### 3.38 $$\int e^{\text{csch}^{-1}(a x^2)} x^4 \, dx$$

Optimal. Leaf size=202 $-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{5 a^{7/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{1}{5} x^5 \sqrt{\frac{1}{a^2 x^4}+1}+\frac{2 x \sqrt{\frac{1}{a^2 x^4}+1}}{5 a^2}-\frac{2 \sqrt{\frac{1}{a^2 x^4}+1}}{5 a^2 x \left (a+\frac{1}{x^2}\right )}+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{x^3}{3 a}$

[Out]

(-2*Sqrt[1 + 1/(a^2*x^4)])/(5*a^2*(a + x^(-2))*x) + (2*Sqrt[1 + 1/(a^2*x^4)]*x)/(5*a^2) + x^3/(3*a) + (Sqrt[1
+ 1/(a^2*x^4)]*x^5)/5 + (2*Sqrt[(a^2 + x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2
])/(5*a^(7/2)*Sqrt[1 + 1/(a^2*x^4)]) - (Sqrt[(a^2 + x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticF[2*ArcCot[Sq
rt[a]*x], 1/2])/(5*a^(7/2)*Sqrt[1 + 1/(a^2*x^4)])

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Rubi [A]  time = 0.124744, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.667, Rules used = {6336, 30, 335, 277, 325, 305, 220, 1196} $\frac{1}{5} x^5 \sqrt{\frac{1}{a^2 x^4}+1}+\frac{2 x \sqrt{\frac{1}{a^2 x^4}+1}}{5 a^2}-\frac{2 \sqrt{\frac{1}{a^2 x^4}+1}}{5 a^2 x \left (a+\frac{1}{x^2}\right )}-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{x^3}{3 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*Sqrt[1 + 1/(a^2*x^4)])/(5*a^2*(a + x^(-2))*x) + (2*Sqrt[1 + 1/(a^2*x^4)]*x)/(5*a^2) + x^3/(3*a) + (Sqrt[1
+ 1/(a^2*x^4)]*x^5)/5 + (2*Sqrt[(a^2 + x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2
])/(5*a^(7/2)*Sqrt[1 + 1/(a^2*x^4)]) - (Sqrt[(a^2 + x^(-4))/(a + x^(-2))^2]*(a + x^(-2))*EllipticF[2*ArcCot[Sq
rt[a]*x], 1/2])/(5*a^(7/2)*Sqrt[1 + 1/(a^2*x^4)])

Rule 6336

Int[E^ArcCsch[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Dist[1/a, Int[x^(m - p), x], x] + Int[x^m*Sqrt[1 + 1/
(a^2*x^(2*p))], x] /; FreeQ[{a, m, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 277

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int e^{\text{csch}^{-1}\left (a x^2\right )} x^4 \, dx &=\frac{\int x^2 \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} x^4 \, dx\\ &=\frac{x^3}{3 a}-\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^4}{a^2}}}{x^6} \, dx,x,\frac{1}{x}\right )\\ &=\frac{x^3}{3 a}+\frac{1}{5} \sqrt{1+\frac{1}{a^2 x^4}} x^5-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 a^2}\\ &=\frac{2 \sqrt{1+\frac{1}{a^2 x^4}} x}{5 a^2}+\frac{x^3}{3 a}+\frac{1}{5} \sqrt{1+\frac{1}{a^2 x^4}} x^5-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 a^4}\\ &=\frac{2 \sqrt{1+\frac{1}{a^2 x^4}} x}{5 a^2}+\frac{x^3}{3 a}+\frac{1}{5} \sqrt{1+\frac{1}{a^2 x^4}} x^5-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 a^3}+\frac{2 \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{a}}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 a^3}\\ &=-\frac{2 \sqrt{1+\frac{1}{a^2 x^4}}}{5 a^2 \left (a+\frac{1}{x^2}\right ) x}+\frac{2 \sqrt{1+\frac{1}{a^2 x^4}} x}{5 a^2}+\frac{x^3}{3 a}+\frac{1}{5} \sqrt{1+\frac{1}{a^2 x^4}} x^5+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{1+\frac{1}{a^2 x^4}}}-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{1+\frac{1}{a^2 x^4}}}\\ \end{align*}

Mathematica [C]  time = 0.248762, size = 112, normalized size = 0.55 $\frac{4 \sqrt{2} x^5 e^{-\text{csch}^{-1}\left (a x^2\right )} \left (\frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{e^{2 \text{csch}^{-1}\left (a x^2\right )}-1}\right )^{5/2} \left (4 \left (1-e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )^{5/2} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{7}{2},\frac{7}{4},e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )+7 e^{2 \text{csch}^{-1}\left (a x^2\right )}-4\right )}{21 \left (a x^2\right )^{5/2}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*Sqrt[2]*(E^ArcCsch[a*x^2]/(-1 + E^(2*ArcCsch[a*x^2])))^(5/2)*x^5*(-4 + 7*E^(2*ArcCsch[a*x^2]) + 4*(1 - E^(2
*ArcCsch[a*x^2]))^(5/2)*Hypergeometric2F1[3/4, 7/2, 7/4, E^(2*ArcCsch[a*x^2])]))/(21*E^ArcCsch[a*x^2]*(a*x^2)^
(5/2))

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Maple [C]  time = 0.211, size = 150, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}}{5\,a \left ({a}^{2}{x}^{4}+1 \right ) }\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( \sqrt{ia}{x}^{7}{a}^{3}+{x}^{3}a\sqrt{ia}+2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{\it EllipticF} \left ( x\sqrt{ia},i \right ) -2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{\it EllipticE} \left ( x\sqrt{ia},i \right ) \right ){\frac{1}{\sqrt{ia}}}}+{\frac{{x}^{3}}{3\,a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*((a^2*x^4+1)/a^2/x^4)^(1/2)*x^2*((I*a)^(1/2)*x^7*a^3+x^3*a*(I*a)^(1/2)+2*I*(1-I*a*x^2)^(1/2)*(1+I*a*x^2)^(
1/2)*EllipticF(x*(I*a)^(1/2),I)-2*I*(1-I*a*x^2)^(1/2)*(1+I*a*x^2)^(1/2)*EllipticE(x*(I*a)^(1/2),I))/a/(a^2*x^4
+1)/(I*a)^(1/2)+1/3*x^3/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x^{3}}{3 \, a} + \frac{\frac{x^{3} \Gamma \left (\frac{3}{4}\right ) \,_2F_1\left (\begin{matrix} -\frac{1}{2},\frac{3}{4} \\ \frac{7}{4} \end{matrix} ; -a^{2} x^{4} \right )}{4 \, \Gamma \left (\frac{7}{4}\right )}}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3/a + integrate(sqrt(a^2*x^4 + 1)*x^2, x)/a

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a x^{4} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + x^{2}}{a}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 3.35218, size = 48, normalized size = 0.24 \begin{align*} - \frac{x^{5} \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 \Gamma \left (- \frac{1}{4}\right )} + \frac{x^{3}}{3 a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**5*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), exp_polar(I*pi)/(a**2*x**4))/(4*gamma(-1/4)) + x**3/(3*a)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4}{\left (\sqrt{\frac{1}{a^{2} x^{4}} + 1} + \frac{1}{a x^{2}}\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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