3.227 \(\int \frac{1}{(d x)^{3/2} (a+b \cos ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2), x]

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Rubi [A]  time = 0.0285917, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2),x]

[Out]

Defer[Int][1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2), x]

Rubi steps

\begin{align*} \int \frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx &=\int \frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 13.9247, size = 0, normalized size = 0. \[ \int \frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2),x]

[Out]

Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2), x]

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Maple [A]  time = 0.233, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\arccos \left ( cx \right ) \right ) ^{2}} \left ( dx \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x)^(3/2)/(a+b*arccos(c*x))^2,x)

[Out]

int(1/(d*x)^(3/2)/(a+b*arccos(c*x))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{1}{2} \,{\left (b^{2} c d^{2} x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a b c d^{2} x^{2}\right )} \sqrt{d} \int \frac{{\left (c^{2} x^{2} - 3\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \sqrt{x}}{a b c^{3} d^{2} x^{5} - a b c d^{2} x^{3} +{\left (b^{2} c^{3} d^{2} x^{5} - b^{2} c d^{2} x^{3}\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )}\,{d x} + \sqrt{c x + 1} \sqrt{-c x + 1} \sqrt{d} \sqrt{x}}{b^{2} c d^{2} x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a b c d^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*c*d^2*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*d^2*x^2)*sqrt(d)*integrate(1/2*(c^2*x^2 - 3
)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(a*b*c^3*d^2*x^5 - a*b*c*d^2*x^3 + (b^2*c^3*d^2*x^5 - b^2*c*d^2*x^3)*ar
ctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)), x) + sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*sqrt(x))/(b^2*c*d^2*x^2*a
rctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*d^2*x^2)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x}}{b^{2} d^{2} x^{2} \arccos \left (c x\right )^{2} + 2 \, a b d^{2} x^{2} \arccos \left (c x\right ) + a^{2} d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*x)/(b^2*d^2*x^2*arccos(c*x)^2 + 2*a*b*d^2*x^2*arccos(c*x) + a^2*d^2*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)**(3/2)/(a+b*acos(c*x))**2,x)

[Out]

Integral(1/((d*x)**(3/2)*(a + b*acos(c*x))**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((d*x)^(3/2)*(b*arccos(c*x) + a)^2), x)