3.234 \(\int \frac {(a+b \tanh ^{-1}(c x^n))^2}{x^2} \, dx\)

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

[Out]

Unintegrable((a+b*arctanh(c*x^n))^2/x^2,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 = {} \[ \int \frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c*x^n])^2/x^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 17.80, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x^{n}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x^{n}\right ) + a^{2}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^n))^2/x^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctanh \left (c \,x^{n}\right )\right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b^2*log(-c*x^n + 1)^2/x - a^2/x - 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) + 2*(2*a*b + (b^2*c*n - 2*a*b*c)*x^n - (b^2*c*x^n - b^2)*log(c*x^n + 1))*log(-c*x^n + 1))/(c*
x^2*x^n - x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^n))^2/x^2,x)

[Out]

int((a + b*atanh(c*x^n))^2/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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