3.94 \(\int (d x)^m (a+b \tanh ^{-1}(c x^2))^3 \, dx\)

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

[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 = {} \[ \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3 \, dx \]

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

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Mathematica [A]  time = 1.85, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3 \, dx \]

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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fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \operatorname {artanh}\left (c x^{2}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (c x^{2}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (c x^{2}\right ) + a^{3}\right )} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctanh(c*x^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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{3} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arctanh(c*x^2))^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctanh \left (c \,x^{2}\right )\right )^{3}\, dx \]

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 \[ -\frac {b^{3} d^{m} x x^{m} \log \left (-c x^{2} + 1\right )^{3}}{8 \, {\left (m + 1\right )}} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} + \int \frac {{\left (b^{3} c d^{m} {\left (m + 1\right )} x^{2} - b^{3} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \left (c x^{2} + 1\right )^{3} + 6 \, {\left (a b^{2} c d^{m} {\left (m + 1\right )} x^{2} - a b^{2} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \left (c x^{2} + 1\right )^{2} + 12 \, {\left (a^{2} b c d^{m} {\left (m + 1\right )} x^{2} - a^{2} b d^{m} {\left (m + 1\right )}\right )} x^{m} \log \left (c x^{2} + 1\right ) + 3 \, {\left ({\left (b^{3} c d^{m} {\left (m + 1\right )} x^{2} - b^{3} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \left (c x^{2} + 1\right ) - 2 \, {\left (a b^{2} d^{m} {\left (m + 1\right )} - {\left (a b^{2} c d^{m} {\left (m + 1\right )} + b^{3} c d^{m}\right )} x^{2}\right )} x^{m}\right )} \log \left (-c x^{2} + 1\right )^{2} - 3 \, {\left ({\left (b^{3} c d^{m} {\left (m + 1\right )} x^{2} - b^{3} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \left (c x^{2} + 1\right )^{2} + 4 \, {\left (a b^{2} c d^{m} {\left (m + 1\right )} x^{2} - a b^{2} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \left (c x^{2} + 1\right ) + 4 \, {\left (a^{2} b c d^{m} {\left (m + 1\right )} x^{2} - a^{2} b d^{m} {\left (m + 1\right )}\right )} x^{m}\right )} \log \left (-c x^{2} + 1\right )}{8 \, {\left (c {\left (m + 1\right )} x^{2} - m - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*atanh(c*x^2))^3,x)

[Out]

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

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

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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