3.2.82 \(\int (d x)^m (a+b \tanh ^{-1}(\frac {c}{x^2}))^3 \, dx\) [182]

Optimal. Leaf size=21 \[ \text {Int}\left ((d x)^m \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^3,x\right ) \]

[Out]

Unintegrable((d*x)^m*(a+b*arctanh(c/x^2))^3,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}\left (\frac {c}{x^2}\right )\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(d*x)^m*(a + b*ArcTanh[c/x^2])^3,x]

[Out]

Defer[Int][(d*x)^m*(a + b*ArcTanh[c/x^2])^3, x]

Rubi steps

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

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Mathematica [A]
time = 1.90, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(d*x)^m*(a + b*ArcTanh[c/x^2])^3,x]

[Out]

Integrate[(d*x)^m*(a + b*ArcTanh[c/x^2])^3, x]

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Maple [A]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*arctanh(c/x^2))^3,x)

[Out]

int((d*x)^m*(a+b*arctanh(c/x^2))^3,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^2))^3,x, algorithm="maxima")

[Out]

-1/8*b^3*d^m*x*x^m*log(x^2 - c)^3/(m + 1) + (d*x)^(m + 1)*a^3/(d*(m + 1)) + integrate(1/8*((b^3*d^m*(m + 1)*x^
2 - b^3*c*d^m*(m + 1))*x^m*log(x^2 + c)^3 + 6*(a*b^2*d^m*(m + 1)*x^2 - a*b^2*c*d^m*(m + 1))*x^m*log(x^2 + c)^2
+ 12*(a^2*b*d^m*(m + 1)*x^2 - a^2*b*c*d^m*(m + 1))*x^m*log(x^2 + c) + 3*((b^3*d^m*(m + 1)*x^2 - b^3*c*d^m*(m
+ 1))*x^m*log(x^2 + c) - 2*(a*b^2*c*d^m*(m + 1) - (a*b^2*d^m*(m + 1) + b^3*d^m)*x^2)*x^m)*log(x^2 - c)^2 - 3*(
(b^3*d^m*(m + 1)*x^2 - b^3*c*d^m*(m + 1))*x^m*log(x^2 + c)^2 + 4*(a*b^2*d^m*(m + 1)*x^2 - a*b^2*c*d^m*(m + 1))
*x^m*log(x^2 + c) + 4*(a^2*b*d^m*(m + 1)*x^2 - a^2*b*c*d^m*(m + 1))*x^m)*log(x^2 - c))/((m + 1)*x^2 - c*(m + 1
)), 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^2))^3,x, algorithm="fricas")

[Out]

integral((b^3*arctanh(c/x^2)^3 + 3*a*b^2*arctanh(c/x^2)^2 + 3*a^2*b*arctanh(c/x^2) + a^3)*(d*x)^m, 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 (\frac {c}{x^{2}} \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*atanh(c/x**2))**3,x)

[Out]

Integral((d*x)**m*(a + b*atanh(c/x**2))**3, 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^2))^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c/x^2) + a)^3*(d*x)^m, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*atanh(c/x^2))^3,x)

[Out]

int((d*x)^m*(a + b*atanh(c/x^2))^3, x)

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