3.6.15 \(\int \frac {\sqrt {d+c^2 d x^2} (a+b \sinh ^{-1}(c x))^n}{x} \, dx\) [515]

Optimal. Leaf size=199 \[ \frac {d e^{-\frac {a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+\frac {d e^{a/b} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+d \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c x)\right )^n}{x \sqrt {d+c^2 d x^2}},x\right ) \]

[Out]

1/2*d*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(-a-b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/exp(a/b)/(((-a-b*arcsinh(c*x))/b
)^n)/(c^2*d*x^2+d)^(1/2)+1/2*d*exp(a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)
/(((a+b*arcsinh(c*x))/b)^n)/(c^2*d*x^2+d)^(1/2)+d*Unintegrable((a+b*arcsinh(c*x))^n/x/(c^2*d*x^2+d)^(1/2),x)

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Rubi [A]
time = 0.36, 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 = {} \begin {gather*} \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x,x]

[Out]

(d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(2*E^(a/b)*Sqrt[d + c^2*d
*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d*E^(a/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b
*ArcSinh[c*x])/b])/(2*Sqrt[d + c^2*d*x^2]*((a + b*ArcSinh[c*x])/b)^n) + d*Defer[Int][(a + b*ArcSinh[c*x])^n/(x
*Sqrt[d + c^2*d*x^2]), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x} \, dx &=\int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x,x]

[Out]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x, x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{n} \sqrt {c^{2} d \,x^{2}+d}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n/x, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n/x, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**n/x, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,\sqrt {d\,c^2\,x^2+d}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(1/2))/x,x)

[Out]

int(((a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(1/2))/x, x)

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