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

Optimal. Leaf size=19 \[ \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^n}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^n}{x} \, dx \]

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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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n}}{x}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n}}{x}\,{d x} \]

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)

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maple [A]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{n}}{x}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n}}{x}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c + d*x))^n/x,x)

[Out]

int((a + b*asinh(c + d*x))^n/x, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}}{x}\, dx \]

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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