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

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

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {(d x)^m}{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{b^{2} \operatorname {artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x^{2}\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(a + b*atanh(c*x^2))^2,x)

[Out]

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

[Out]

Timed out

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