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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {\sqrt {d x}}{\left (a+b \cos ^{-1}(c x)\right )^2},x\right ) \]

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

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*}

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Mathematica [A]  time = 5.44, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x}}{\left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

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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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x}}{b^{2} \arccos \left (c x\right )^{2} + 2 \, a b \arccos \left (c x\right ) + a^{2}}, x\right ) \]

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

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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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 \[ -\frac {\frac {1}{2} \, {\left (b^{2} c \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) + a b c\right )} \sqrt {d} \int \frac {{\left (3 \, c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {x}}{a b c^{3} x^{3} - a b c x + {\left (b^{2} c^{3} x^{3} - b^{2} c x\right )} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )}\,{d x} - \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {d} \sqrt {x}}{b^{2} c \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) + a b c} \]

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

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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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x}}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]

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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