3.3.40 \(\int (d x)^m (a+b \tanh ^{-1}(c x^n))^2 \, dx\) [240]

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

[Out]

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

________________________________________________________________________________________

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 (c x^n\right )\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 5.10, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \arctanh \left (c \,x^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^n))^2,x, algorithm="maxima")

[Out]

1/4*b^2*d^m*x*x^m*log(-c*x^n + 1)^2/(m + 1) + (d*x)^(m + 1)*a^2/(d*(m + 1)) - integrate(-1/4*((b^2*c*d^m*(m +
1)*e^(m*log(x) + n*log(x)) - b^2*d^m*(m + 1)*x^m)*log(c*x^n + 1)^2 + 4*(a*b*c*d^m*(m + 1)*e^(m*log(x) + n*log(
x)) - a*b*d^m*(m + 1)*x^m)*log(c*x^n + 1) + 2*(2*a*b*d^m*(m + 1)*x^m - (2*a*b*c*d^m*(m + 1) + b^2*c*d^m*n)*e^(
m*log(x) + n*log(x)) - (b^2*c*d^m*(m + 1)*e^(m*log(x) + n*log(x)) - 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)

________________________________________________________________________________________

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^n))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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^n))^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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 (c\,x^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*atanh(c*x^n))^2,x)

[Out]

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

________________________________________________________________________________________