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

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

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int (d x)^m \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx \]

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

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Mathematica [A]  time = 1.72, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx \]

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 = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \operatorname {artanh}\left (\frac {c}{x^{2}}\right )^{2} + 2 \, a b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a^{2}\right )} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.40, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}\, dx \]

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

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]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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