3.3.30 \(\int \frac {a+b \tanh ^{-1}(c x^n)}{x^4} \, dx\) [230]

Optimal. Leaf size=72 \[ -\frac {a+b \tanh ^{-1}\left (c x^n\right )}{3 x^3}-\frac {b c n x^{-3+n} \, _2F_1\left (1,-\frac {3-n}{2 n};-\frac {3 (1-n)}{2 n};c^2 x^{2 n}\right )}{3 (3-n)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 72, 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 )}{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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^n])/x^4,x]

[Out]

-1/3*(a + b*ArcTanh[c*x^n])/x^3 - (b*c*n*x^(-3 + n)*Hypergeometric2F1[1, -1/2*(3 - n)/n, (-3*(1 - n))/(2*n), c
^2*x^(2*n)])/(3*(3 - 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^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 = 0.03, size = 73, normalized size = 1.01 \begin {gather*} -\frac {a}{3 x^3}-\frac {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};1+\frac {-3+n}{2 n};c^2 x^{2 n}\right )}{3 (-3+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a/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))

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \arctanh \left (c \,x^{n}\right )}{x^{4}}\, dx\]

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)

________________________________________________________________________________________

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^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

________________________________________________________________________________________

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^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^4,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^n))/x^4,x)

[Out]

int((a + b*atanh(c*x^n))/x^4, x)

________________________________________________________________________________________