\(\int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx\) [65]

Optimal result

Integrand size = 18, antiderivative size = 33 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\text {csch}^{-1}(c x)}{c}+\frac {\log (x)}{c}-\frac {\log \left (1+c^2 x^2\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6475, 342, 221, 272, 36, 29, 31} \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\log \left (c^2 x^2+1\right )}{2 c}+\frac {\log (x)}{c}-\frac {\text {csch}^{-1}(c x)}{c} \]

[In]

Int[E^ArcCsch[c*x]/(1 + c^2*x^2),x]

[Out]

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

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]

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 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 272

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 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 6475

Int[E^ArcCsch[(c_.)*(x_)]/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[1/(a*c^2), Int[1/(x^2*Sqrt[1 + 1/(c^2*x^2)]
), 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{c^2}+\frac {\int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{c} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c^2}+\frac {\text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 c} \\ & = -\frac {\text {csch}^{-1}(c x)}{c}+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c}-\frac {1}{2} c \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {\text {csch}^{-1}(c x)}{c}+\frac {\log (x)}{c}-\frac {\log \left (1+c^2 x^2\right )}{2 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\text {arcsinh}\left (\frac {1}{c x}\right )}{c}+\frac {\log (x)}{c}-\frac {\log \left (1+c^2 x^2\right )}{2 c} \]

[In]

Integrate[E^ArcCsch[c*x]/(1 + c^2*x^2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs. \(2(31)=62\).

Time = 0.88 (sec) , antiderivative size = 170, normalized size of antiderivative = 5.15

method	result	size
default	\(\frac {\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \left (\sqrt {\frac {1}{c^{2}}}\, \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, c^{2}-\sqrt {-\frac {\left (-c^{2} x +\sqrt {-c^{2}}\right ) \left (c^{2} x +\sqrt {-c^{2}}\right )}{c^{4}}}\, c^{2} \sqrt {\frac {1}{c^{2}}}-\ln \left (\frac {2 \sqrt {\frac {1}{c^{2}}}\, \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, c^{2}+2}{c^{2} x}\right )\right )}{\sqrt {\frac {1}{c^{2}}}\, \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, c^{2}}+\frac {\ln \left (x \right )-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}}{c}\)	\(170\)

[In]

int((1/c/x+(1+1/c^2/x^2)^(1/2))/(c^2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (31) = 62\).

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\log \left (c^{2} x^{2} + 1\right ) + 2 \, \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 2 \, \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 2 \, \log \left (x\right )}{2 \, c} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=\frac {\int \frac {c x \sqrt {1 + \frac {1}{c^{2} x^{2}}}}{c^{2} x^{3} + x}\, dx + \int \frac {1}{c^{2} x^{3} + x}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=\int { \frac {\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}}{c^{2} x^{2} + 1} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (31) = 62\).

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\frac {\log \left (c^{2} x^{2} + 1\right )}{2 \, c} - \frac {{\left ({\left | c \right |} \mathrm {sgn}\left (x\right ) - c\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{2 \, c^{2}} + \frac {{\left ({\left | c \right |} \mathrm {sgn}\left (x\right ) + c\right )} \log \left (\sqrt {c^{2} x^{2} + 1} - 1\right )}{2 \, c^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.41 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{1+c^2 x^2} \, dx=-\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{c^2}}}{x}\right )\,\sqrt {\frac {1}{c^2}}-\frac {\ln \left (c^2\,x^2+1\right )-2\,\ln \left (x\right )}{2\,c} \]

[In]

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

[Out]

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