3.6 \(\int x (a+b \text {csch}^{-1}(c x)) \, dx\)

Optimal. Leaf size=38 \[ \frac {1}{2} x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c} \]

[Out]

1/2*x^2*(a+b*arccsch(c*x))+1/2*b*x*(1+1/c^2/x^2)^(1/2)/c

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6284, 191} \[ \frac {1}{2} x^2 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCsch[c*x]),x]

[Out]

(b*Sqrt[1 + 1/(c^2*x^2)]*x)/(2*c) + (x^2*(a + b*ArcCsch[c*x]))/2

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 6284

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsch[c*
x]))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.32 \[ \frac {a x^2}{2}+\frac {b x \sqrt {\frac {c^2 x^2+1}{c^2 x^2}}}{2 c}+\frac {1}{2} b x^2 \text {csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcCsch[c*x]),x]

[Out]

(a*x^2)/2 + (b*x*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(2*c) + (b*x^2*ArcCsch[c*x])/2

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fricas [B]  time = 0.74, size = 70, normalized size = 1.84 \[ \frac {b c x^{2} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + a c x^{2} + b x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x)),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x)),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x, x)

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maple [A]  time = 0.04, size = 65, normalized size = 1.71 \[ \frac {\frac {c^{2} x^{2} a}{2}+b \left (\frac {c^{2} x^{2} \mathrm {arccsch}\left (c x \right )}{2}+\frac {c^{2} x^{2}+1}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccsch(c*x)),x)

[Out]

1/c^2*(1/2*c^2*x^2*a+b*(1/2*c^2*x^2*arccsch(c*x)+1/2/((c^2*x^2+1)/c^2/x^2)^(1/2)/c/x*(c^2*x^2+1)))

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maxima [A]  time = 0.38, size = 35, normalized size = 0.92 \[ \frac {1}{2} \, a x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x)),x, algorithm="maxima")

[Out]

1/2*a*x^2 + 1/2*(x^2*arccsch(c*x) + x*sqrt(1/(c^2*x^2) + 1)/c)*b

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mupad [B]  time = 2.22, size = 39, normalized size = 1.03 \[ \frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{2}+\frac {b\,x\,\sqrt {\frac {1}{c^2\,x^2}+1}}{2\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asinh(1/(c*x))),x)

[Out]

(a*x^2)/2 + (b*x^2*asinh(1/(c*x)))/2 + (b*x*(1/(c^2*x^2) + 1)^(1/2))/(2*c)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*acsch(c*x)),x)

[Out]

Integral(x*(a + b*acsch(c*x)), x)

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