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

Optimal. Leaf size=30 \[ -\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{c x}-\tan ^{-1}(c x) \]

[Out]

-1/c/x-arctan(c*x)-(1+1/c^2/x^2)^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6342, 261, 325, 203} \[ -\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{c x}-\tan ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 203

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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 6342

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

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 30, normalized size = 1.00 \[ -\sqrt {\frac {1}{c^2 x^2}+1}-\frac {1}{c x}-\tan ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 41, normalized size = 1.37 \[ -\frac {c x \arctan \left (c x\right ) + c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + c x + 1}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 43, normalized size = 1.43 \[ \frac {2 \, \mathrm {sgn}\relax (x)}{{\left (x {\left | c \right |} - \sqrt {c^{2} x^{2} + 1}\right )}^{2} - 1} - \frac {1}{c x} - \arctan \left (c x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.07, size = 154, normalized size = 5.13 \[ -\frac {\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, \left (\left (\frac {c^{2} x^{2}+1}{c^{2}}\right )^{\frac {3}{2}} c^{2}-\sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, x^{2} c^{2}+\ln \left (x +\sqrt {-\frac {\left (-c^{2} x +\sqrt {-c^{2}}\right ) \left (c^{2} x +\sqrt {-c^{2}}\right )}{c^{4}}}\right ) x -\ln \left (x +\sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\right ) x \right )}{\sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}}-\frac {1}{c x}-\arctan \left (c x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 34, normalized size = 1.13 \[ -\frac {\sqrt {c^{2} x^{2} + 1}}{c x} - \frac {1}{c x} - \arctan \left (c x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.15, size = 29, normalized size = 0.97 \[ -\mathrm {atan}\left (c\,x\right )-\frac {x\,\sqrt {\frac {1}{c^2\,x^2}+1}+\frac {1}{c}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan(c*x) - (x*(1/(c^2*x^2) + 1)^(1/2) + 1/c)/x

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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