\(\int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx\) [97]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx=\text {Int}\left (\frac {(a+b \text {arcsinh}(c+d x))^n}{x},x\right ) \]

[Out]

Unintegrable((a+b*arcsinh(d*x+c))^n/x,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx \]

[In]

Int[(a + b*ArcSinh[c + d*x])^n/x,x]

[Out]

Defer[Subst][Defer[Int][(a + b*ArcSinh[x])^n/(-(c/d) + x/d), x], x, c + d*x]/d

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^n}{-\frac {c}{d}+\frac {x}{d}} \, dx,x,c+d x\right )}{d} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx=\int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx \]

[In]

Integrate[(a + b*ArcSinh[c + d*x])^n/x,x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^n/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

int((a+b*arcsinh(d*x+c))^n/x,x)

[Out]

int((a+b*arcsinh(d*x+c))^n/x,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^n/x,x, algorithm="fricas")

[Out]

integral((b*arcsinh(d*x + c) + a)^n/x, x)

Sympy [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}}{x}\, dx \]

[In]

integrate((a+b*asinh(d*x+c))**n/x,x)

[Out]

Integral((a + b*asinh(c + d*x))**n/x, x)

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^n/x,x, algorithm="maxima")

[Out]

integrate((b*arcsinh(d*x + c) + a)^n/x, x)

Giac [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^n/x,x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^n/x, x)

Mupad [N/A]

Not integrable

Time = 2.66 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^n}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n}{x} \,d x \]

[In]

int((a + b*asinh(c + d*x))^n/x,x)

[Out]

int((a + b*asinh(c + d*x))^n/x, x)