3.49 \(\int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx\)

Optimal. Leaf size=85 \[ \frac {2 x^3}{3 a^2}+\frac {x^4 \sqrt {\frac {1}{a^2 x^2}+1}}{2 a}-\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{4 a^5}+\frac {x^2 \sqrt {\frac {1}{a^2 x^2}+1}}{4 a^3}+\frac {x^5}{5} \]

[Out]

2/3*x^3/a^2+1/5*x^5-1/4*arctanh((1+1/a^2/x^2)^(1/2))/a^5+1/4*x^2*(1+1/a^2/x^2)^(1/2)/a^3+1/2*x^4*(1+1/a^2/x^2)
^(1/2)/a

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Rubi [A]  time = 0.25, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6338, 6742, 266, 47, 51, 63, 208} \[ \frac {x^4 \sqrt {\frac {1}{a^2 x^2}+1}}{2 a}+\frac {2 x^3}{3 a^2}+\frac {x^2 \sqrt {\frac {1}{a^2 x^2}+1}}{4 a^3}-\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{4 a^5}+\frac {x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcCsch[a*x])*x^4,x]

[Out]

(Sqrt[1 + 1/(a^2*x^2)]*x^2)/(4*a^3) + (2*x^3)/(3*a^2) + (Sqrt[1 + 1/(a^2*x^2)]*x^4)/(2*a) + x^5/5 - ArcTanh[Sq
rt[1 + 1/(a^2*x^2)]]/(4*a^5)

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

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

Rule 6338

Int[E^(ArcCsch[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x] /; FreeQ[m, x] && Int
egerQ[n]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx &=\int \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2 x^4 \, dx\\ &=\int \left (\frac {2 x^2}{a^2}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}} x^3}{a}+x^4\right ) \, dx\\ &=\frac {2 x^3}{3 a^2}+\frac {x^5}{5}+\frac {2 \int \sqrt {1+\frac {1}{a^2 x^2}} x^3 \, dx}{a}\\ &=\frac {2 x^3}{3 a^2}+\frac {x^5}{5}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x^3} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^3}\\ &=\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{4 a^3}+\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{8 a^5}\\ &=\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{4 a^3}+\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}+\frac {\operatorname {Subst}\left (\int \frac {1}{-a^2+a^2 x^2} \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}\right )}{4 a^3}\\ &=\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{4 a^3}+\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}-\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{4 a^5}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 84, normalized size = 0.99 \[ \frac {a^2 x^2 \left (12 a^3 x^3+30 a^2 x^2 \sqrt {\frac {1}{a^2 x^2}+1}+15 \sqrt {\frac {1}{a^2 x^2}+1}+40 a x\right )-15 \log \left (x \left (\sqrt {\frac {1}{a^2 x^2}+1}+1\right )\right )}{60 a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcCsch[a*x])*x^4,x]

[Out]

(a^2*x^2*(15*Sqrt[1 + 1/(a^2*x^2)] + 40*a*x + 30*a^2*Sqrt[1 + 1/(a^2*x^2)]*x^2 + 12*a^3*x^3) - 15*Log[(1 + Sqr
t[1 + 1/(a^2*x^2)])*x])/(60*a^5)

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fricas [A]  time = 0.67, size = 87, normalized size = 1.02 \[ \frac {12 \, a^{5} x^{5} + 40 \, a^{3} x^{3} + 15 \, {\left (2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 15 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{60 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2*x^4,x, algorithm="fricas")

[Out]

1/60*(12*a^5*x^5 + 40*a^3*x^3 + 15*(2*a^4*x^4 + a^2*x^2)*sqrt((a^2*x^2 + 1)/(a^2*x^2)) + 15*log(a*x*sqrt((a^2*
x^2 + 1)/(a^2*x^2)) - a*x))/a^5

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giac [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/a/x+(1+1/a^2/x^2)^(1/2))^2*x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Error: Bad Argument Type

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maple [A]  time = 0.04, size = 114, normalized size = 1.34 \[ \frac {x^{5}}{5}+\frac {2 x^{3}}{3 a^{2}}-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (-2 x \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} a^{4}+x \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right )\right )}{4 a^{5} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x+(1+1/x^2/a^2)^(1/2))^2*x^4,x)

[Out]

1/5*x^5+2/3*x^3/a^2-1/4/a^5*((a^2*x^2+1)/a^2/x^2)^(1/2)*x*(-2*x*((a^2*x^2+1)/a^2)^(3/2)*a^4+x*((a^2*x^2+1)/a^2
)^(1/2)*a^2+ln(x+((a^2*x^2+1)/a^2)^(1/2)))/((a^2*x^2+1)/a^2)^(1/2)

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maxima [A]  time = 0.31, size = 117, normalized size = 1.38 \[ \frac {1}{5} \, x^{5} + \frac {2 \, x^{3}}{3 \, a^{2}} + \frac {\frac {2 \, {\left ({\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \sqrt {\frac {1}{a^{2} x^{2}} + 1}\right )}}{a^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{2} - 2 \, a^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} + a^{4}} - \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{a^{4}} + \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a^{4}}}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2*x^4,x, algorithm="maxima")

[Out]

1/5*x^5 + 2/3*x^3/a^2 + 1/8*(2*((1/(a^2*x^2) + 1)^(3/2) + sqrt(1/(a^2*x^2) + 1))/(a^4*(1/(a^2*x^2) + 1)^2 - 2*
a^4*(1/(a^2*x^2) + 1) + a^4) - log(sqrt(1/(a^2*x^2) + 1) + 1)/a^4 + log(sqrt(1/(a^2*x^2) + 1) - 1)/a^4)/a

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mupad [B]  time = 2.17, size = 73, normalized size = 0.86 \[ \frac {x^5}{5}+\frac {2\,x^3}{3\,a^2}+\frac {x^4\,\sqrt {\frac {1}{a^2\,x^2}+1}}{2\,a}+\frac {x^2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{4\,a^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*((1/(a^2*x^2) + 1)^(1/2) + 1/(a*x))^2,x)

[Out]

(atan((1/(a^2*x^2) + 1)^(1/2)*1i)*1i)/(4*a^5) + x^5/5 + (2*x^3)/(3*a^2) + (x^4*(1/(a^2*x^2) + 1)^(1/2))/(2*a)
+ (x^2*(1/(a^2*x^2) + 1)^(1/2))/(4*a^3)

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sympy [A]  time = 4.72, size = 82, normalized size = 0.96 \[ \frac {x^{5}}{5} + \frac {x^{5}}{2 \sqrt {a^{2} x^{2} + 1}} + \frac {2 x^{3}}{3 a^{2}} + \frac {3 x^{3}}{4 a^{2} \sqrt {a^{2} x^{2} + 1}} + \frac {x}{4 a^{4} \sqrt {a^{2} x^{2} + 1}} - \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x+(1+1/a**2/x**2)**(1/2))**2*x**4,x)

[Out]

x**5/5 + x**5/(2*sqrt(a**2*x**2 + 1)) + 2*x**3/(3*a**2) + 3*x**3/(4*a**2*sqrt(a**2*x**2 + 1)) + x/(4*a**4*sqrt
(a**2*x**2 + 1)) - asinh(a*x)/(4*a**5)

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