3.5.27 \(\int \frac {1}{\sqrt {a-b \text {ArcSin}(1-d x^2)}} \, dx\) [427]

Optimal. Leaf size=201 \[ -\frac {\sqrt {\pi } x S\left (\frac {\sqrt {a-b \text {ArcSin}\left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\sqrt {-b} \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )}-\frac {\sqrt {\pi } x \text {FresnelC}\left (\frac {\sqrt {a-b \text {ArcSin}\left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\sqrt {-b} \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )} \]

[Out]

-x*FresnelS((a+b*arcsin(d*x^2-1))^(1/2)/(-b)^(1/2)/Pi^(1/2))*(cos(1/2*a/b)-sin(1/2*a/b))*Pi^(1/2)/(cos(1/2*arc
sin(d*x^2-1))+sin(1/2*arcsin(d*x^2-1)))/(-b)^(1/2)-x*FresnelC((a+b*arcsin(d*x^2-1))^(1/2)/(-b)^(1/2)/Pi^(1/2))
*(cos(1/2*a/b)+sin(1/2*a/b))*Pi^(1/2)/(cos(1/2*arcsin(d*x^2-1))+sin(1/2*arcsin(d*x^2-1)))/(-b)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {4903} \begin {gather*} -\frac {\sqrt {\pi } x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \text {FresnelC}\left (\frac {\sqrt {a-b \text {ArcSin}\left (1-d x^2\right )}}{\sqrt {\pi } \sqrt {-b}}\right )}{\sqrt {-b} \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 {\pi } x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) S\left (\frac {\sqrt {a-b \text {ArcSin}\left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right )}{\sqrt {-b} \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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a - b*ArcSin[1 - d*x^2]],x]

[Out]

-((Sqrt[Pi]*x*FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt
[-b]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))) - (Sqrt[Pi]*x*FresnelC[Sqrt[a - b*ArcSin[1 - d*x^
2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[-b]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d
*x^2]/2]))

Rule 4903

Int[1/Sqrt[(a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(-Sqrt[Pi])*x*(Cos[a/(2*b)] - c*Sin[a
/(2*b)])*(FresnelC[(1/(Sqrt[b*c]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2
] - c*Sin[ArcSin[c + d*x^2]/2]))), x] - Simp[Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/(2*b)])*(FresnelS[(1/(Sqrt[b*c
]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Sqrt[b*c]*(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}{\sqrt {a-b \sin ^{-1}\left (1-d x^2\right )}} \, dx &=-\frac {\sqrt {\pi } x S\left (\frac {\sqrt {a-b \sin ^{-1}\left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\sqrt {-b} \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 {\sqrt {\pi } x C\left (\frac {\sqrt {a-b \sin ^{-1}\left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\sqrt {-b} \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 = 0.04, size = 155, normalized size = 0.77 \begin {gather*} \frac {b \sqrt {\pi } x \left (S\left (\frac {\sqrt {a-b \text {ArcSin}\left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )+\text {FresnelC}\left (\frac {\sqrt {a-b \text {ArcSin}\left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )\right )}{(-b)^{3/2} \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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a - b*ArcSin[1 - d*x^2]],x]

[Out]

(b*Sqrt[Pi]*x*(FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]) + Fre
snelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)])))/((-b)^(3/2)*(Cos[ArcS
in[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a +b \arcsin \left (d \,x^{2}-1\right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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