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

Optimal. Leaf size=71 \[ -\frac{\log \left (1-c^2 x^2\right )}{2 c}+\frac{\log (x)}{c}-\frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right )}{c} \]

[Out]

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

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Rubi [A]  time = 0.119409, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6339, 1956, 92, 208, 266, 36, 29, 31} \[ -\frac{\log \left (1-c^2 x^2\right )}{2 c}+\frac{\log (x)}{c}-\frac{\sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right )}{c} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSech[c*x]/(1 - c^2*x^2),x]

[Out]

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

Rule 6339

Int[E^ArcSech[(c_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[1/(a*c), Int[Sqrt[1/(1 + c*x)]/(x*Sqrt[1 -
c*x]), x], x] + Dist[1/c, Int[1/(x*(a + b*x^2)), x], x] /; FreeQ[{a, b, c}, 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 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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0398584, size = 73, normalized size = 1.03 \[ -\frac{\log \left (1-c^2 x^2\right )}{2 c}+\frac{2 \log (x)}{c}-\frac{\log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )}{c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.204, size = 87, normalized size = 1.2 \begin{align*} -{x\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{\frac{\ln \left ( cx-1 \right ) }{2\,c}}+{\frac{\ln \left ( x \right ) }{c}}-{\frac{\ln \left ( cx+1 \right ) }{2\,c}} \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))/(-c^2*x^2+1),x)

[Out]

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

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

[Out]

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

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Fricas [B]  time = 2.07632, size = 213, normalized size = 3. \begin{align*} -\frac{\log \left (c^{2} x^{2} - 1\right ) + \log \left (c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + 1\right ) - \log \left (c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} - 1\right ) - 2 \, \log \left (x\right )}{2 \, 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))/(-c^2*x^2+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 15.4875, size = 48, normalized size = 0.68 \begin{align*} - \frac{\log{\left (-1 + \frac{1}{c x} \right )}}{2 c} - \frac{\log{\left (\sqrt{1 + \frac{1}{c x}} \right )}}{c} - \frac{2 \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{1 + \frac{1}{c x}}}{2} \right )}}{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))/(-c**2*x**2+1),x)

[Out]

-log(-1 + 1/(c*x))/(2*c) - log(sqrt(1 + 1/(c*x)))/c - 2*acosh(sqrt(2)*sqrt(1 + 1/(c*x))/2)/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}}{c^{2} x^{2} - 1}\,{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))/(-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)