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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{d+e x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.118826, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{d+e x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 45.2428, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{d+e x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.33, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}}{ex+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} \log \left (e x + d\right )}{e} + \int \frac{b^{2}{\left (\log \left (c x^{2} + 1\right ) - \log \left (-c x^{2} + 1\right )\right )}^{2}}{4 \,{\left (e x + d\right )}} + \frac{a b{\left (\log \left (c x^{2} + 1\right ) - \log \left (-c x^{2} + 1\right )\right )}}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x^{2}\right ) + a^{2}}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )}^{2}}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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