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

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

[Out]

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

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Rubi [A]  time = 0.0233537, 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 = {} \[ \int \frac{\sqrt{d x}}{\left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.95259, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x}}{\left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.246, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\arccos \left ( cx \right ) \right ) ^{2}}\sqrt{dx}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*x)^(1/2)/(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 \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a b c\right )} \sqrt{d} \int \frac{{\left (3 \, c^{2} x^{2} - 1\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \sqrt{x}}{a b c^{3} x^{3} - a b c x +{\left (b^{2} c^{3} x^{3} - b^{2} c x\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 \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x}}{\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((d*x)**(1/2)/(a+b*acos(c*x))**2,x)

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x}}{{\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((d*x)^(1/2)/(a+b*arccos(c*x))^2,x, algorithm="giac")

[Out]

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