3.96 \(\int \frac{e^{\text{sech}^{-1}(c x)}}{x^3 (1-c^2 x^2)} \, dx\)

Optimal. Leaf size=85 \[ c^2 \tanh ^{-1}(c x)-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{c x+1}}}-\frac{1}{3 c x^3}-\frac{2 c \sqrt{1-c x}}{3 x \sqrt{\frac{1}{c x+1}}}-\frac{c}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.165316, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6341, 1956, 103, 12, 95, 325, 206} \[ c^2 \tanh ^{-1}(c x)-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{c x+1}}}-\frac{1}{3 c x^3}-\frac{2 c \sqrt{1-c x}}{3 x \sqrt{\frac{1}{c x+1}}}-\frac{c}{x} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSech[c*x]/(x^3*(1 - c^2*x^2)),x]

[Out]

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

Rule 6341

Int[(E^ArcSech[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[d/(a*c), Int[((d*x)^(m
- 1)*Sqrt[1/(1 + c*x)])/Sqrt[1 - c*x], x], x] + Dist[d/c, Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a,
b, c, d, m}, x] && EqQ[b + a*c^2, 0]

Rule 1956

Int[(x_)^(m_.)*((e_.)*((a_) + (b_.)*(x_)^(n_.))^(r_.))^(p_)*((f_.)*((c_) + (d_.)*(x_)^(n_.))^(s_))^(q_), x_Sym
bol] :> Dist[((e*(a + b*x^n)^r)^p*(f*(c + d*x^n)^s)^q)/((a + b*x^n)^(p*r)*(c + d*x^n)^(q*s)), Int[x^m*(a + b*x
^n)^(p*r)*(c + d*x^n)^(q*s), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r, s}, x]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)}}{x^3 \left (1-c^2 x^2\right )} \, dx &=\frac{\int \frac{\sqrt{\frac{1}{1+c x}}}{x^4 \sqrt{1-c x}} \, dx}{c}+\frac{\int \frac{1}{x^4 \left (1-c^2 x^2\right )} \, dx}{c}\\ &=-\frac{1}{3 c x^3}+c \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^4 \sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=-\frac{1}{3 c x^3}-\frac{c}{x}-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{1+c x}}}+c^3 \int \frac{1}{1-c^2 x^2} \, dx-\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{2 c^2}{x^2 \sqrt{1-c x} \sqrt{1+c x}} \, dx}{3 c}\\ &=-\frac{1}{3 c x^3}-\frac{c}{x}-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{1+c x}}}+c^2 \tanh ^{-1}(c x)+\frac{1}{3} \left (2 c \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=-\frac{1}{3 c x^3}-\frac{c}{x}-\frac{\sqrt{1-c x}}{3 c x^3 \sqrt{\frac{1}{1+c x}}}-\frac{2 c \sqrt{1-c x}}{3 x \sqrt{\frac{1}{1+c x}}}+c^2 \tanh ^{-1}(c x)\\ \end{align*}

Mathematica [A]  time = 0.241577, size = 90, normalized size = 1.06 \[ -\frac{6 c^2 x^2+2 \sqrt{\frac{1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )+3 c^3 x^3 \log (1-c x)-3 c^3 x^3 \log (c x+1)+2}{6 c x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[c*x]/(x^3*(1 - c^2*x^2)),x]

[Out]

-(2 + 6*c^2*x^2 + 2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3) + 3*c^3*x^3*Log[1 - c*x] - 3*c
^3*x^3*Log[1 + c*x])/(6*c*x^3)

________________________________________________________________________________________

Maple [C]  time = 0.22, size = 86, normalized size = 1. \begin{align*} -{\frac{ \left ({\it csgn} \left ( c \right ) \right ) ^{2} \left ( 2\,{c}^{2}{x}^{2}+1 \right ) }{3\,{x}^{2}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}-{\frac{{c}^{2}\ln \left ( cx-1 \right ) }{2}}-{\frac{1}{3\,c{x}^{3}}}-{\frac{c}{x}}+{\frac{{c}^{2}\ln \left ( cx+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, c^{2} \log \left (c x + 1\right ) - \frac{1}{2} \, c^{2} \log \left (c x - 1\right ) + c \int \frac{1}{x^{2}}\,{d x} + \frac{-\frac{1}{3 \, x^{3}}}{c} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1}}{c^{3} x^{6} - c x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.20589, size = 197, normalized size = 2.32 \begin{align*} \frac{3 \, c^{3} x^{3} \log \left (c x + 1\right ) - 3 \, c^{3} x^{3} \log \left (c x - 1\right ) - 6 \, c^{2} x^{2} - 2 \,{\left (2 \, c^{3} x^{3} + c x\right )} \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} - 2}{6 \, c x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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