3.3.28 \(\int \frac {a+b \tanh ^{-1}(c x^n)}{x^2} \, dx\) [228]

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

[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 = 0.02, 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 = {6037, 371} \begin {gather*} -\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} \end {gather*}

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, -1/2*(1 - n)/n, (3 - n^(-1))/2, c^2*x^(2*
n)])/(1 - n)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && 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 = 0.05, size = 66, normalized size = 0.99 \begin {gather*} -\frac {a}{x}-\frac {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};1+\frac {-1+n}{2 n};c^2 x^{2 n}\right )}{-1+n} \end {gather*}

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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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
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*arctanh(c*x^n))/x^2,x, algorithm="fricas")

[Out]

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

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

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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Giac [F]
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*arctanh(c*x^n))/x^2,x, algorithm="giac")

[Out]

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

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

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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