3.414 \(\int \frac{1}{(a-b \sin ^{-1}(1-d x^2))^3} \, dx\)

Optimal. Leaf size=240 \[ -\frac{x \left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) \text{CosIntegral}\left (-\frac{a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right )}{16 b^3 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}+\frac{x \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right ) \text{Si}\left (\frac{a}{2 b}-\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )}{16 b^3 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}+\frac{x}{8 b^2 \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}-\frac{\sqrt{2 d x^2-d^2 x^4}}{4 b d x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2} \]

[Out]

-Sqrt[2*d*x^2 - d^2*x^4]/(4*b*d*x*(a - b*ArcSin[1 - d*x^2])^2) + x/(8*b^2*(a - b*ArcSin[1 - d*x^2])) - (x*CosI
ntegral[-(a - b*ArcSin[1 - d*x^2])/(2*b)]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(16*b^3*(Cos[ArcSin[1 - d*x^2]/2] - S
in[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])/(16*b
^3*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))

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Rubi [A]  time = 0.0470692, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4828, 4816} \[ -\frac{x \left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) \text{CosIntegral}\left (-\frac{a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right )}{16 b^3 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}+\frac{x \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right ) \text{Si}\left (\frac{a}{2 b}-\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )}{16 b^3 \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}+\frac{x}{8 b^2 \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}-\frac{\sqrt{2 d x^2-d^2 x^4}}{4 b d x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*ArcSin[1 - d*x^2])^(-3),x]

[Out]

-Sqrt[2*d*x^2 - d^2*x^4]/(4*b*d*x*(a - b*ArcSin[1 - d*x^2])^2) + x/(8*b^2*(a - b*ArcSin[1 - d*x^2])) - (x*CosI
ntegral[-(a - b*ArcSin[1 - d*x^2])/(2*b)]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(16*b^3*(Cos[ArcSin[1 - d*x^2]/2] - S
in[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])/(16*b
^3*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))

Rule 4828

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[(x*(a + b*ArcSin[c + d*x^2])^(n + 2))/
(4*b^2*(n + 1)*(n + 2)), x] + (-Dist[1/(4*b^2*(n + 1)*(n + 2)), Int[(a + b*ArcSin[c + d*x^2])^(n + 2), x], x]
+ Simp[(Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcSin[c + d*x^2])^(n + 1))/(2*b*d*(n + 1)*x), x]) /; FreeQ[{a, b, c
, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]

Rule 4816

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(-1), x_Symbol] :> -Simp[(x*(c*Cos[a/(2*b)] - Sin[a/(2*b)])*Co
sIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])])/(2*b*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2])),
x] - Simp[(x*(c*Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(c/(2*b))*(a + b*ArcSin[c + d*x^2])])/(2*b*(Cos[ArcS
in[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 )^3} \, dx &=-\frac{\sqrt{2 d x^2-d^2 x^4}}{4 b d x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2}+\frac{x}{8 b^2 \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}-\frac{\int \frac{1}{a-b \sin ^{-1}\left (1-d x^2\right )} \, dx}{8 b^2}\\ &=-\frac{\sqrt{2 d x^2-d^2 x^4}}{4 b d x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2}+\frac{x}{8 b^2 \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 )}{16 b^3 \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 )}{16 b^3 \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*}

Mathematica [A]  time = 0.476581, size = 195, normalized size = 0.81 \[ -\frac{\frac{4 b^2 \sqrt{-d x^2 \left (d x^2-2\right )}}{d \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2}+\frac{\left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right ) \left (\left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) \text{CosIntegral}\left (\frac{1}{2} \left (\sin ^{-1}\left (1-d x^2\right )-\frac{a}{b}\right )\right )+\left (\sin \left (\frac{a}{2 b}\right )-\cos \left (\frac{a}{2 b}\right )\right ) \text{Si}\left (\frac{a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right )\right )}{d}-\frac{2 b x^2}{a-b \sin ^{-1}\left (1-d x^2\right )}}{16 b^3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*ArcSin[1 - d*x^2])^(-3),x]

[Out]

-((4*b^2*Sqrt[-(d*x^2*(-2 + d*x^2))])/(d*(a - b*ArcSin[1 - d*x^2])^2) - (2*b*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)])*SinIntegral[(a - b*ArcSin[1 - d*x^2])/(2*b)]))/d)/(16*b^3
*x)

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arcsin \left ( d{x}^{2}-1 \right ) \right ) ^{-3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arcsin(d*x^2-1))^3,x)

[Out]

int(1/(a+b*arcsin(d*x^2-1))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b d x \arctan \left (d x^{2} - 1, \sqrt{-d x^{2} + 2} \sqrt{d} x\right ) + a d x - 2 \, \sqrt{-d x^{2} + 2} b \sqrt{d} -{\left (b^{4} d \arctan \left (d x^{2} - 1, \sqrt{-d x^{2} + 2} \sqrt{d} x\right )^{2} + 2 \, a b^{3} d \arctan \left (d x^{2} - 1, \sqrt{-d x^{2} + 2} \sqrt{d} x\right ) + a^{2} b^{2} d\right )} \int \frac{1}{b^{3} \arctan \left (d x^{2} - 1, \sqrt{-d x^{2} + 2} \sqrt{d} x\right ) + a b^{2}}\,{d x}}{8 \,{\left (b^{4} d \arctan \left (d x^{2} - 1, \sqrt{-d x^{2} + 2} \sqrt{d} x\right )^{2} + 2 \, a b^{3} d \arctan \left (d x^{2} - 1, \sqrt{-d x^{2} + 2} \sqrt{d} x\right ) + a^{2} b^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arcsin(d*x^2-1))^3,x, algorithm="maxima")

[Out]

1/8*(b*d*x*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x) + a*d*x - 2*sqrt(-d*x^2 + 2)*b*sqrt(d) - 8*(b^4*d*ar
ctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x)^2 + 2*a*b^3*d*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x) + a^
2*b^2*d)*integrate(1/8/(b^3*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x) + a*b^2), x))/(b^4*d*arctan2(d*x^2
- 1, sqrt(-d*x^2 + 2)*sqrt(d)*x)^2 + 2*a*b^3*d*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x) + a^2*b^2*d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} \arcsin \left (d x^{2} - 1\right )^{3} + 3 \, a b^{2} \arcsin \left (d x^{2} - 1\right )^{2} + 3 \, a^{2} b \arcsin \left (d x^{2} - 1\right ) + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arcsin(d*x^2-1))^3,x, algorithm="fricas")

[Out]

integral(1/(b^3*arcsin(d*x^2 - 1)^3 + 3*a*b^2*arcsin(d*x^2 - 1)^2 + 3*a^2*b*arcsin(d*x^2 - 1) + a^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{asin}{\left (d x^{2} - 1 \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*asin(d*x**2-1))**3,x)

[Out]

Integral((a + b*asin(d*x**2 - 1))**(-3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arcsin(d*x^2-1))^3,x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x^2 - 1) + a)^(-3), x)