3.100 \(\int \frac{1}{(a+b \cos ^{-1}(-1+d x^2))^{7/2}} \, dx\)

Optimal. Leaf size=269 \[ -\frac{\sqrt{2 d x^2-d^2 x^4}}{15 b^3 d x \sqrt{a+b \cos ^{-1}\left (d x^2-1\right )}}+\frac{x}{15 b^2 \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^{3/2}}+\frac{\sqrt{2 d x^2-d^2 x^4}}{5 b d x \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^{5/2}}+\frac{2 \sqrt{\pi } \left (\frac{1}{b}\right )^{7/2} \cos \left (\frac{a}{2 b}\right ) \cos \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left (d x^2-1\right )}}{\sqrt{\pi }}\right )}{15 d x}+\frac{2 \sqrt{\pi } \left (\frac{1}{b}\right )^{7/2} \sin \left (\frac{a}{2 b}\right ) \cos \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right ) S\left (\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left (d x^2-1\right )}}{\sqrt{\pi }}\right )}{15 d x} \]

[Out]

Sqrt[2*d*x^2 - d^2*x^4]/(5*b*d*x*(a + b*ArcCos[-1 + d*x^2])^(5/2)) + x/(15*b^2*(a + b*ArcCos[-1 + d*x^2])^(3/2
)) - Sqrt[2*d*x^2 - d^2*x^4]/(15*b^3*d*x*Sqrt[a + b*ArcCos[-1 + d*x^2]]) + (2*(b^(-1))^(7/2)*Sqrt[Pi]*Cos[a/(2
*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(15*d*x) + (2
*(b^(-1))^(7/2)*Sqrt[Pi]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt
[Pi]]*Sin[a/(2*b)])/(15*d*x)

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Rubi [A]  time = 0.0500292, antiderivative size = 269, 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 = {4829, 4824} \[ -\frac{\sqrt{2 d x^2-d^2 x^4}}{15 b^3 d x \sqrt{a+b \cos ^{-1}\left (d x^2-1\right )}}+\frac{x}{15 b^2 \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^{3/2}}+\frac{\sqrt{2 d x^2-d^2 x^4}}{5 b d x \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^{5/2}}+\frac{2 \sqrt{\pi } \left (\frac{1}{b}\right )^{7/2} \cos \left (\frac{a}{2 b}\right ) \cos \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left (d x^2-1\right )}}{\sqrt{\pi }}\right )}{15 d x}+\frac{2 \sqrt{\pi } \left (\frac{1}{b}\right )^{7/2} \sin \left (\frac{a}{2 b}\right ) \cos \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right ) S\left (\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left (d x^2-1\right )}}{\sqrt{\pi }}\right )}{15 d x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[-1 + d*x^2])^(-7/2),x]

[Out]

Sqrt[2*d*x^2 - d^2*x^4]/(5*b*d*x*(a + b*ArcCos[-1 + d*x^2])^(5/2)) + x/(15*b^2*(a + b*ArcCos[-1 + d*x^2])^(3/2
)) - Sqrt[2*d*x^2 - d^2*x^4]/(15*b^3*d*x*Sqrt[a + b*ArcCos[-1 + d*x^2]]) + (2*(b^(-1))^(7/2)*Sqrt[Pi]*Cos[a/(2
*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(15*d*x) + (2
*(b^(-1))^(7/2)*Sqrt[Pi]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt
[Pi]]*Sin[a/(2*b)])/(15*d*x)

Rule 4829

Int[((a_.) + ArcCos[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[(x*(a + b*ArcCos[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*ArcCos[c + d*x^2])^(n + 2), x], x]
- Simp[(Sqrt[-2*c*d*x^2 - d^2*x^4]*(a + b*ArcCos[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 4824

Int[((a_.) + ArcCos[-1 + (d_.)*(x_)^2]*(b_.))^(-3/2), x_Symbol] :> Simp[Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a
+ b*ArcCos[-1 + d*x^2]]), x] + (-Simp[(2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Cos[ArcCos[-1 + d*x^2]/2]*FresnelC[
Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]])/(d*x), x] - Simp[(2*(1/b)^(3/2)*Sqrt[Pi]*Sin[a/(2*b)]*Cos[ArcC
os[-1 + d*x^2]/2]*FresnelS[Sqrt[1/(Pi*b)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]])/(d*x), x]) /; FreeQ[{a, b, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^{7/2}} \, dx &=\frac{\sqrt{2 d x^2-d^2 x^4}}{5 b d x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^{5/2}}+\frac{x}{15 b^2 \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^{3/2}}-\frac{\int \frac{1}{\left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^{3/2}} \, dx}{15 b^2}\\ &=\frac{\sqrt{2 d x^2-d^2 x^4}}{5 b d x \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^{5/2}}+\frac{x}{15 b^2 \left (a+b \cos ^{-1}\left (-1+d x^2\right )\right )^{3/2}}-\frac{\sqrt{2 d x^2-d^2 x^4}}{15 b^3 d x \sqrt{a+b \cos ^{-1}\left (-1+d x^2\right )}}+\frac{2 \left (\frac{1}{b}\right )^{7/2} \sqrt{\pi } \cos \left (\frac{a}{2 b}\right ) \cos \left (\frac{1}{2} \cos ^{-1}\left (-1+d x^2\right )\right ) C\left (\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left (-1+d x^2\right )}}{\sqrt{\pi }}\right )}{15 d x}+\frac{2 \left (\frac{1}{b}\right )^{7/2} \sqrt{\pi } \cos \left (\frac{1}{2} \cos ^{-1}\left (-1+d x^2\right )\right ) S\left (\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left (-1+d x^2\right )}}{\sqrt{\pi }}\right ) \sin \left (\frac{a}{2 b}\right )}{15 d x}\\ \end{align*}

Mathematica [A]  time = 0.553429, size = 309, normalized size = 1.15 \[ \frac{2 \cos \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right ) \left (-a^2 \sin \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right )+\sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left (\frac{a}{2 b}\right ) \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^{5/2} \text{FresnelC}\left (\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left (d x^2-1\right )}}{\sqrt{\pi }}\right )+\sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left (\frac{a}{2 b}\right ) \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^{5/2} S\left (\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left (d x^2-1\right )}}{\sqrt{\pi }}\right )+a b \cos \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right )-2 a b \cos ^{-1}\left (d x^2-1\right ) \sin \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right )+b^2 \cos ^{-1}\left (d x^2-1\right ) \cos \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right )+3 b^2 \sin \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right )-b^2 \cos ^{-1}\left (d x^2-1\right )^2 \sin \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right )\right )}{15 b^3 d x \left (a+b \cos ^{-1}\left (d x^2-1\right )\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[-1 + d*x^2])^(-7/2),x]

[Out]

(2*Cos[ArcCos[-1 + d*x^2]/2]*(a*b*Cos[ArcCos[-1 + d*x^2]/2] + b^2*ArcCos[-1 + d*x^2]*Cos[ArcCos[-1 + d*x^2]/2]
+ Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[-1 + d*x^2])^(5/2)*Cos[a/(2*b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCo
s[-1 + d*x^2]])/Sqrt[Pi]] + Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[-1 + d*x^2])^(5/2)*FresnelS[(Sqrt[b^(-1)]*Sqrt
[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] - a^2*Sin[ArcCos[-1 + d*x^2]/2] + 3*b^2*Sin[ArcCos[-1 + d*x
^2]/2] - 2*a*b*ArcCos[-1 + d*x^2]*Sin[ArcCos[-1 + d*x^2]/2] - b^2*ArcCos[-1 + d*x^2]^2*Sin[ArcCos[-1 + d*x^2]/
2]))/(15*b^3*d*x*(a + b*ArcCos[-1 + d*x^2])^(5/2))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arccos \left ( d{x}^{2}-1 \right ) \right ) ^{-{\frac{7}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arccos(d*x^2-1))^(7/2),x)

[Out]

int(1/(a+b*arccos(d*x^2-1))^(7/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \arccos \left (d x^{2} - 1\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(d*x^2-1))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*arccos(d*x^2 - 1) + a)^(-7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(d*x^2-1))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*acos(d*x**2-1))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \arccos \left (d x^{2} - 1\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(d*x^2-1))^(7/2),x, algorithm="giac")

[Out]

integrate((b*arccos(d*x^2 - 1) + a)^(-7/2), x)