3.4.56 \(\int \frac {a+b \text {ArcSin}(c x^2)}{x^2} \, dx\) [356]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {4926, 12, 227} \begin {gather*} 2 b \sqrt {c} F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )-\frac {a+b \text {ArcSin}\left (c x^2\right )}{x} \end {gather*}

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 227

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]

Rule 4926

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]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 44, normalized size = 1.29 \begin {gather*} -\frac {a+b \text {ArcSin}\left (c x^2\right )-2 i b \sqrt {-c} x F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )}{x} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (32 ) = 64\).
time = 0.01, size = 66, normalized size = 1.94

method	result	size
default	\(-\frac {a}{x}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{x}+\frac {2 \sqrt {c}\, \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \EllipticF \left (x \sqrt {c}, i\right )}{\sqrt {-c^{2} x^{4}+1}}\right )\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*c*x*integrate(e^(1/2*log(c*x^2 + 1) + 1/2*log(-c*x^2 + 1))/(c^4*x^8 - c^2*x^4 + (c^2*x^4 - 1)*e^(log(c*x^2
+ 1) + log(-c*x^2 + 1))), x) + arctan2(c*x^2, sqrt(c*x^2 + 1)*sqrt(-c*x^2 + 1)))*b/x - a/x

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Fricas [A]
time = 0.71, size = 45, normalized size = 1.32 \begin {gather*} \frac {b x \arctan \left (\frac {\sqrt {-c^{2} x^{4} + 1}}{c x^{2}}\right ) + {\left (b x - b\right )} \arcsin \left (c x^{2}\right ) - a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*arctan(sqrt(-c^2*x^4 + 1)/(c*x^2)) + (b*x - b)*arcsin(c*x^2) - a)/x

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Sympy [A]
time = 0.81, size = 49, normalized size = 1.44 \begin {gather*} - \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x^2))/x^2,x)

[Out]

int((a + b*asin(c*x^2))/x^2, x)

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