3.239 \(\int (d x)^m (a+b \tanh ^{-1}(c x^n))^3 \, dx\)

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

[Out]

Unintegrable((d*x)^m*(a+b*arctanh(c*x^n))^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 = {} \[ \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 8.42, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} b^{3} \operatorname {artanh}\left (c x^{n}\right )^{3} + 3 \, \left (d x\right )^{m} a b^{2} \operatorname {artanh}\left (c x^{n}\right )^{2} + 3 \, \left (d x\right )^{m} a^{2} b \operatorname {artanh}\left (c x^{n}\right ) + \left (d x\right )^{m} a^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctanh(c*x^n))^3,x, algorithm="fricas")

[Out]

integral((d*x)^m*b^3*arctanh(c*x^n)^3 + 3*(d*x)^m*a*b^2*arctanh(c*x^n)^2 + 3*(d*x)^m*a^2*b*arctanh(c*x^n) + (d
*x)^m*a^3, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )}^{3} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctanh(c*x^n))^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctanh \left (c \,x^{n}\right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{3} d^{m} x x^{m} \log \left (-c x^{n} + 1\right )^{3}}{8 \, {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} + \int \frac {{\left (b^{3} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - b^{3} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )^{3} + 6 \, {\left (a b^{2} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - a b^{2} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )^{2} - 3 \, {\left (2 \, a b^{2} d^{m} {\left (m + 1\right )} x^{m} - {\left (2 \, a b^{2} c d^{m} {\left (m + 1\right )} + b^{3} c d^{m} n\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - {\left (b^{3} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - b^{3} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )\right )} \log \left (-c x^{n} + 1\right )^{2} + 12 \, {\left (a^{2} b c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - a^{2} b d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right ) - 3 \, {\left (4 \, a^{2} b c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - 4 \, a^{2} b d^{m} {\left (m + 1\right )} x^{m} + {\left (b^{3} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - b^{3} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )^{2} + 4 \, {\left (a b^{2} c d^{m} {\left (m + 1\right )} e^{\left (m \log \relax (x) + n \log \relax (x)\right )} - a b^{2} d^{m} {\left (m + 1\right )} x^{m}\right )} \log \left (c x^{n} + 1\right )\right )} \log \left (-c x^{n} + 1\right )}{8 \, {\left (c {\left (m + 1\right )} x^{n} - m - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctanh(c*x^n))^3,x, algorithm="maxima")

[Out]

-1/8*b^3*d^m*x*x^m*log(-c*x^n + 1)^3/(m + 1) + (d*x)^(m + 1)*a^3/(d*(m + 1)) + integrate(1/8*((b^3*c*d^m*(m +
1)*e^(m*log(x) + n*log(x)) - b^3*d^m*(m + 1)*x^m)*log(c*x^n + 1)^3 + 6*(a*b^2*c*d^m*(m + 1)*e^(m*log(x) + n*lo
g(x)) - a*b^2*d^m*(m + 1)*x^m)*log(c*x^n + 1)^2 - 3*(2*a*b^2*d^m*(m + 1)*x^m - (2*a*b^2*c*d^m*(m + 1) + b^3*c*
d^m*n)*e^(m*log(x) + n*log(x)) - (b^3*c*d^m*(m + 1)*e^(m*log(x) + n*log(x)) - b^3*d^m*(m + 1)*x^m)*log(c*x^n +
1))*log(-c*x^n + 1)^2 + 12*(a^2*b*c*d^m*(m + 1)*e^(m*log(x) + n*log(x)) - a^2*b*d^m*(m + 1)*x^m)*log(c*x^n +
1) - 3*(4*a^2*b*c*d^m*(m + 1)*e^(m*log(x) + n*log(x)) - 4*a^2*b*d^m*(m + 1)*x^m + (b^3*c*d^m*(m + 1)*e^(m*log(
x) + n*log(x)) - b^3*d^m*(m + 1)*x^m)*log(c*x^n + 1)^2 + 4*(a*b^2*c*d^m*(m + 1)*e^(m*log(x) + n*log(x)) - a*b^
2*d^m*(m + 1)*x^m)*log(c*x^n + 1))*log(-c*x^n + 1))/(c*(m + 1)*x^n - m - 1), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*atanh(c*x^n))^3,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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