3.2 \(\int x^5 (a+b \text{csch}^{-1}(c x)) \, dx\)

Optimal. Leaf size=86 \[ \frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^5 \sqrt{\frac{1}{c^2 x^2}+1}}{30 c}-\frac{2 b x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^3}+\frac{4 b x \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^5} \]

[Out]

(4*b*Sqrt[1 + 1/(c^2*x^2)]*x)/(45*c^5) - (2*b*Sqrt[1 + 1/(c^2*x^2)]*x^3)/(45*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x
^5)/(30*c) + (x^6*(a + b*ArcCsch[c*x]))/6

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Rubi [A]  time = 0.0441296, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6284, 271, 191} \[ \frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^5 \sqrt{\frac{1}{c^2 x^2}+1}}{30 c}-\frac{2 b x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^3}+\frac{4 b x \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^5} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcCsch[c*x]),x]

[Out]

(4*b*Sqrt[1 + 1/(c^2*x^2)]*x)/(45*c^5) - (2*b*Sqrt[1 + 1/(c^2*x^2)]*x^3)/(45*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x
^5)/(30*c) + (x^6*(a + b*ArcCsch[c*x]))/6

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]

Rule 271

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

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]

Rubi steps

\begin{align*} \int x^5 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b \int \frac{x^4}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{6 c}\\ &=\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(2 b) \int \frac{x^2}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{15 c^3}\\ &=-\frac{2 b \sqrt{1+\frac{1}{c^2 x^2}} x^3}{45 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{(4 b) \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{45 c^5}\\ &=\frac{4 b \sqrt{1+\frac{1}{c^2 x^2}} x}{45 c^5}-\frac{2 b \sqrt{1+\frac{1}{c^2 x^2}} x^3}{45 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.127229, size = 72, normalized size = 0.84 \[ \frac{a x^6}{6}+b \sqrt{\frac{c^2 x^2+1}{c^2 x^2}} \left (-\frac{2 x^3}{45 c^3}+\frac{4 x}{45 c^5}+\frac{x^5}{30 c}\right )+\frac{1}{6} b x^6 \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcCsch[c*x]),x]

[Out]

(a*x^6)/6 + b*Sqrt[(1 + c^2*x^2)/(c^2*x^2)]*((4*x)/(45*c^5) - (2*x^3)/(45*c^3) + x^5/(30*c)) + (b*x^6*ArcCsch[
c*x])/6

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Maple [A]  time = 0.184, size = 83, normalized size = 1. \begin{align*}{\frac{1}{{c}^{6}} \left ({\frac{{c}^{6}{x}^{6}a}{6}}+b \left ({\frac{{c}^{6}{x}^{6}{\rm arccsch} \left (cx\right )}{6}}+{\frac{ \left ({c}^{2}{x}^{2}+1 \right ) \left ( 3\,{c}^{4}{x}^{4}-4\,{c}^{2}{x}^{2}+8 \right ) }{90\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arccsch(c*x)),x)

[Out]

1/c^6*(1/6*c^6*x^6*a+b*(1/6*c^6*x^6*arccsch(c*x)+1/90*(c^2*x^2+1)*(3*c^4*x^4-4*c^2*x^2+8)/((c^2*x^2+1)/c^2/x^2
)^(1/2)/c/x))

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Maxima [A]  time = 1.00425, size = 104, normalized size = 1.21 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{90} \,{\left (15 \, x^{6} \operatorname{arcsch}\left (c x\right ) + \frac{3 \, c^{4} x^{5}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 10 \, c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccsch(c*x)),x, algorithm="maxima")

[Out]

1/6*a*x^6 + 1/90*(15*x^6*arccsch(c*x) + (3*c^4*x^5*(1/(c^2*x^2) + 1)^(5/2) - 10*c^2*x^3*(1/(c^2*x^2) + 1)^(3/2
) + 15*x*sqrt(1/(c^2*x^2) + 1))/c^5)*b

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Fricas [A]  time = 2.19157, size = 215, normalized size = 2.5 \begin{align*} \frac{15 \, b c^{5} x^{6} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 15 \, a c^{5} x^{6} +{\left (3 \, b c^{4} x^{5} - 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{90 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccsch(c*x)),x, algorithm="fricas")

[Out]

1/90*(15*b*c^5*x^6*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 15*a*c^5*x^6 + (3*b*c^4*x^5 - 4*b*c^2*
x^3 + 8*b*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/c^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \left (a + b \operatorname{acsch}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*acsch(c*x)),x)

[Out]

Integral(x**5*(a + b*acsch(c*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccsch(c*x)),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x^5, x)