3.3.25 \(\int x (a+b \tanh ^{-1}(c x^n)) \, dx\) [225]

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

[Out]

1/2*x^2*(a+b*arctanh(c*x^n))-1/2*b*c*n*x^(2+n)*hypergeom([1, 1/2*(2+n)/n],[3/2+1/n],c^2*x^(2*n))/(2+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 = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6037, 371} \begin {gather*} \frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac {b c n x^{n+2} \, _2F_1\left (1,\frac {n+2}{2 n};\frac {1}{2} \left (3+\frac {2}{n}\right );c^2 x^{2 n}\right )}{2 (n+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcTanh[c*x^n]),x]

[Out]

(x^2*(a + b*ArcTanh[c*x^n]))/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/(2*n), (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 x \left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac {1}{2} (b c n) \int \frac {x^{1+n}}{1-c^2 x^{2 n}} \, dx\\ &=\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-\frac {b c n x^{2+n} \, _2F_1\left (1,\frac {2+n}{2 n};\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 = 1.09 \begin {gather*} \frac {a x^2}{2}+\frac {1}{2} b x^2 \tanh ^{-1}\left (c x^n\right )-\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[x*(a + b*ArcTanh[c*x^n]),x]

[Out]

(a*x^2)/2 + (b*x^2*ArcTanh[c*x^n])/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 \[\int x \left (a +b \arctanh \left (c \,x^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/2*a*x^2 + 1/4*(x^2*log(c*x^n + 1) - x^2*log(-c*x^n + 1) + 2*n*integrate(1/2*x/(c*x^n + 1), x) + 2*n*integrat
e(1/2*x/(c*x^n - 1), x))*b

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

[Out]

integral(b*x*arctanh(c*x^n) + a*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*atanh(c*x^n)),x)

[Out]

int(x*(a + b*atanh(c*x^n)), x)

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