\(\int \frac {1}{x \arcsin (a+b x)^3} \, dx\) [154]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {1}{x \arcsin (a+b x)^3} \, dx=\text {Int}\left (\frac {1}{x \arcsin (a+b x)^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \arcsin (a+b x)^3} \, dx=\int \frac {1}{x \arcsin (a+b x)^3} \, dx \]

[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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \arcsin (x)^3} \, dx,x,a+b x\right )}{b} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.87 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \arcsin (a+b x)^3} \, dx=\int \frac {1}{x \arcsin (a+b x)^3} \, dx \]

[In]

Integrate[1/(x*ArcSin[a + b*x]^3),x]

[Out]

Integrate[1/(x*ArcSin[a + b*x]^3), x]

Maple [N/A] (verified)

Not integrable

Time = 9.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \arcsin \left (b x +a \right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \arcsin (a+b x)^3} \, dx=\int { \frac {1}{x \arcsin \left (b x + a\right )^{3}} \,d x } \]

[In]

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

[Out]

integral(1/(x*arcsin(b*x + a)^3), x)

Sympy [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \arcsin (a+b x)^3} \, dx=\int \frac {1}{x \operatorname {asin}^{3}{\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(x*asin(a + b*x)**3), x)

Maxima [N/A]

Not integrable

Time = 57.49 (sec) , antiderivative size = 172, normalized size of antiderivative = 14.33 \[ \int \frac {1}{x \arcsin (a+b x)^3} \, dx=\int { \frac {1}{x \arcsin \left (b x + a\right )^{3}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \arcsin (a+b x)^3} \, dx=\int { \frac {1}{x \arcsin \left (b x + a\right )^{3}} \,d x } \]

[In]

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

[Out]

integrate(1/(x*arcsin(b*x + a)^3), x)

Mupad [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \arcsin (a+b x)^3} \, dx=\int \frac {1}{x\,{\mathrm {asin}\left (a+b\,x\right )}^3} \,d x \]

[In]

int(1/(x*asin(a + b*x)^3),x)

[Out]

int(1/(x*asin(a + b*x)^3), x)