + 1)^2 + 112*(a*x - 1)^3/(a*x + 1)^3 - 168*(a*x - 1)^4/(a
*x + 1)^4 + 1)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(9/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(7/2)) + 84*log(sqrt((a*x
- 1)/(a*x + 1)) + 1)/(a^2*c^3) - 84*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3))

________________________________________________________________________________________

Fricas [A]  time = 1.55957, size = 460, normalized size = 2.79 \begin{align*} \frac{42 \,{\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 ) - 42 \,{\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 (7 \, a^{5} x^{5} - 109 \, a^{4} x^{4} + 145 \, a^{3} x^{3} + 39 \, a^{2} x^{2} - 156 \, a x + 66\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{7 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="fricas")

[Out]

1/7*(42*(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) - 42*(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) + (7*a^5*x^5 - 109*a^4*x^4 + 145*a^3*x^3 + 39
*a^2*x^2 - 156*a*x + 66)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x
+ a*c^3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.3338, size = 246, normalized size = 1.49 \begin{align*} \frac{1}{14} \, a{\left (\frac{84 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{84 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{3}} - \frac{{\left (a x + 1\right )}^{3}{\left (\frac{7 \,{\left (a x - 1\right )}}{a x + 1} + \frac{28 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{140 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{28 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}{\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))^(3/2)/(c-c/a/x)^3,x, algorithm="giac")

[Out]

1/14*a*(84*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 84*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2*c^3)
- (a*x + 1)^3*(7*(a*x - 1)/(a*x + 1) + 28*(a*x - 1)^2/(a*x + 1)^2 + 140*(a*x - 1)^3/(a*x + 1)^3 + 1)/((a*x -
1)^3*a^2*c^3*sqrt((a*x - 1)/(a*x + 1))) - 28*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3*((a*x - 1)/(a*x + 1) - 1)))