3.154 \(\int \frac {1}{x \sin ^{-1}(a+b x)^3} \, dx\)

Optimal. Leaf size=15 \[ \text {Int}\left (\frac {1}{x \sin ^{-1}(a+b x)^3},x\right ) \]

[Out]

Unintegrable(1/x/arcsin(b*x+a)^3,x)

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {1}{x \sin ^{-1}(a+b x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*ArcSin[a + b*x]^3),x]

[Out]

Defer[Subst][Defer[Int][1/((-(a/b) + x/b)*ArcSin[x]^3), x], x, a + b*x]/b

Rubi steps

\begin {align*} \int \frac {1}{x \sin ^{-1}(a+b x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ \end {align*}

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Mathematica [A]  time = 2.71, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sin ^{-1}(a+b x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*ArcSin[a + b*x]^3),x]

[Out]

Integrate[1/(x*ArcSin[a + b*x]^3), x]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \arcsin \left (b x + a\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arcsin(b*x+a)^3,x, algorithm="fricas")

[Out]

integral(1/(x*arcsin(b*x + a)^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \arcsin \left (b x + a\right )^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arcsin(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(1/(x*arcsin(b*x + a)^3), x)

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maple [A]  time = 1.49, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \arcsin \left (b x +a \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/arcsin(b*x+a)^3,x)

[Out]

int(1/x/arcsin(b*x+a)^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {x^{2} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2} \int \frac {a b x + 2 \, a^{2} - 2}{x^{3} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}\,{d x} + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + {\left (a b x + a^{2} - 1\right )} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}{2 \, b^{2} x^{2} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arcsin(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{x\,{\mathrm {asin}\left (a+b\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*asin(a + b*x)^3),x)

[Out]

int(1/(x*asin(a + b*x)^3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \operatorname {asin}^{3}{\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/asin(b*x+a)**3,x)

[Out]

Integral(1/(x*asin(a + b*x)**3), x)

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