3.356 \(\int \frac{a+b \sin ^{-1}(c x^2)}{x^2} \, dx\)

Optimal. Leaf size=34 \[ 2 b \sqrt{c} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{c} x\right ),-1\right )-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x} \]

[Out]

-((a + b*ArcSin[c*x^2])/x) + 2*b*Sqrt[c]*EllipticF[ArcSin[Sqrt[c]*x], -1]

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Rubi [A]  time = 0.0224676, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4842, 12, 221} \[ 2 b \sqrt{c} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x^2])/x^2,x]

[Out]

-((a + b*ArcSin[c*x^2])/x) + 2*b*Sqrt[c]*EllipticF[ArcSin[Sqrt[c]*x], -1]

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), 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 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}\left (c x^2\right )}{x^2} \, dx &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x}+b \int \frac{2 c}{\sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x}+(2 b c) \int \frac{1}{\sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{x}+2 b \sqrt{c} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0521546, size = 44, normalized size = 1.29 \[ -\frac{-2 i b \sqrt{-c} x \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c} x\right ),-1\right )+a+b \sin ^{-1}\left (c x^2\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x^2])/x^2,x]

[Out]

-((a + b*ArcSin[c*x^2] - (2*I)*b*Sqrt[-c]*x*EllipticF[I*ArcSinh[Sqrt[-c]*x], -1])/x)

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Maple [B]  time = 0.007, size = 66, normalized size = 1.9 \begin{align*} -{\frac{a}{x}}+b \left ( -{\frac{\arcsin \left ( c{x}^{2} \right ) }{x}}+2\,{\frac{\sqrt{c}\sqrt{-c{x}^{2}+1}\sqrt{c{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{c},i \right ) }{\sqrt{-{c}^{2}{x}^{4}+1}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x^2))/x^2,x)

[Out]

-a/x+b*(-1/x*arcsin(c*x^2)+2*c^(1/2)*(-c*x^2+1)^(1/2)*(c*x^2+1)^(1/2)/(-c^2*x^4+1)^(1/2)*EllipticF(x*c^(1/2),I
))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^2,x, algorithm="fricas")

[Out]

integral((b*arcsin(c*x^2) + a)/x^2, x)

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Sympy [A]  time = 1.39979, size = 49, normalized size = 1.44 \begin{align*} - \frac{a}{x} + \frac{b c x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{c^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{5}{4}\right )} - \frac{b \operatorname{asin}{\left (c x^{2} \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x**2))/x**2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^2,x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x^2) + a)/x^2, x)