3.230 \(\int \frac{a+b \tanh ^{-1}(c x^n)}{x^4} \, dx\)

Optimal. Leaf size=72 \[ -\frac{b c n x^{n-3} \text{Hypergeometric2F1}\left (1,-\frac{3-n}{2 n},-\frac{3 (1-n)}{2 n},c^2 x^{2 n}\right )}{3 (3-n)}-\frac{a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3} \]

[Out]

-(a + b*ArcTanh[c*x^n])/(3*x^3) - (b*c*n*x^(-3 + n)*Hypergeometric2F1[1, -(3 - n)/(2*n), (-3*(1 - n))/(2*n), c
^2*x^(2*n)])/(3*(3 - n))

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Rubi [A]  time = 0.0342286, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6097, 364} \[ -\frac{a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3}-\frac{b c n x^{n-3} \, _2F_1\left (1,-\frac{3-n}{2 n};-\frac{3 (1-n)}{2 n};c^2 x^{2 n}\right )}{3 (3-n)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^n])/x^4,x]

[Out]

-(a + b*ArcTanh[c*x^n])/(3*x^3) - (b*c*n*x^(-3 + n)*Hypergeometric2F1[1, -(3 - n)/(2*n), (-3*(1 - n))/(2*n), c
^2*x^(2*n)])/(3*(3 - n))

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

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])

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^n\right )}{x^4} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3}+\frac{1}{3} (b c n) \int \frac{x^{-4+n}}{1-c^2 x^{2 n}} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3}-\frac{b c n x^{-3+n} \, _2F_1\left (1,-\frac{3-n}{2 n};-\frac{3 (1-n)}{2 n};c^2 x^{2 n}\right )}{3 (3-n)}\\ \end{align*}

Mathematica [A]  time = 0.0416435, size = 73, normalized size = 1.01 \[ \frac{b c n x^{n-3} \text{Hypergeometric2F1}\left (1,\frac{n-3}{2 n},\frac{n-3}{2 n}+1,c^2 x^{2 n}\right )}{3 (n-3)}-\frac{a}{3 x^3}-\frac{b \tanh ^{-1}\left (c x^n\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^n])/x^4,x]

[Out]

-a/(3*x^3) - (b*ArcTanh[c*x^n])/(3*x^3) + (b*c*n*x^(-3 + n)*Hypergeometric2F1[1, (-3 + n)/(2*n), 1 + (-3 + n)/
(2*n), c^2*x^(2*n)])/(3*(-3 + n))

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\it Artanh} \left ( c{x}^{n} \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^n))/x^4,x)

[Out]

int((a+b*arctanh(c*x^n))/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{6} \,{\left (3 \, n \int \frac{1}{3 \,{\left (c x^{4} x^{n} + x^{4}\right )}}\,{d x} + 3 \, n \int \frac{1}{3 \,{\left (c x^{4} x^{n} - x^{4}\right )}}\,{d x} + \frac{\log \left (c x^{n} + 1\right ) - \log \left (-c x^{n} + 1\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))/x^4,x, algorithm="maxima")

[Out]

-1/6*(3*n*integrate(1/3/(c*x^4*x^n + x^4), x) + 3*n*integrate(1/3/(c*x^4*x^n - x^4), x) + (log(c*x^n + 1) - lo
g(-c*x^n + 1))/x^3)*b - 1/3*a/x^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{artanh}\left (c x^{n}\right ) + a}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))/x^4,x, algorithm="fricas")

[Out]

integral((b*arctanh(c*x^n) + a)/x^4, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**n))/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{artanh}\left (c x^{n}\right ) + a}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))/x^4,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^n) + a)/x^4, x)