### 3.865 $$\int e^{-2 \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^{3/2} \, dx$$

Optimal. Leaf size=213 $-\frac{5 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{a x+1}+\frac{x (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 (a x+1)}-\frac{2 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}+\frac{a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}}$

[Out]

-((a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 + a*x)) - (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) + (
(c - c/(a^2*x^2))^(3/2)*x*(1 - a*x))/(2*(1 + a*x)) - (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcSin[a*x])/((1 - a*x
)^(3/2)*(1 + a*x)^(3/2)) + (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*(1 - a*x)
^(3/2)*(1 + a*x)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.446075, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.458, Rules used = {6167, 6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} $-\frac{5 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{a x+1}+\frac{x (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{3/2}}{2 (a x+1)}-\frac{2 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}+\frac{a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c - c/(a^2*x^2))^(3/2)/E^(2*ArcCoth[a*x]),x]

[Out]

-((a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 + a*x)) - (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) + (
(c - c/(a^2*x^2))^(3/2)*x*(1 - a*x))/(2*(1 + a*x)) - (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcSin[a*x])/((1 - a*x
)^(3/2)*(1 + a*x)^(3/2)) + (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*(1 - a*x)
^(3/2)*(1 + a*x)^(3/2))

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6159

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

Rule 6129

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

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 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 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

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

Mathematica [A]  time = 0.108662, size = 115, normalized size = 0.54 $\frac{c \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (2 a^2 x^2+4 a x-1\right )-4 a^2 x^2 \log \left (\sqrt{a^2 x^2-1}+a x\right )+a^2 x^2 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{2 a^2 x \sqrt{a^2 x^2-1}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a^2*x^2))^(3/2)/E^(2*ArcCoth[a*x]),x]

[Out]

(c*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-1 + 4*a*x + 2*a^2*x^2) + a^2*x^2*ArcTan[1/Sqrt[-1 + a^2*x^2]] -
4*a^2*x^2*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(2*a^2*x*Sqrt[-1 + a^2*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.177, size = 454, normalized size = 2.1 \begin{align*} -{\frac{x}{6\,{a}^{2}c} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 12\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{x}^{3}{a}^{5}c-12\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{5/2}x{a}^{5}-4\,\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}} \right ) ^{3/2}{x}^{2}{a}^{4}c+\sqrt{-{\frac{c}{{a}^{2}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{{\frac{3}{2}}}{x}^{2}{a}^{4}c+6\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}{x}^{3}{a}^{3}{c}^{2}+3\,{a}^{4} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{5/2}\sqrt{-{\frac{c}{{a}^{2}}}}-18\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{3}{a}^{3}{c}^{2}+18\,{c}^{5/2}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) \sqrt{-{\frac{c}{{a}^{2}}}}{x}^{2}a-6\,{c}^{5/2}\sqrt{-{\frac{c}{{a}^{2}}}}\ln \left ({\frac{1}{\sqrt{c}} \left ( \sqrt{c}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}+cx \right ) } \right ){x}^{2}a-3\,\sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{x}^{2}{a}^{2}{c}^{2}-3\,\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{-{\frac{c}{{a}^{2}}}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}{a}^{2}-c \right ) } \right ){x}^{2}{c}^{3} \right ){\frac{1}{\sqrt{-{\frac{c}{{a}^{2}}}}}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(c*(a^2*x^2-1)/a^2/x^2)^(3/2)*x/a^2*(12*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^3*a^5*c-12*(-c/a^2)^(1
/2)*(c*(a^2*x^2-1)/a^2)^(5/2)*x*a^5-4*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x^2*a^4*c+(-c/a^2)^(1/2)*(c
*(a^2*x^2-1)/a^2)^(3/2)*x^2*a^4*c+6*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*x^3*a^3*c^2+3*a^4*(c*(a^2*x^2
-1)/a^2)^(5/2)*(-c/a^2)^(1/2)-18*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^3*a^3*c^2+18*c^(5/2)*ln(x*c^(1/2)+
(c*(a^2*x^2-1)/a^2)^(1/2))*(-c/a^2)^(1/2)*x^2*a-6*c^(5/2)*(-c/a^2)^(1/2)*ln((c^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(
1/2)+c*x)/c^(1/2))*x^2*a-3*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^2*a^2*c^2-3*ln(2*((-c/a^2)^(1/2)*(c*(a^2
*x^2-1)/a^2)^(1/2)*a^2-c)/x/a^2)*x^2*c^3)/(-c/a^2)^(1/2)/(c*(a^2*x^2-1)/a^2)^(3/2)/c

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x - 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{3}{2}}}{a x + 1}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.86562, size = 693, normalized size = 3.25 \begin{align*} \left [\frac{8 \, a \sqrt{-c} c x \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + a \sqrt{-c} c x \log \left (-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \,{\left (2 \, a^{2} c x^{2} + 4 \, a c x - c\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, \frac{a c^{\frac{3}{2}} x \arctan \left (\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 2 \, a c^{\frac{3}{2}} x \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) +{\left (2 \, a^{2} c x^{2} + 4 \, a c x - c\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(8*a*sqrt(-c)*c*x*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + a*sqrt(-c)*c
*x*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) + 2*(2*a^2*c*x^2 + 4*a*c*x - c
)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x), 1/2*(a*c^(3/2)*x*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)
)/(a^2*c*x^2 - c)) + 2*a*c^(3/2)*x*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) +
(2*a^2*c*x^2 + 4*a*c*x - c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x)]

________________________________________________________________________________________

Sympy [C]  time = 13.2434, size = 376, normalized size = 1.77 \begin{align*} c \left (\begin{cases} \frac{\sqrt{c} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{i \sqrt{c} \log{\left (a x \right )}}{a} + \frac{i \sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2 a} + \frac{\sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{i \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{a} + \frac{i \sqrt{c} \log{\left (a^{2} x^{2} \right )}}{2 a} - \frac{i \sqrt{c} \log{\left (\sqrt{- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text{otherwise} \end{cases}\right ) - \frac{2 c \left (\begin{cases} - \frac{a \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + \sqrt{c} \operatorname{acosh}{\left (a x \right )} + \frac{\sqrt{c}}{a x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{i a \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - i \sqrt{c} \operatorname{asin}{\left (a x \right )} - \frac{i \sqrt{c}}{a x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right )}{a} + \frac{c \left (\begin{cases} \frac{i a \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} + \frac{i \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{i \sqrt{c}}{2 a^{2} x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{a \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{\sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \text{otherwise} \end{cases}\right )}{a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*Piecewise((sqrt(c)*sqrt(a**2*x**2 - 1)/a - I*sqrt(c)*log(a*x)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) + sqrt(c)*a
sin(1/(a*x))/a, Abs(a**2*x**2) > 1), (I*sqrt(c)*sqrt(-a**2*x**2 + 1)/a + I*sqrt(c)*log(a**2*x**2)/(2*a) - I*sq
rt(c)*log(sqrt(-a**2*x**2 + 1) + 1)/a, True)) - 2*c*Piecewise((-a*sqrt(c)*x/sqrt(a**2*x**2 - 1) + sqrt(c)*acos
h(a*x) + sqrt(c)/(a*x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (I*a*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - I*sqrt(
c)*asin(a*x) - I*sqrt(c)/(a*x*sqrt(-a**2*x**2 + 1)), True))/a + c*Piecewise((I*a*sqrt(c)*acosh(1/(a*x))/2 + I*
sqrt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - I*sqrt(c)/(2*a**2*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) >
1), (-a*sqrt(c)*asin(1/(a*x))/2 - sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**2

________________________________________________________________________________________

Giac [A]  time = 1.56796, size = 359, normalized size = 1.69 \begin{align*} -{\left (\frac{c^{\frac{3}{2}} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right )}{a^{2}} - \frac{2 \, c^{\frac{3}{2}} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a{\left | a \right |}} - \frac{\sqrt{a^{2} c x^{2} - c} c \mathrm{sgn}\left (x\right )}{a^{2}} - \frac{{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} c^{2}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 4 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} a c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) -{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} c^{3}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 4 \, a c^{\frac{7}{2}} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a^{2}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^(3/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^2 - 2*c^(3/2)*log(abs(-sqrt(a^2*c)*x
+ sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs(a)) - sqrt(a^2*c*x^2 - c)*c*sgn(x)/a^2 - ((sqrt(a^2*c)*x - sqrt(a^2*c*x
^2 - c))^3*c^2*abs(a)*sgn(x) + 4*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^(5/2)*sgn(x) - (sqrt(a^2*c)*x - s
qrt(a^2*c*x^2 - c))*c^3*abs(a)*sgn(x) + 4*a*c^(7/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^2*a
^2*abs(a)))*abs(a)