3.241 \(\int (d x)^m (a+b \tanh ^{-1}(c x^n)) \, dx\)

Optimal. Leaf size=84 \[ \frac{x (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )}{m+1}-\frac{b c n x^{n+1} (d x)^m \text{Hypergeometric2F1}\left (1,\frac{m+n+1}{2 n},\frac{m+3 n+1}{2 n},c^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0441134, antiderivative size = 88, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6097, 20, 364} \[ \frac{(d x)^{m+1} \left (a+b \tanh ^{-1}\left (c x^n\right )\right )}{d (m+1)}-\frac{b c n x^{n+1} (d x)^m \, _2F_1\left (1,\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};c^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*ArcTanh[c*x^n]),x]

[Out]

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

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]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + 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])

Rubi steps

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

Mathematica [A]  time = 0.0870418, size = 77, normalized size = 0.92 \[ \frac{x (d x)^m \left ((m+n+1) \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-b c n x^n \text{Hypergeometric2F1}\left (1,\frac{m+n+1}{2 n},\frac{m+3 n+1}{2 n},c^2 x^{2 n}\right )\right )}{(m+1) (m+n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*ArcTanh[c*x^n]),x]

[Out]

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

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Maple [F]  time = 0.279, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b{\it Artanh} \left ( c{x}^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m} b \operatorname{artanh}\left (c x^{n}\right ) + \left (d x\right )^{m} a, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x)^m*b*arctanh(c*x^n) + (d*x)^m*a, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x^{n}\right ) + a\right )} \left (d x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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