### 3.430 $$\int e^{-3 \coth ^{-1}(a x)} (c-\frac{c}{a x})^3 \, dx$$

Optimal. Leaf size=135 $\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^3 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{6 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}$

[Out]

(6*c^3*Sqrt[1 - 1/(a^2*x^2)])/a + (32*c^3*(a - x^(-1)))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) - (c^3*Sqrt[1 - 1/(a^2*x^2
)])/(2*a^2*x) + c^3*Sqrt[1 - 1/(a^2*x^2)]*x + (33*c^3*ArcCsc[a*x])/(2*a) - (6*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)
]])/a

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Rubi [A]  time = 0.429622, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.409, Rules used = {6177, 1805, 1807, 1809, 844, 216, 266, 63, 208} $\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^3 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{6 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6*c^3*Sqrt[1 - 1/(a^2*x^2)])/a + (32*c^3*(a - x^(-1)))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) - (c^3*Sqrt[1 - 1/(a^2*x^2
)])/(2*a^2*x) + c^3*Sqrt[1 - 1/(a^2*x^2)]*x + (33*c^3*ArcCsc[a*x])/(2*a) - (6*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)
]])/a

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 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 1807

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

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

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 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 e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^6}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-c^6+\frac{6 c^6 x}{a}+\frac{16 c^6 x^2}{a^2}-\frac{6 c^6 x^3}{a^3}+\frac{c^6 x^4}{a^4}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\operatorname{Subst}\left (\int \frac{-\frac{6 c^6}{a}-\frac{16 c^6 x}{a^2}+\frac{6 c^6 x^2}{a^3}-\frac{c^6 x^3}{a^4}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{a^2 \operatorname{Subst}\left (\int \frac{\frac{12 c^6}{a^3}+\frac{33 c^6 x}{a^4}-\frac{12 c^6 x^2}{a^5}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 c^3}\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{12 c^6}{a^5}-\frac{33 c^6 x}{a^6}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 c^3}\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\left (33 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}+\frac{\left (6 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a}\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}-\left (6 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{32 c^3 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}-\frac{6 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}

Mathematica [C]  time = 0.462017, size = 663, normalized size = 4.91 $\frac{c^3 \left (70 \sqrt{2} a^6 x^6 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )-280 \sqrt{2} a^5 x^5 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+350 \sqrt{2} a^4 x^4 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+126 \sqrt{2} a^2 x^2 (a x-1)^3 (a x+1) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{5}{2},\frac{7}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )-350 \sqrt{2} a^2 x^2 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+90 \sqrt{2} a x (a x-1)^4 (a x+1) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{7}{2},\frac{9}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+280 \sqrt{2} a x \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )-70 \sqrt{2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+630 a^7 x^7 \sqrt{\frac{1}{a x}+1}-32340 a^6 x^6 \sqrt{\frac{1}{a x}+1}+17955 a^5 x^5 \sqrt{\frac{1}{a x}+1}+16800 a^4 x^4 \sqrt{\frac{1}{a x}+1}-3465 a^3 x^3 \sqrt{\frac{1}{a x}+1}+420 a^2 x^2 \sqrt{\frac{1}{a x}+1}+44730 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-2520 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )+44730 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-2520 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )-3780 a^6 x^6 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )\right )}{630 a^6 x^5 \sqrt{1-\frac{1}{a x}} (a x+1)}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(420*a^2*Sqrt[1 + 1/(a*x)]*x^2 - 3465*a^3*Sqrt[1 + 1/(a*x)]*x^3 + 16800*a^4*Sqrt[1 + 1/(a*x)]*x^4 + 17955
*a^5*Sqrt[1 + 1/(a*x)]*x^5 - 32340*a^6*Sqrt[1 + 1/(a*x)]*x^6 + 630*a^7*Sqrt[1 + 1/(a*x)]*x^7 + 44730*a^5*Sqrt[
1 - 1/(a*x)]*x^5*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 44730*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[Sqrt[1 - 1/(a*x)]/
Sqrt[2]] - 2520*a^5*Sqrt[1 - 1/(a*x)]*x^5*ArcSin[1/(a*x)] - 2520*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[1/(a*x)] - 3
780*a^6*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^6*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]] + 126*Sqrt[2]*a^2*x^2*(-1 +
a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - 1/(a*x))/2] + 90*Sqrt[2]*a*x*(-1 + a*x)^4*(1 + a*x)*Hy
pergeometric2F1[3/2, 7/2, 9/2, (1 - 1/(a*x))/2] - 70*Sqrt[2]*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2
] + 280*Sqrt[2]*a*x*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] - 350*Sqrt[2]*a^2*x^2*Hypergeometric2F1
[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 350*Sqrt[2]*a^4*x^4*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] - 2
80*Sqrt[2]*a^5*x^5*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 70*Sqrt[2]*a^6*x^6*Hypergeometric2F1[3
/2, 9/2, 11/2, (1 - 1/(a*x))/2]))/(630*a^6*Sqrt[1 - 1/(a*x)]*x^5*(1 + a*x))

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Maple [B]  time = 0.134, size = 450, normalized size = 3.3 \begin{align*} -{\frac{{c}^{3}}{2\,{x}^{2}{a}^{3} \left ( ax-1 \right ) } \left ( -12\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{5}{a}^{5}+12\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-57\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+12\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}-33\,{a}^{4}{x}^{4}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +23\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-78\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}-66\,{a}^{3}{x}^{3}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +32\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}+10\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-33\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+12\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-33\,{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) - \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-12*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^5*a^5+12*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^3*a^3-57*(a^2*x^2-1)^(1/2)*
(a^2)^(1/2)*x^4*a^4+12*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^4*a^5-33*a^4*x^4*(a^2)^(1/2)*ar
ctan(1/(a^2*x^2-1)^(1/2))+23*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*x^2*a^2-78*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^3*a^3+24
*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^3*a^4-66*a^3*x^3*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/
2))+32*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+10*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a-33*(a^2*x^2-1)^(1/2)*(
a^2)^(1/2)*x^2*a^2+12*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-33*a^2*x^2*(a^2)^(1/2)*arc
tan(1/(a^2*x^2-1)^(1/2))-(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/a^3*c^3*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/x^2/((a*x-
1)*(a*x+1))^(1/2)/(a*x-1)

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Maxima [A]  time = 1.58562, size = 304, normalized size = 2.25 \begin{align*} -{\left (\frac{33 \, c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{6 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{6 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{32 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}} + \frac{11 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 6 \, c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 13 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac{{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(33*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 6*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 6*c^3*log(sqrt
((a*x - 1)/(a*x + 1)) - 1)/a^2 - 32*c^3*sqrt((a*x - 1)/(a*x + 1))/a^2 + (11*c^3*((a*x - 1)/(a*x + 1))^(5/2) -
6*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 13*c^3*sqrt((a*x - 1)/(a*x + 1)))/((a*x - 1)*a^2/(a*x + 1) - (a*x - 1)^2*a
^2/(a*x + 1)^2 - (a*x - 1)^3*a^2/(a*x + 1)^3 + a^2))*a

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Fricas [A]  time = 1.95096, size = 339, normalized size = 2.51 \begin{align*} -\frac{66 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 12 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 12 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (2 \, a^{3} c^{3} x^{3} + 78 \, a^{2} c^{3} x^{2} + 11 \, a c^{3} x - c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(66*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) + 12*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 1
2*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (2*a^3*c^3*x^3 + 78*a^2*c^3*x^2 + 11*a*c^3*x - c^3)*sqrt((a
*x - 1)/(a*x + 1)))/(a^3*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef