\(\int \frac {1}{x \text {arcsinh}(a+b x)} \, dx\) [84]

Optimal result

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

[Out]

Unintegrable(1/x/arcsinh(b*x+a),x)

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {1}{x \text {arcsinh}(a+b x)} \, dx=\int \frac {1}{x \text {arcsinh}(a+b x)} \, dx \]

[In]

Int[1/(x*ArcSinh[a + b*x]),x]

[Out]

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

Mathematica [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \text {arcsinh}(a+b x)} \, dx=\int \frac {1}{x \text {arcsinh}(a+b x)} \, dx \]

[In]

Integrate[1/(x*ArcSinh[a + b*x]),x]

[Out]

Integrate[1/(x*ArcSinh[a + b*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.98 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

integral(1/(x*arcsinh(b*x + a)), x)

Sympy [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \text {arcsinh}(a+b x)} \, dx=\int \frac {1}{x \operatorname {asinh}{\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(x*asinh(a + b*x)), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \text {arcsinh}(a+b x)} \, dx=\int { \frac {1}{x \operatorname {arsinh}\left (b x + a\right )} \,d x } \]

[In]

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

[Out]

integrate(1/(x*arcsinh(b*x + a)), x)

Giac [N/A]

Not integrable

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

[In]

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

[Out]

integrate(1/(x*arcsinh(b*x + a)), x)

Mupad [N/A]

Not integrable

Time = 2.73 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \text {arcsinh}(a+b x)} \, dx=\int \frac {1}{x\,\mathrm {asinh}\left (a+b\,x\right )} \,d x \]

[In]

int(1/(x*asinh(a + b*x)),x)

[Out]

int(1/(x*asinh(a + b*x)), x)