\(\int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx\) [88]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx=\text {Int}\left (\frac {1}{x \text {arcsinh}(a+b x)^2},x\right ) \]

[Out]

Unintegrable(1/x/arcsinh(b*x+a)^2,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 = {} \[ \int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx=\int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx \]

[In]

Int[1/(x*ArcSinh[a + b*x]^2),x]

[Out]

Defer[Subst][Defer[Int][1/((-(a/b) + x/b)*ArcSinh[x]^2), 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 ) \text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx=\int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx \]

[In]

Integrate[1/(x*ArcSinh[a + b*x]^2),x]

[Out]

Integrate[1/(x*ArcSinh[a + b*x]^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.85 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \operatorname {arcsinh}\left (b x +a \right )^{2}}d x\]

[In]

int(1/x/arcsinh(b*x+a)^2,x)

[Out]

int(1/x/arcsinh(b*x+a)^2,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx=\int { \frac {1}{x \operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]

[In]

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

[Out]

integral(1/(x*arcsinh(b*x + a)^2), x)

Sympy [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx=\int \frac {1}{x \operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]

[In]

integrate(1/x/asinh(b*x+a)**2,x)

[Out]

Integral(1/(x*asinh(a + b*x)**2), x)

Maxima [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 527, normalized size of antiderivative = 43.92 \[ \int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx=\int { \frac {1}{x \operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]

[In]

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

[Out]

-(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b + b)*x + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + a)/((b^3*x^3 + 2*a*b^2
*x^2 + (a^2*b + b)*x + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*x^2 + a*b*x))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b
*x + a^2 + 1))) - integrate((a*b^4*x^4 + 4*a^2*b^3*x^3 + a^5 + 2*a^3 + 2*(3*a^3*b^2 + a*b^2)*x^2 + (a*b^2*x^2
+ a^3 + 2*(a^2*b + b)*x + a)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 4*(a^4*b + a^2*b)*x + (2*a*b^3*x^3 + 2*a^4 + 2*(3
*a^2*b^2 + b^2)*x^2 + 3*a^2 + (6*a^3*b + 5*a*b)*x + 1)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + a)/((b^5*x^6 + 4*a*
b^4*x^5 + 2*(3*a^2*b^3 + b^3)*x^4 + 4*(a^3*b^2 + a*b^2)*x^3 + (a^4*b + 2*a^2*b + b)*x^2 + (b^3*x^4 + 2*a*b^2*x
^3 + a^2*b*x^2)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*(b^4*x^5 + 3*a*b^3*x^4 + (3*a^2*b^2 + b^2)*x^3 + (a^3*b + a*
b)*x^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx=\int { \frac {1}{x \operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]

[In]

integrate(1/x/arcsinh(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(1/(x*arcsinh(b*x + a)^2), x)

Mupad [N/A]

Not integrable

Time = 2.71 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \text {arcsinh}(a+b x)^2} \, dx=\int \frac {1}{x\,{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

int(1/(x*asinh(a + b*x)^2),x)

[Out]

int(1/(x*asinh(a + b*x)^2), x)