3.4 \(\int x^3 (a+b \text {csch}^{-1}(c x)) \, dx\)

Optimal. Leaf size=62 \[ \frac {1}{4} x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{12 c}-\frac {b x \sqrt {\frac {1}{c^2 x^2}+1}}{6 c^3} \]

[Out]

1/4*x^4*(a+b*arccsch(c*x))-1/6*b*x*(1+1/c^2/x^2)^(1/2)/c^3+1/12*b*x^3*(1+1/c^2/x^2)^(1/2)/c

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Rubi [A]  time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6284, 271, 191} \[ \frac {1}{4} x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{12 c}-\frac {b x \sqrt {\frac {1}{c^2 x^2}+1}}{6 c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcCsch[c*x]),x]

[Out]

-(b*Sqrt[1 + 1/(c^2*x^2)]*x)/(6*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^3)/(12*c) + (x^4*(a + b*ArcCsch[c*x]))/4

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 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 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^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \int \frac {x^2}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{4 c}\\ &=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{4} x^4 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{6 c^3}\\ &=-\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{4} x^4 \left (a+b \text {csch}^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 62, normalized size = 1.00 \[ \frac {a x^4}{4}+b \sqrt {\frac {c^2 x^2+1}{c^2 x^2}} \left (\frac {x^3}{12 c}-\frac {x}{6 c^3}\right )+\frac {1}{4} b x^4 \text {csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*ArcCsch[c*x]),x]

[Out]

(a*x^4)/4 + b*Sqrt[(1 + c^2*x^2)/(c^2*x^2)]*(-1/6*x/c^3 + x^3/(12*c)) + (b*x^4*ArcCsch[c*x])/4

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fricas [A]  time = 3.31, size = 87, normalized size = 1.40 \[ \frac {3 \, b c^{3} x^{4} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 3 \, a c^{3} x^{4} + {\left (b c^{2} x^{3} - 2 \, b x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{12 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsch(c*x)),x, algorithm="fricas")

[Out]

1/12*(3*b*c^3*x^4*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 3*a*c^3*x^4 + (b*c^2*x^3 - 2*b*x)*sqrt(
(c^2*x^2 + 1)/(c^2*x^2)))/c^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsch(c*x)),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x^3, x)

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maple [A]  time = 0.04, size = 74, normalized size = 1.19 \[ \frac {\frac {c^{4} x^{4} a}{4}+b \left (\frac {c^{4} x^{4} \mathrm {arccsch}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}+1\right ) \left (c^{2} x^{2}-2\right )}{12 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccsch(c*x)),x)

[Out]

1/c^4*(1/4*c^4*x^4*a+b*(1/4*c^4*x^4*arccsch(c*x)+1/12*(c^2*x^2+1)*(c^2*x^2-2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/c/x)
)

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maxima [A]  time = 0.32, size = 57, normalized size = 0.92 \[ \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsch(c*x)),x, algorithm="maxima")

[Out]

1/4*a*x^4 + 1/12*(3*x^4*arccsch(c*x) + (c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) - 3*x*sqrt(1/(c^2*x^2) + 1))/c^3)*b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*asinh(1/(c*x))),x)

[Out]

int(x^3*(a + b*asinh(1/(c*x))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*acsch(c*x)),x)

[Out]

Integral(x**3*(a + b*acsch(c*x)), x)

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