3.1.49 \(\int (d x)^m (a+b \tanh ^{-1}(c x))^p \, dx\) [49]

Optimal. Leaf size=19 \[ \text {Int}\left ((d x)^m \left (a+b \tanh ^{-1}(c x)\right )^p,x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (d x)^m \left (a+b \tanh ^{-1}(c x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(d*x)^m*(a + b*ArcTanh[c*x])^p,x]

[Out]

Defer[Int][(d*x)^m*(a + b*ArcTanh[c*x])^p, x]

Rubi steps

\begin {align*} \int (d x)^m \left (a+b \tanh ^{-1}(c x)\right )^p \, dx &=\int (d x)^m \left (a+b \tanh ^{-1}(c x)\right )^p \, dx\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a+b \tanh ^{-1}(c x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \arctanh \left (c x \right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
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))^p,x, algorithm="maxima")

[Out]

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

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Fricas [A]
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))^p,x, algorithm="fricas")

[Out]

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

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Sympy [A]
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 \right )}\right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
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))^p,x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^p\,{\left (d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))^p*(d*x)^m,x)

[Out]

int((a + b*atanh(c*x))^p*(d*x)^m, x)

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