3.286 \(\int \sinh ^{-1}(a x^2) \, dx\)

Optimal. Leaf size=162 \[ -\frac{\left (a x^2+1\right ) \sqrt{\frac{a^2 x^4+1}{\left (a x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{\sqrt{a} \sqrt{a^2 x^4+1}}-\frac{2 x \sqrt{a^2 x^4+1}}{a x^2+1}+\frac{2 \left (a x^2+1\right ) \sqrt{\frac{a^2 x^4+1}{\left (a x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{\sqrt{a} \sqrt{a^2 x^4+1}}+x \sinh ^{-1}\left (a x^2\right ) \]

[Out]

(-2*x*Sqrt[1 + a^2*x^4])/(1 + a*x^2) + x*ArcSinh[a*x^2] + (2*(1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]*Ell
ipticE[2*ArcTan[Sqrt[a]*x], 1/2])/(Sqrt[a]*Sqrt[1 + a^2*x^4]) - ((1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]
*EllipticF[2*ArcTan[Sqrt[a]*x], 1/2])/(Sqrt[a]*Sqrt[1 + a^2*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.054459, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5900, 12, 305, 220, 1196} \[ -\frac{2 x \sqrt{a^2 x^4+1}}{a x^2+1}-\frac{\left (a x^2+1\right ) \sqrt{\frac{a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{\sqrt{a} \sqrt{a^2 x^4+1}}+\frac{2 \left (a x^2+1\right ) \sqrt{\frac{a^2 x^4+1}{\left (a x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{\sqrt{a} \sqrt{a^2 x^4+1}}+x \sinh ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x^2],x]

[Out]

(-2*x*Sqrt[1 + a^2*x^4])/(1 + a*x^2) + x*ArcSinh[a*x^2] + (2*(1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]*Ell
ipticE[2*ArcTan[Sqrt[a]*x], 1/2])/(Sqrt[a]*Sqrt[1 + a^2*x^4]) - ((1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]
*EllipticF[2*ArcTan[Sqrt[a]*x], 1/2])/(Sqrt[a]*Sqrt[1 + a^2*x^4])

Rule 5900

Int[ArcSinh[u_], x_Symbol] :> Simp[x*ArcSinh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 + u^2], x], x]
/; InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 \sinh ^{-1}\left (a x^2\right ) \, dx &=x \sinh ^{-1}\left (a x^2\right )-\int \frac{2 a x^2}{\sqrt{1+a^2 x^4}} \, dx\\ &=x \sinh ^{-1}\left (a x^2\right )-(2 a) \int \frac{x^2}{\sqrt{1+a^2 x^4}} \, dx\\ &=x \sinh ^{-1}\left (a x^2\right )-2 \int \frac{1}{\sqrt{1+a^2 x^4}} \, dx+2 \int \frac{1-a x^2}{\sqrt{1+a^2 x^4}} \, dx\\ &=-\frac{2 x \sqrt{1+a^2 x^4}}{1+a x^2}+x \sinh ^{-1}\left (a x^2\right )+\frac{2 \left (1+a x^2\right ) \sqrt{\frac{1+a^2 x^4}{\left (1+a x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{\sqrt{a} \sqrt{1+a^2 x^4}}-\frac{\left (1+a x^2\right ) \sqrt{\frac{1+a^2 x^4}{\left (1+a x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{\sqrt{a} \sqrt{1+a^2 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0047616, size = 35, normalized size = 0.22 \[ x \sinh ^{-1}\left (a x^2\right )-\frac{2}{3} a x^3 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-a^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x^2],x]

[Out]

x*ArcSinh[a*x^2] - (2*a*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -(a^2*x^4)])/3

________________________________________________________________________________________

Maple [C]  time = 0.005, size = 77, normalized size = 0.5 \begin{align*} x{\it Arcsinh} \left ( a{x}^{2} \right ) -{2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}} \left ({\it EllipticF} \left ( x\sqrt{ia},i \right ) -{\it EllipticE} \left ( x\sqrt{ia},i \right ) \right ){\frac{1}{\sqrt{ia}}}{\frac{1}{\sqrt{{a}^{2}{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -2 \, a \int \frac{x^{2}}{a^{3} x^{6} + a x^{2} +{\left (a^{2} x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} + x \log \left (a x^{2} + \sqrt{a^{2} x^{4} + 1}\right ) - \frac{i \, \sqrt{2}{\left (\log \left (\frac{i \, \sqrt{2}{\left (2 \, \sqrt{a^{2}} x + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + 1\right ) - \log \left (-\frac{i \, \sqrt{2}{\left (2 \, \sqrt{a^{2}} x + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + 1\right )\right )}}{4 \,{\left (a^{2}\right )}^{\frac{1}{4}}} - \frac{i \, \sqrt{2}{\left (\log \left (\frac{i \, \sqrt{2}{\left (2 \, \sqrt{a^{2}} x - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + 1\right ) - \log \left (-\frac{i \, \sqrt{2}{\left (2 \, \sqrt{a^{2}} x - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + 1\right )\right )}}{4 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + \frac{\sqrt{2} \log \left (\sqrt{a^{2}} x^{2} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x + 1\right )}{4 \,{\left (a^{2}\right )}^{\frac{1}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{a^{2}} x^{2} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x + 1\right )}{4 \,{\left (a^{2}\right )}^{\frac{1}{4}}} - 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x^2),x, algorithm="maxima")

[Out]

-2*a*integrate(x^2/(a^3*x^6 + a*x^2 + (a^2*x^4 + 1)^(3/2)), x) + x*log(a*x^2 + sqrt(a^2*x^4 + 1)) - 1/4*I*sqrt
(2)*(log(1/2*I*sqrt(2)*(2*sqrt(a^2)*x + sqrt(2)*(a^2)^(1/4))/(a^2)^(1/4) + 1) - log(-1/2*I*sqrt(2)*(2*sqrt(a^2
)*x + sqrt(2)*(a^2)^(1/4))/(a^2)^(1/4) + 1))/(a^2)^(1/4) - 1/4*I*sqrt(2)*(log(1/2*I*sqrt(2)*(2*sqrt(a^2)*x - s
qrt(2)*(a^2)^(1/4))/(a^2)^(1/4) + 1) - log(-1/2*I*sqrt(2)*(2*sqrt(a^2)*x - sqrt(2)*(a^2)^(1/4))/(a^2)^(1/4) +
1))/(a^2)^(1/4) + 1/4*sqrt(2)*log(sqrt(a^2)*x^2 + sqrt(2)*(a^2)^(1/4)*x + 1)/(a^2)^(1/4) - 1/4*sqrt(2)*log(sqr
t(a^2)*x^2 - sqrt(2)*(a^2)^(1/4)*x + 1)/(a^2)^(1/4) - 2*x

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x^2),x, algorithm="fricas")

[Out]

integral(arcsinh(a*x^2), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{asinh}{\left (a x^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x**2),x)

[Out]

Integral(asinh(a*x**2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x^2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x^2), x)