3.3.29 \(\int \frac {a+b \tanh ^{-1}(c x^n)}{x^3} \, dx\) [229]

Optimal. Leaf size=70 \[ -\frac {a+b \tanh ^{-1}\left (c x^n\right )}{2 x^2}-\frac {b c n x^{-2+n} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {2}{n}\right );\frac {1}{2} \left (3-\frac {2}{n}\right );c^2 x^{2 n}\right )}{2 (2-n)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 70, 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 )}{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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^n])/x^3,x]

[Out]

-1/2*(a + b*ArcTanh[c*x^n])/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 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^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*}

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Mathematica [A]
time = 0.03, size = 73, normalized size = 1.04 \begin {gather*} -\frac {a}{2 x^2}-\frac {b \tanh ^{-1}\left (c x^n\right )}{2 x^2}+\frac {b c n x^{-2+n} \, _2F_1\left (1,\frac {-2+n}{2 n};1+\frac {-2+n}{2 n};c^2 x^{2 n}\right )}{2 (-2+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*a/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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \arctanh \left (c \,x^{n}\right )}{x^{3}}\, dx\]

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 = 0.00, 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^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 = 0.00, 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^3,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^n))/x^3,x)

[Out]

int((a + b*atanh(c*x^n))/x^3, x)

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