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

Optimal. Leaf size=108 \[ -\frac{1}{2} c \log \left (1-c^2 x^2\right )-\frac{\sqrt{1-c x}}{2 c x^2 \sqrt{\frac{1}{c x+1}}}-\frac{1}{2 c x^2}+c \log (x)-\frac{1}{2} c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right ) \]

[Out]

-1/(2*c*x^2) - Sqrt[1 - c*x]/(2*c*x^2*Sqrt[(1 + c*x)^(-1)]) - (c*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sq
rt[1 - c*x]*Sqrt[1 + c*x]])/2 + c*Log[x] - (c*Log[1 - c^2*x^2])/2

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Rubi [A]  time = 0.176461, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6341, 1956, 103, 12, 92, 208, 266, 44} \[ -\frac{1}{2} c \log \left (1-c^2 x^2\right )-\frac{\sqrt{1-c x}}{2 c x^2 \sqrt{\frac{1}{c x+1}}}-\frac{1}{2 c x^2}+c \log (x)-\frac{1}{2} c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*c*x^2) - Sqrt[1 - c*x]/(2*c*x^2*Sqrt[(1 + c*x)^(-1)]) - (c*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sq
rt[1 - c*x]*Sqrt[1 + c*x]])/2 + c*Log[x] - (c*Log[1 - c^2*x^2])/2

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 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]

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.151469, size = 108, normalized size = 1. \[ \frac{1}{2} \left (-c \log \left (1-c^2 x^2\right )-\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c x^2}-\frac{1}{c x^2}+3 c \log (x)-c \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.227, size = 111, normalized size = 1. \begin{align*} -{\frac{1}{2\,x}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ({c}^{2}{x}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) +\sqrt{-{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{\frac{c\ln \left ( cx-1 \right ) }{2}}-{\frac{1}{2\,c{x}^{2}}}+c\ln \left ( x \right ) -{\frac{c\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^2/(-c^2*x^2+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c \int \frac{1}{x}\,{d x} - \frac{1}{2} \, c \log \left (c x + 1\right ) - \frac{1}{2} \, c \log \left (c x - 1\right ) + \frac{-\frac{1}{2 \, x^{2}}}{c} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1}}{c^{3} x^{5} - c 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^2/(-c^2*x^2+1),x, algorithm="maxima")

[Out]

c*integrate(1/x, x) - 1/2*c*log(c*x + 1) - 1/2*c*log(c*x - 1) + integrate(x^(-3), x)/c - integrate(sqrt(c*x +
1)*sqrt(-c*x + 1)/(c^3*x^5 - c*x^3), x)

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Fricas [B]  time = 2.12026, size = 344, normalized size = 3.19 \begin{align*} -\frac{2 \, c^{2} x^{2} \log \left (c^{2} x^{2} - 1\right ) + c^{2} x^{2} \log \left (c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + 1\right ) - c^{2} x^{2} \log \left (c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} - 1\right ) - 4 \, c^{2} x^{2} \log \left (x\right ) + 2 \, c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + 2}{4 \, c x^{2}} \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^2/(-c^2*x^2+1),x, algorithm="fricas")

[Out]

-1/4*(2*c^2*x^2*log(c^2*x^2 - 1) + c^2*x^2*log(c*x*sqrt((c*x + 1)/(c*x))*sqrt(-(c*x - 1)/(c*x)) + 1) - c^2*x^2
*log(c*x*sqrt((c*x + 1)/(c*x))*sqrt(-(c*x - 1)/(c*x)) - 1) - 4*c^2*x^2*log(x) + 2*c*x*sqrt((c*x + 1)/(c*x))*sq
rt(-(c*x - 1)/(c*x)) + 2)/(c*x^2)

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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^{5} - x^{3}}\, dx + \int \frac{1}{c^{2} x^{5} - x^{3}}\, 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**2/(-c**2*x**2+1),x)

[Out]

-(Integral(c*x*sqrt(-1 + 1/(c*x))*sqrt(1 + 1/(c*x))/(c**2*x**5 - x**3), x) + Integral(1/(c**2*x**5 - x**3), x)
)/c

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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^{2}}\,{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^2/(-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^2), x)