3.3.41 \(\int (d x)^m (a+b \tanh ^{-1}(c x^n)) \, dx\) [241]

Optimal. Leaf size=84 \[ \frac {x (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )}{1+m}-\frac {b c n x^{1+n} (d x)^m \, _2F_1\left (1,\frac {1+m+n}{2 n};\frac {1+m+3 n}{2 n};c^2 x^{2 n}\right )}{(1+m) (1+m+n)} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 = {6051, 6037, 371} \begin {gather*} \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 \, _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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*ArcTanh[c*x^n]),x]

[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))

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
]

Rule 6051

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_)*(x_))^(m_), x_Symbol] :> Dist[d^IntPart[m]*((d*x)^Fr
acPart[m]/x^FracPart[m]), Int[x^m*(a + b*ArcTanh[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p,
0] && (EqQ[p, 1] || RationalQ[m, n])

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*}

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Mathematica [A]
time = 0.06, size = 77, normalized size = 0.92 \begin {gather*} \frac {x (d x)^m \left ((1+m+n) \left (a+b \tanh ^{-1}\left (c x^n\right )\right )-b c n x^n \, _2F_1\left (1,\frac {1+m+n}{2 n};\frac {1+m+3 n}{2 n};c^2 x^{2 n}\right )\right )}{(1+m) (1+m+n)} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \arctanh \left (c \,x^{n}\right )\right )\, dx\]

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

[Out]

1/2*(d^m*n*integrate(x^m/(c*(m + 1)*x^n + m + 1), x) + d^m*n*integrate(x^m/(c*(m + 1)*x^n - m - 1), x) + (d^m*
x*x^m*log(c*x^n + 1) - d^m*x*x^m*log(-c*x^n + 1))/(m + 1))*b + (d*x)^(m + 1)*a/(d*(m + 1))

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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((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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{m} \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((d*x)**m*(a+b*atanh(c*x**n)),x)

[Out]

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

[Out]

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

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

[Out]

int((d*x)^m*(a + b*atanh(c*x^n)), x)

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