3.228 \(\int \frac {a+b \tanh ^{-1}(c x^n)}{x^2} \, dx\)

Optimal. Leaf size=67 \[ -\frac {a+b \tanh ^{-1}\left (c x^n\right )}{x}-\frac {b c n x^{n-1} \, _2F_1\left (1,-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );c^2 x^{2 n}\right )}{1-n} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, 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} \[ -\frac {a+b \tanh ^{-1}\left (c x^n\right )}{x}-\frac {b c n x^{n-1} \, _2F_1\left (1,-\frac {1-n}{2 n};\frac {1}{2} \left (3-\frac {1}{n}\right );c^2 x^{2 n}\right )}{1-n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^n])/x^2,x]

[Out]

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

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 66, normalized size = 0.99 \[ -\frac {a}{x}+\frac {b c n x^{n-1} \, _2F_1\left (1,\frac {n-1}{2 n};\frac {n-1}{2 n}+1;c^2 x^{2 n}\right )}{n-1}-\frac {b \tanh ^{-1}\left (c x^n\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arctanh(c*x^n) + a)/x^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctanh(c*x^n) + a)/x^2, x)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arctanh \left (c \,x^{n}\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^n))/x^2,x)

[Out]

int((a+b*arctanh(c*x^n))/x^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^n\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^n))/x^2,x)

[Out]

int((a + b*atanh(c*x^n))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {atanh}{\left (c x^{n} \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**n))/x**2,x)

[Out]

Integral((a + b*atanh(c*x**n))/x**2, x)

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