3.97 \(\int \frac{(a+b \sinh ^{-1}(c+d x))^n}{x} \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^n}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0595191, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^n}{x} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^n}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^n}{-\frac{c}{d}+\frac{x}{d}} \, dx,x,c+d x\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.170924, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^n}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.122, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{n}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{n}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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