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

Optimal. Leaf size=72 \[ \frac {(d x)^{m+1} \left (a+b \tanh ^{-1}(c x)\right )}{d (m+1)}-\frac {b c (d x)^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{d^2 (m+1) (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.03, 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 = {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 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])

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]

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*}

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Mathematica [A]  time = 0.07, size = 59, normalized size = 0.82 \[ -\frac {x (d x)^m \left (b c x \, _2F_1\left (1,\frac {m}{2}+1;\frac {m}{2}+2;c^2 x^2\right )-(m+2) \left (a+b \tanh ^{-1}(c x)\right )\right )}{(m+1) (m+2)} \]

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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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {artanh}\left (c x\right ) + a\right )} \left (d x\right )^{m}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} \left (d x\right )^{m}\,{d x} \]

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)

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maple [F]  time = 1.47, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctanh \left (c x \right )\right )\, dx \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (2 \, c d^{m} \int \frac {x x^{m}}{c^{2} {\left (m + 1\right )} x^{2} - m - 1}\,{d x} + \frac {d^{m} x x^{m} \log \left (c x + 1\right ) - d^{m} x x^{m} \log \left (-c x + 1\right )}{m + 1}\right )} b + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))*(d*x)^m,x)

[Out]

int((a + b*atanh(c*x))*(d*x)^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )\, dx \]

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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