### 3.380 $$\int e^{\coth ^{-1}(a x)} (c-\frac{c}{a x})^3 \, dx$$

Optimal. Leaf size=88 $\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{2 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c^3 \csc ^{-1}(a x)}{2 a}$

[Out]

(c^3*Sqrt[1 - 1/(a^2*x^2)]*(4*a + x^(-1)))/(2*a^2) + c^3*(1 - 1/(a^2*x^2))^(3/2)*x + (c^3*ArcCsc[a*x])/(2*a) -
(2*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a

________________________________________________________________________________________

Rubi [A]  time = 0.190626, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {6177, 1807, 815, 844, 216, 266, 63, 208} $\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (4 a+\frac{1}{x}\right )}{2 a^2}+c^3 x \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-\frac{2 c^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c^3 \csc ^{-1}(a x)}{2 a}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[a*x]*(c - c/(a*x))^3,x]

[Out]

(c^3*Sqrt[1 - 1/(a^2*x^2)]*(4*a + x^(-1)))/(2*a^2) + c^3*(1 - 1/(a^2*x^2))^(3/2)*x + (c^3*ArcCsc[a*x])/(2*a) -
(2*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 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 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

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

Mathematica [A]  time = 0.141962, size = 167, normalized size = 1.9 $\frac{c^3 \left (2 a^4 x^4+4 a^3 x^3-3 a^2 x^2+2 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )+2 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{1}{a x}\right )-4 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-4 a x+1\right )}{2 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(1 - 4*a*x - 3*a^2*x^2 + 4*a^3*x^3 + 2*a^4*x^4 + 2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcSin[Sqrt[1 - 1/(a*x)]
/Sqrt[2]] + 2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcSin[1/(a*x)] - 4*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcTanh[Sqrt[1 -
1/(a^2*x^2)]]))/(2*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3)

________________________________________________________________________________________

Maple [B]  time = 0.131, size = 200, normalized size = 2.3 \begin{align*} -{\frac{ \left ( ax-1 \right ){c}^{3}}{2\,{x}^{2}{a}^{3}} \left ( -4\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+4\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+4\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-{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 ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \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)^3,x)

[Out]

-1/2*(a*x-1)*c^3*(-4*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*x^3*a^3+4*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*x*a-(a^2*x^2-1)^(1/
2)*(a^2)^(1/2)*x^2*a^2+4*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*x^2*a^3-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*x-1)/(a*x+1))^(1/2)/((a*x-1)*(a*x+1))^(1/2)/a^3/x^
2/(a^2)^(1/2)

________________________________________________________________________________________

Maxima [B]  time = 1.63146, size = 271, normalized size = 3.08 \begin{align*} -{\left (\frac{c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{2 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{2 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac{3 \, 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}} - 5 \, 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(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^3,x, algorithm="maxima")

[Out]

-(c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 2*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 2*c^3*log(sqrt((a
*x - 1)/(a*x + 1)) - 1)/a^2 + (3*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 6*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 5*c^3*s
qrt((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

________________________________________________________________________________________

Fricas [A]  time = 1.6649, size = 332, normalized size = 3.77 \begin{align*} -\frac{2 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 4 \, a^{2} c^{3} x^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 4 \, 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} + 6 \, a^{2} c^{3} x^{2} + 3 \, 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(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^3,x, algorithm="fricas")

[Out]

-1/2*(2*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) + 4*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 4*a
^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (2*a^3*c^3*x^3 + 6*a^2*c^3*x^2 + 3*a*c^3*x - c^3)*sqrt((a*x -
1)/(a*x + 1)))/(a^3*x^2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.21314, size = 263, normalized size = 2.99 \begin{align*} -{\left (\frac{c^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{2 \, c^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{2 \, c^{3} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{2 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} - \frac{\frac{5 \,{\left (a x - 1\right )} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + 3 \, c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \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)^3,x, algorithm="giac")

[Out]

-(c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 + 2*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 2*c^3*log(abs(sqr
t((a*x - 1)/(a*x + 1)) - 1))/a^2 + 2*c^3*sqrt((a*x - 1)/(a*x + 1))/(a^2*((a*x - 1)/(a*x + 1) - 1)) - (5*(a*x -
1)*c^3*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) + 3*c^3*sqrt((a*x - 1)/(a*x + 1)))/(a^2*((a*x - 1)/(a*x + 1) + 1)^
2))*a