### 3.63 $$\int \frac{e^{\text{csch}^{-1}(c x)} x^2}{1+c^2 x^2} \, dx$$

Optimal. Leaf size=36 $\frac{x \sqrt{\frac{1}{c^2 x^2}+1}}{c^2}+\frac{\log \left (c^2 x^2+1\right )}{2 c^3}$

[Out]

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

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Rubi [A]  time = 0.0698466, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {6342, 191, 260} $\frac{x \sqrt{\frac{1}{c^2 x^2}+1}}{c^2}+\frac{\log \left (c^2 x^2+1\right )}{2 c^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.0671598, size = 35, normalized size = 0.97 $\frac{2 c x \sqrt{\frac{1}{c^2 x^2}+1}+\log \left (c^2 x^2+1\right )}{2 c^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.151, size = 89, normalized size = 2.5 \begin{align*}{\frac{x}{{c}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}\sqrt{-{\frac{1}{{c}^{4}} \left ( x{c}^{2}+\sqrt{-{c}^{2}} \right ) \left ( -x{c}^{2}+\sqrt{-{c}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}}}}}}}+{\frac{\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{c}^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03118, size = 42, normalized size = 1.17 \begin{align*} \frac{\log \left (c^{3} x^{2} + c\right )}{2 \, c^{3}} + \frac{\sqrt{c^{2} x^{2} + 1}}{c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.60734, size = 89, normalized size = 2.47 \begin{align*} \frac{2 \, c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + \log \left (c^{2} x^{2} + 1\right )}{2 \, c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13129, size = 61, normalized size = 1.69 \begin{align*} -\frac{{\left | c \right |} \mathrm{sgn}\left (x\right )}{c^{4}} + \frac{2 \, \sqrt{c^{2} x^{2} + 1}{\left | c \right |} \mathrm{sgn}\left (x\right ) + c \log \left (c^{2} x^{2} + 1\right )}{2 \, c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-abs(c)*sgn(x)/c^4 + 1/2*(2*sqrt(c^2*x^2 + 1)*abs(c)*sgn(x) + c*log(c^2*x^2 + 1))/c^4