3.307 \(\int x^m \sinh ^{-1}(a x^n) \, dx\)

Optimal. Leaf size=77 \[ \frac {x^{m+1} \sinh ^{-1}\left (a x^n\right )}{m+1}-\frac {a n x^{m+n+1} \, _2F_1\left (\frac {1}{2},\frac {m+n+1}{2 n};\frac {m+3 n+1}{2 n};-a^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]

[Out]

x^(1+m)*arcsinh(a*x^n)/(1+m)-a*n*x^(1+m+n)*hypergeom([1/2, 1/2*(1+m+n)/n],[1/2*(1+m+3*n)/n],-a^2*x^(2*n))/(1+m
)/(1+m+n)

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Rubi [A]  time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5902, 12, 364} \[ \frac {x^{m+1} \sinh ^{-1}\left (a x^n\right )}{m+1}-\frac {a n x^{m+n+1} \, _2F_1\left (\frac {1}{2},\frac {m+n+1}{2 n};\frac {m+3 n+1}{2 n};-a^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m*ArcSinh[a*x^n],x]

[Out]

(x^(1 + m)*ArcSinh[a*x^n])/(1 + m) - (a*n*x^(1 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/(2*n), (1 + m + 3*n
)/(2*n), -(a^2*x^(2*n))])/((1 + m)*(1 + m + n))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 5902

Int[((a_.) + ArcSinh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin
h[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 + u^2],
x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)
^(m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps

\begin {align*} \int x^m \sinh ^{-1}\left (a x^n\right ) \, dx &=\frac {x^{1+m} \sinh ^{-1}\left (a x^n\right )}{1+m}-\frac {\int \frac {a n x^{m+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx}{1+m}\\ &=\frac {x^{1+m} \sinh ^{-1}\left (a x^n\right )}{1+m}-\frac {(a n) \int \frac {x^{m+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx}{1+m}\\ &=\frac {x^{1+m} \sinh ^{-1}\left (a x^n\right )}{1+m}-\frac {a n x^{1+m+n} \, _2F_1\left (\frac {1}{2},\frac {1+m+n}{2 n};\frac {1+m+3 n}{2 n};-a^2 x^{2 n}\right )}{(1+m) (1+m+n)}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 74, normalized size = 0.96 \[ \frac {x^{m+1} \left ((m+n+1) \sinh ^{-1}\left (a x^n\right )-a n x^n \, _2F_1\left (\frac {1}{2},\frac {m+n+1}{2 n};\frac {m+3 n+1}{2 n};-a^2 x^{2 n}\right )\right )}{(m+1) (m+n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*ArcSinh[a*x^n],x]

[Out]

(x^(1 + m)*((1 + m + n)*ArcSinh[a*x^n] - a*n*x^n*Hypergeometric2F1[1/2, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n)
, -(a^2*x^(2*n))]))/((1 + m)*(1 + m + n))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arcsinh(a*x^n),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {arsinh}\left (a x^{n}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arcsinh(a*x^n),x, algorithm="giac")

[Out]

integrate(x^m*arcsinh(a*x^n), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int x^{m} \arcsinh \left (a \,x^{n}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*arcsinh(a*x^n),x)

[Out]

int(x^m*arcsinh(a*x^n),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -a n \int \frac {e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{a^{3} {\left (m + 1\right )} x^{3 \, n} + a {\left (m + 1\right )} x^{n} + {\left (a^{2} {\left (m + 1\right )} x^{2 \, n} + m + 1\right )} \sqrt {a^{2} x^{2 \, n} + 1}}\,{d x} + n \int \frac {x^{m}}{a^{2} {\left (m + 1\right )} x^{2 \, n} + m + 1}\,{d x} + \frac {{\left (m + 1\right )} x x^{m} \log \left (a x^{n} + \sqrt {a^{2} x^{2 \, n} + 1}\right ) - n x x^{m}}{m^{2} + 2 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arcsinh(a*x^n),x, algorithm="maxima")

[Out]

-a*n*integrate(e^(m*log(x) + n*log(x))/(a^3*(m + 1)*x^(3*n) + a*(m + 1)*x^n + (a^2*(m + 1)*x^(2*n) + m + 1)*sq
rt(a^2*x^(2*n) + 1)), x) + n*integrate(x^m/(a^2*(m + 1)*x^(2*n) + m + 1), x) + ((m + 1)*x*x^m*log(a*x^n + sqrt
(a^2*x^(2*n) + 1)) - n*x*x^m)/(m^2 + 2*m + 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,\mathrm {asinh}\left (a\,x^n\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*asinh(a*x^n),x)

[Out]

int(x^m*asinh(a*x^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {asinh}{\left (a x^{n} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*asinh(a*x**n),x)

[Out]

Integral(x**m*asinh(a*x**n), x)

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