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

Optimal. Leaf size=72 \[ \frac{(d x)^{m+1} \left (a+b \tanh ^{-1}(c x)\right )}{d (m+1)}-\frac{b c (d x)^{m+2} \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},c^2 x^2\right )}{d^2 (m+1) (m+2)} \]

[Out]

((d*x)^(1 + m)*(a + b*ArcTanh[c*x]))/(d*(1 + m)) - (b*c*(d*x)^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/
2, c^2*x^2])/(d^2*(1 + m)*(2 + m))

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Rubi [A]  time = 0.032328, antiderivative size = 72, normalized size of antiderivative = 1., 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 = {5916, 364} \[ \frac{(d x)^{m+1} \left (a+b \tanh ^{-1}(c x)\right )}{d (m+1)}-\frac{b c (d x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right )}{d^2 (m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*ArcTanh[c*x]),x]

[Out]

((d*x)^(1 + m)*(a + b*ArcTanh[c*x]))/(d*(1 + m)) - (b*c*(d*x)^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/
2, c^2*x^2])/(d^2*(1 + m)*(2 + m))

Rule 5916

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcT
anh[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTanh[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

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}(c x)\right ) \, dx &=\frac{(d x)^{1+m} \left (a+b \tanh ^{-1}(c x)\right )}{d (1+m)}-\frac{(b c) \int \frac{(d x)^{1+m}}{1-c^2 x^2} \, dx}{d (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \tanh ^{-1}(c x)\right )}{d (1+m)}-\frac{b c (d x)^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};c^2 x^2\right )}{d^2 (1+m) (2+m)}\\ \end{align*}

Mathematica [A]  time = 0.0667753, size = 59, normalized size = 0.82 \[ -\frac{x (d x)^m \left (b c x \text{Hypergeometric2F1}\left (1,\frac{m}{2}+1,\frac{m}{2}+2,c^2 x^2\right )-(m+2) \left (a+b \tanh ^{-1}(c x)\right )\right )}{(m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.925, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b{\it Artanh} \left ( cx \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)),x)

[Out]

int((d*x)^m*(a+b*arctanh(c*x)),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)),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 (b \operatorname{artanh}\left (c x\right ) + a\right )} \left (d x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (c x\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)),x, algorithm="giac")

[Out]

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