3.3.25 \(\int \frac {\sqrt {d x}}{(a+b \text {ArcCos}(c x))^2} \, dx\) [225]

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {\sqrt {d x}}{(a+b \text {ArcCos}(c x))^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {d x}}{(a+b \text {ArcCos}(c x))^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 3.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d x}}{(a+b \text {ArcCos}(c x))^2} \, dx \end {gather*}

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 = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {d x}}{\left (a +b \arccos \left (c x \right )\right )^{2}}\, dx\]

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 = 0.00, 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((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 = 0.00, 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((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d x}}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}

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 = 0.00, 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((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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {d\,x}}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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