3.5.13 \(\int \frac {1}{(a-b \text {ArcSin}(1-d x^2))^2} \, dx\) [413]

Optimal. Leaf size=216 \[ -\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )}-\frac {x \text {CosIntegral}\left (-\frac {a-b \text {ArcSin}\left (1-d x^2\right )}{2 b}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )}-\frac {x \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a}{2 b}-\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )} \]

[Out]

-1/4*x*Ci(1/2*(-a-b*arcsin(d*x^2-1))/b)*(cos(1/2*a/b)-sin(1/2*a/b))/b^2/(cos(1/2*arcsin(d*x^2-1))+sin(1/2*arcs
in(d*x^2-1)))-1/4*x*Si(1/2*a/b+1/2*arcsin(d*x^2-1))*(cos(1/2*a/b)+sin(1/2*a/b))/b^2/(cos(1/2*arcsin(d*x^2-1))+
sin(1/2*arcsin(d*x^2-1)))-1/2*(-d^2*x^4+2*d*x^2)^(1/2)/b/d/x/(a+b*arcsin(d*x^2-1))

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Rubi [A]
time = 0.02, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4909} \begin {gather*} -\frac {x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) \text {CosIntegral}\left (-\frac {a-b \text {ArcSin}\left (1-d x^2\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )}-\frac {x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a}{2 b}-\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )}-\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - b*ArcSin[1 - d*x^2])^(-2),x]

[Out]

-1/2*Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*(a - b*ArcSin[1 - d*x^2])) - (x*CosIntegral[-1/2*(a - b*ArcSin[1 - d*x^2])
/b]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(4*b^2*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])) - (x*(Cos[a/(
2*b)] + Sin[a/(2*b)])*SinIntegral[a/(2*b) - ArcSin[1 - d*x^2]/2])/(4*b^2*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSi
n[1 - d*x^2]/2]))

Rule 4909

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(-2), x_Symbol] :> Simp[-Sqrt[-2*c*d*x^2 - d^2*x^4]/(2*b*d*x*(
a + b*ArcSin[c + d*x^2])), x] + (-Simp[x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*(CosIntegral[(c/(2*b))*(a + b*ArcSin[
c + d*x^2])]/(4*b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))), x] + Simp[x*(Cos[a/(2*b)] - c*S
in[a/(2*b)])*(SinIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])]/(4*b^2*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin
[c + d*x^2]/2]))), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2} \, dx &=-\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}-\frac {x \text {Ci}\left (-\frac {a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}-\frac {x \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a}{2 b}-\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 183, normalized size = 0.85 \begin {gather*} \frac {2 b \sqrt {d x^2 \left (2-d x^2\right )}+\left (a-b \text {ArcSin}\left (1-d x^2\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right ) \left (\text {CosIntegral}\left (\frac {1}{2} \left (-\frac {a}{b}+\text {ArcSin}\left (1-d x^2\right )\right )\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )+\left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a-b \text {ArcSin}\left (1-d x^2\right )}{2 b}\right )\right )}{4 b^2 d x \left (-a+b \text {ArcSin}\left (1-d x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a - b*ArcSin[1 - d*x^2])^(-2),x]

[Out]

(2*b*Sqrt[d*x^2*(2 - d*x^2)] + (a - b*ArcSin[1 - d*x^2])*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])
*(CosIntegral[(-(a/b) + ArcSin[1 - d*x^2])/2]*(Cos[a/(2*b)] - Sin[a/(2*b)]) + (Cos[a/(2*b)] + Sin[a/(2*b)])*Si
nIntegral[(a - b*ArcSin[1 - d*x^2])/(2*b)]))/(4*b^2*d*x*(-a + b*ArcSin[1 - d*x^2]))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arcsin \left (d \,x^{2}-1\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arcsin(d*x^2-1))^2,x)

[Out]

int(1/(a+b*arcsin(d*x^2-1))^2,x)

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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(1/(a+b*arcsin(d*x^2-1))^2,x, algorithm="maxima")

[Out]

1/2*(2*(b^2*d*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x) + a*b*d)*sqrt(d)*integrate(1/2*sqrt(-d*x^2 + 2)*x
/(a*b*d*x^2 - 2*a*b + (b^2*d*x^2 - 2*b^2)*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x)), x) - sqrt(-d*x^2 +
2)*sqrt(d))/(b^2*d*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x) + a*b*d)

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Fricas [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(1/(a+b*arcsin(d*x^2-1))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*arcsin(d*x^2 - 1)^2 + 2*a*b*arcsin(d*x^2 - 1) + a^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*asin(d*x**2-1))**2,x)

[Out]

Integral((a + b*asin(d*x**2 - 1))**(-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(1/(a+b*arcsin(d*x^2-1))^2,x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x^2 - 1) + a)^(-2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*asin(d*x^2 - 1))^2,x)

[Out]

int(1/(a + b*asin(d*x^2 - 1))^2, x)

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