3.229 \(\int \frac{a+b \tanh ^{-1}(c x^n)}{x^3} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b c n x^{n-2} \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (1-\frac{2}{n}\right ),\frac{1}{2} \left (3-\frac{2}{n}\right ),c^2 x^{2 n}\right )}{2 (2-n)}-\frac{a+b \tanh ^{-1}\left (c x^n\right )}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0311192, antiderivative size = 70, 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} \[ -\frac{a+b \tanh ^{-1}\left (c x^n\right )}{2 x^2}-\frac{b c n x^{n-2} \, _2F_1\left (1,\frac{1}{2} \left (1-\frac{2}{n}\right );\frac{1}{2} \left (3-\frac{2}{n}\right );c^2 x^{2 n}\right )}{2 (2-n)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^n])/x^3,x]

[Out]

-(a + b*ArcTanh[c*x^n])/(2*x^2) - (b*c*n*x^(-2 + n)*Hypergeometric2F1[1, (1 - 2/n)/2, (3 - 2/n)/2, c^2*x^(2*n)
])/(2*(2 - 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^3} \, dx &=-\frac{a+b \tanh ^{-1}\left (c x^n\right )}{2 x^2}+\frac{1}{2} (b c n) \int \frac{x^{-3+n}}{1-c^2 x^{2 n}} \, dx\\ &=-\frac{a+b \tanh ^{-1}\left (c x^n\right )}{2 x^2}-\frac{b c n x^{-2+n} \, _2F_1\left (1,\frac{1}{2} \left (1-\frac{2}{n}\right );\frac{1}{2} \left (3-\frac{2}{n}\right );c^2 x^{2 n}\right )}{2 (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.0411021, size = 73, normalized size = 1.04 \[ \frac{b c n x^{n-2} \text{Hypergeometric2F1}\left (1,\frac{n-2}{2 n},\frac{n-2}{2 n}+1,c^2 x^{2 n}\right )}{2 (n-2)}-\frac{a}{2 x^2}-\frac{b \tanh ^{-1}\left (c x^n\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^n])/x^3,x]

[Out]

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

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\it Artanh} \left ( c{x}^{n} \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^n))/x^3,x)

[Out]

int((a+b*arctanh(c*x^n))/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \,{\left (2 \, n \int \frac{1}{2 \,{\left (c x^{3} x^{n} + x^{3}\right )}}\,{d x} + 2 \, n \int \frac{1}{2 \,{\left (c x^{3} x^{n} - x^{3}\right )}}\,{d x} + \frac{\log \left (c x^{n} + 1\right ) - \log \left (-c x^{n} + 1\right )}{x^{2}}\right )} b - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))/x^3,x, algorithm="maxima")

[Out]

-1/4*(2*n*integrate(1/2/(c*x^3*x^n + x^3), x) + 2*n*integrate(1/2/(c*x^3*x^n - x^3), x) + (log(c*x^n + 1) - lo
g(-c*x^n + 1))/x^2)*b - 1/2*a/x^2

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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^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))/x^3,x, algorithm="fricas")

[Out]

integral((b*arctanh(c*x^n) + a)/x^3, 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**3,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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))/x^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^n) + a)/x^3, x)