3.3.35 \(\int \frac {(a+b \tanh ^{-1}(c x^n))^2}{x^3} \, dx\) [235]

Optimal. Leaf size=19 \[ \text {Int}\left (\frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^3},x\right ) \]

[Out]

Unintegrable((a+b*arctanh(c*x^n))^2/x^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 \frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*x^n])^2/x^3,x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 10.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c \,x^{n}\right )\right )^{2}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/8*b^2*log(-c*x^n + 1)^2/x^2 - 1/2*a^2/x^2 - integrate(-1/4*((b^2*c*x^n - b^2)*log(c*x^n + 1)^2 + 4*(a*b*c*x
^n - a*b)*log(c*x^n + 1) + (4*a*b + (b^2*c*n - 4*a*b*c)*x^n - 2*(b^2*c*x^n - b^2)*log(c*x^n + 1))*log(-c*x^n +
1))/(c*x^3*x^n - x^3), 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((a+b*arctanh(c*x^n))^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atanh}{\left (c x^{n} \right )}\right )^{2}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^n))^2/x^3,x)

[Out]

int((a + b*atanh(c*x^n))^2/x^3, x)

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